Comparisons of major routing strategies.
\\n\\n
IntechOpen was founded by scientists, for scientists, in order to make book publishing accessible around the globe. Over the last two decades, this has driven Open Access (OA) book publishing whilst levelling the playing field for global academics. Through our innovative publishing model and the support of the research community, we have now published over 5,700 Open Access books and are visited online by over three million academics every month. These researchers are increasingly working in broad technology-based subjects, driving multidisciplinary academic endeavours into human health, environment, and technology.
\\n\\nBy listening to our community, and in order to serve these rapidly growing areas which lie at the core of IntechOpen's expertise, we are launching a portfolio of Open Science journals:
\\n\\nAll three journals will publish under an Open Access model and embrace Open Science policies to help support the changing needs of academics in these fast-moving research areas. There will be direct links to preprint servers and data repositories, allowing full reproducibility and rapid dissemination of published papers to help accelerate the pace of research. Each journal has renowned Editors in Chief who will work alongside a global Editorial Board, delivering robust single-blind peer review. Supported by our internal editorial teams, this will ensure our authors will receive a quick, user-friendly, and personalised publishing experience.
\\n\\n"By launching our journals portfolio we are introducing new, dedicated homes for interdisciplinary technology-focused researchers to publish their work, whilst embracing Open Science and creating a unique global home for academics to disseminate their work. We are taking a leap toward Open Science continuing and expanding our fundamental commitment to openly sharing scientific research across the world, making it available for the benefit of all." Dr. Sara Uhac, IntechOpen CEO
\\n\\n"Our aim is to promote and create better science for a better world by increasing access to information and the latest scientific developments to all scientists, innovators, entrepreneurs and students and give them the opportunity to learn, observe and contribute to knowledge creation. Open Science promotes a swifter path from research to innovation to produce new products and services." Alex Lazinica, IntechOpen founder
\\n\\nIn conclusion, Natalia Reinic Babic, Head of Journal Publishing and Open Science at IntechOpen adds:
\\n\\n“On behalf of the journal team I’d like to thank all our Editors in Chief, Editorial Boards, internal supporting teams, and our scientific community for their continuous support in making this portfolio a reality - we couldn’t have done it without you! With your support in place, we are confident these journals will become as impactful and successful as our book publishing program and bring us closer to a more open (science) future.”
\\n\\nWe invite you to visit the journals homepage and learn more about the journal’s Editorial Boards, scope and vision as all three journals are now open for submissions.
\\n\\nFeel free to share this news on social media and help us mark this memorable moment!
\\n\\n\\n"}]',published:!0,mainMedia:{caption:"",originalUrl:"/media/original/237"}},components:[{type:"htmlEditorComponent",content:'
After years of being acknowledged as the world's leading publisher of Open Access books, today, we are proud to announce we’ve successfully launched a portfolio of Open Science journals covering rapidly expanding areas of interdisciplinary research.
\n\n\n\nIntechOpen was founded by scientists, for scientists, in order to make book publishing accessible around the globe. Over the last two decades, this has driven Open Access (OA) book publishing whilst levelling the playing field for global academics. Through our innovative publishing model and the support of the research community, we have now published over 5,700 Open Access books and are visited online by over three million academics every month. These researchers are increasingly working in broad technology-based subjects, driving multidisciplinary academic endeavours into human health, environment, and technology.
\n\nBy listening to our community, and in order to serve these rapidly growing areas which lie at the core of IntechOpen's expertise, we are launching a portfolio of Open Science journals:
\n\nAll three journals will publish under an Open Access model and embrace Open Science policies to help support the changing needs of academics in these fast-moving research areas. There will be direct links to preprint servers and data repositories, allowing full reproducibility and rapid dissemination of published papers to help accelerate the pace of research. Each journal has renowned Editors in Chief who will work alongside a global Editorial Board, delivering robust single-blind peer review. Supported by our internal editorial teams, this will ensure our authors will receive a quick, user-friendly, and personalised publishing experience.
\n\n"By launching our journals portfolio we are introducing new, dedicated homes for interdisciplinary technology-focused researchers to publish their work, whilst embracing Open Science and creating a unique global home for academics to disseminate their work. We are taking a leap toward Open Science continuing and expanding our fundamental commitment to openly sharing scientific research across the world, making it available for the benefit of all." Dr. Sara Uhac, IntechOpen CEO
\n\n"Our aim is to promote and create better science for a better world by increasing access to information and the latest scientific developments to all scientists, innovators, entrepreneurs and students and give them the opportunity to learn, observe and contribute to knowledge creation. Open Science promotes a swifter path from research to innovation to produce new products and services." Alex Lazinica, IntechOpen founder
\n\nIn conclusion, Natalia Reinic Babic, Head of Journal Publishing and Open Science at IntechOpen adds:
\n\n“On behalf of the journal team I’d like to thank all our Editors in Chief, Editorial Boards, internal supporting teams, and our scientific community for their continuous support in making this portfolio a reality - we couldn’t have done it without you! With your support in place, we are confident these journals will become as impactful and successful as our book publishing program and bring us closer to a more open (science) future.”
\n\nWe invite you to visit the journals homepage and learn more about the journal’s Editorial Boards, scope and vision as all three journals are now open for submissions.
\n\nFeel free to share this news on social media and help us mark this memorable moment!
\n\n\n'}],latestNews:[{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"},{slug:"intechopen-identified-as-one-of-the-most-significant-contributor-to-oa-book-growth-in-doab-20210809",title:"IntechOpen Identified as One of the Most Significant Contributors to OA Book Growth in DOAB"}]},book:{item:{type:"book",id:"1677",leadTitle:null,fullTitle:"Current Topics in Phylogenetics and Phylogeography of Terrestrial and Aquatic Systems",title:"Current Topics in Phylogenetics and Phylogeography of Terrestrial and Aquatic Systems",subtitle:null,reviewType:"peer-reviewed",abstract:"Mapping phylogenetics on geographical scales is one of the most important scientific aspects of bioscience research. 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\r\n\tNeurosurgery is one of the most rapidly evolving areas in medicine. Every day our understanding of the complex biology of tumors evolves, and this expanding knowledge shapes our everyday treatment routines. The surgical approaches are becoming more minimally invasive. The microscope is still the essence of neurosurgery; however, the endoscope and exoscope also find room in the surgical management of the tumors. Medical treatment alternatives are promising, and every day we are getting closer to better medical alternatives. Radiosurgery also keeps its place in the management of some tumors, and every day it is also becoming much safer. Staying up-to-date in the field of neurosurgery is becoming much more important than ever before. In this book, we will focus on the recent advances in neurosurgery, and we would like to invite researchers to share their valuable experiences.
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!0,isSalesforceBook:!1,hash:"932191f01172af37c1842cead04da142",bookSignature:"Prof. Bora Gürer and M.D. Pinar Kuru Bektaşoğlu",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/11692.jpg",keywords:"Neuro-Oncology, Embolization, Hydrocephalus, Functional Neurosurgery, Deep Brain Stimulation, Pediatric Neurosurgery, Brain Injury, Concussion, Management, Diagnosis, Neuroanesthesia, Neurorehabilitation",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 6th 2022",dateEndSecondStepPublish:"June 3rd 2022",dateEndThirdStepPublish:"August 2nd 2022",dateEndFourthStepPublish:"October 21st 2022",dateEndFifthStepPublish:"December 20th 2022",remainingDaysToSecondStep:"18 days",secondStepPassed:!1,currentStepOfPublishingProcess:2,editedByType:null,kuFlag:!1,biosketch:"Youngest Professor of neurosurgery in his country with a keen interest in complex neurovascular, neuro-oncological, and skull base surgeries. Dr. Gürer previously worked as Research Fellow at the University of Wisconsin, Department of Neurological Surgery and he is currently a professor of neurosurgery at Istinye University and continuing his clinical practice in VM Medical Park Mersin Hospital.",coeditorOneBiosketch:"Dr. Kuru Bektaşoğlu is a neurosurgeon with an interest in neurovascular and neurooncological surgeries who has more than 70 presentations both in international and national meetings and 40 published papers in international and national peer-reviewed journals.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"95341",title:"Prof.",name:"Bora",middleName:null,surname:"Gürer",slug:"bora-gurer",fullName:"Bora Gürer",profilePictureURL:"https://mts.intechopen.com/storage/users/95341/images/system/95341.png",biography:"Dr. Gürer graduated from the Faculty of Medicine, Dokuz Eylul University, İzmir, Turkey, in 2007. He completed his neurosurgical residency in 2013. For six months, he worked as a research fellow in the Department of Neurological Surgery, University of Wisconsin, USA, where he studied cranial base and neurovascular surgery. \r\n\r\nIn 2016, he became the youngest associate professor of neurosurgery in his country. Until 2021 he worked as an educational supervisor at the University of Health Sciences, Istanbul, Turkey. In May 2021, he was appointed Professor of Neurosurgery, Department of Neurosurgery, Istinye University, Istanbul, Turkey.\r\n\r\nDr. Gürer has published more than eighty papers in international and national journals. He has also written three books and seven book chapters. He has a keen interest in complex neurovascular, neuro-oncological, and skull base surgeries.\r\n\r\nHe is currently continuing his clinical practice at VM Medical Park Mersin Hospital, Turkey.",institutionString:"Istinye University School of Medicine",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"3",totalChapterViews:"0",totalEditedBooks:"3",institution:null}],coeditorOne:{id:"244230",title:"M.D.",name:"Pinar",middleName:null,surname:"Kuru Bektaşoğlu",slug:"pinar-kuru-bektasoglu",fullName:"Pinar Kuru Bektaşoğlu",profilePictureURL:"https://mts.intechopen.com/storage/users/244230/images/system/244230.jpg",biography:"Dr. Kuru Bektaşoğlu was graduated from Marmara University, School of Medicine in 2014. She had completed her neurosurgical residency in 2021. Also, she had completed her PhD in 2021 in the field of Physiology at Marmara University Institute of Health Sciences. She had worked as Research fellow at Yeditepe University, Department of Neurological Surgery for 6 months, and studied white matter fiber dissection. \n\tShe had more than 70 presentations both in international and national meetings and 40 published papers both in international and national peer-reviewed journals. Till today she had 2 written book and 8 book chapters. These publications were cited more than 500 times and her h-index is 11.\n\tShe has a keen interest in neurovascular and neurooncological surgeries. Besides, she would like to pursue her research career in translational neuroscience with special emphasis on vascular and tumor biology.",institutionString:"Sivas Numune Hospital",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:null},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"16",title:"Medicine",slug:"medicine"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"429343",firstName:"Martina",lastName:"Ivancic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/429343/images/19998_n.jpg",email:"martina@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. 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There are several challenges for underwater acoustic communication. First, because electromagnetic wave does not perform well in underwater environments due to the serious attenuation, acoustic communication is used as a major communication technology in underwater networks [2, 3]. The acoustic channel is asymmetric with large end‐to‐end propagation delays and limited bandwidth as well as high bit error rates due to multi‐path fading. Second, underwater sensor nodes’ mobility caused by ocean currents brings intermittent connective links. Third, underwater sensor nodes are prone to failure because of corrosion, fouling, and limited battery power that is available. Furthermore, the speed of the sound can change with water temperature, which leads to changes in the transmission path and may be the cause of the data not being forwarded to the sea surface on time. Therefore, the routing protocols available for terrestrial wireless sensor networks are not suitable for underwater acoustic sensor networks (UASNs), and many new routing protocols have been studied.
There are several surveys on routing protocols for UASNs reported in the literature, which provide overviews about the basic ideas of the related protocols mainly following the taxonomy for terrestrial wireless sensor networks. In Ref. [4], the authors investigated the characteristics and algorithms of routing protocols and classified them into the non–cross‐layer design, the traditional cross‐layer design, and the intelligent algorithm design. The work in Ref. [5] discussed the current state of the art on the UASN protocol design and provided a detailed overview on the current solutions for medium access control, network, and transport layer protocols. A review and comparison of different algorithms was proposed in Ref. [6] to fulfill different application requirements with dynamic environmental conditions. All these investigations show that it is almost impossible to conclude that any particular routing strategy can cost‐effectively support all underwater applications because each of them has certain strengths and weaknesses and is only applicable to specific situations.
In this chapter, we conduct a review mainly from an application perspective. An optimal protocol design must take into account different favorable features available in different scenarios to maximize routing performance, especially the characteristics of various network topologies. Therefore, it is interesting to investigate how each protocol exploits these features to maximize protocol performance and the feasibility of the proposed schemes.
The rest of the chapter is organized as follows. In Section 2, we briefly introduce the characteristics of application scenarios and routing strategies. In Sections 3–5, we discuss typical routing protocols of different strategies in single‐sink–, multi‐sink–, as well as zero‐sink–based UASNs. In Sections 6–8, we discuss typical routing protocols based on cross‐layer and reinforcement learning as well as opportunistic routing protocols. Finally, in Section 9, we draw the conclusions.
Most applications of UASNs are related to underwater collection, where the surface units are used to collect data transmitted by underwater sensor nodes. Therefore, application scenarios can be classified into single‐sink, multi‐sink, and zero‐sink according to the number of sinks that a UASN uses. As shown in Figure 1, in a single‐sink–based UASN, there is only one sink node which can be static or mobile. A static sink node is fixed on the surface (in some cases on the bottom) of the ocean as shown in Figure 1 [1]. A mobile sink node moves to collect data from the sensor nodes deployed in the ocean as shown in Figure 2. The topology of a multi‐sink–based UASN is shown in Figure 3, where two or more sink nodes are used to receive packets collected by sensor nodes [1]. A sensor node only needs to transmit the packets to one of the sink nodes close to the node. Sensor nodes in a multi‐sink–based UASN can also be static or mobile. In a zero‐sink–based UASN, several functionally identical autonomous underwater vehicles (AUVs) usually work as a team collaboratively, which requires communicating with each other, as illustrated in Figure 4.
A static single‐sink–based UASN.
A mobile single‐sink–based UASN.
A multi‐sink–based UASN.
A zero‐sink–based UASN.
Major factors that can affect the design of a routing protocol include the number of sink nodes and the topology of the corresponding UASN. Arranging multiple sink nodes in the network can improve the routing performance by shortening transmission path. A sensor node only needs to transmit the packets to the sink closer to it [1]. Deploying a UASN with static topology or a single‐sink can simplify the design of a routing protocol. A zero‐sink–based UASN may have several AUVs to work as a team, and it is more difficult to design routing protocols for this kind of UASN to achieve good performance.
Table 1 lists routing strategies under reviewing.
Routing strategy | Characteristics | Application scenarios | Advantages | Disadvantages |
---|---|---|---|---|
Geo‐based routing | It is based on GPS | All scenarios | Network architecture is simple | Positioning may be not accurate |
Source routing | The transmission path is determined by the source node | Single‐sink–based UASN with a static sink | It reduces the cost of route maintenance | It increases the packet overhead and routing cost |
Hop‐by‐hop routing | The next‐hop is determined by relay nodes | Except zero‐sink–based UASN | Network is more scalable and flexible | The final path may be not optimal |
Clustered routing | The nodes are usually divided into groups | Single‐sink–based UASN | It has a good adaptability and can reduce data redundancy | It needs to select the cluster head |
Cross‐layer routing | It considers the characteristics of other layers | Single‐sink–based UASN with a static sink and zero‐sink–based UASN | It can minimize energy costs while maximizing the performance of the entire network | Routing protocol is more complex |
Reinforcement learning scheme | Each node selects next hop via an iterative calculation of the reinforcement function | Single‐sink–based UASN | It extends the lifetime of the network | More powerful nodes are required |
Opportunistic routing | Each node selects several suitable neighbor nodes as a set of candidate‐forwarding nodes | Multi‐sink–based UASN with mobile topology | It improves the end‐to‐end success delivery rate of packet forwarding. | More powerful nodes are required |
Comparisons of major routing strategies.
Geo‐based routing uses the position information of sensor nodes to find the best route from a source node to a destination node. Each node has to be aware of the target area, its own position, and all neighbors’ positions. A node forwards packets to the target area in accordance with a certain policy based on the location information. At present, two common ways to obtain position information are using signal strength to estimate the relative coordinates and GPS. However, GPS cannot work well in underwater environments, and the relative coordinate of a node estimated by signal strength suffers from signal attenuation and noise interference.
Source routing allows the sender of a packet to specify the route that the packet takes in the network partially or completely. The transmission path in is determined by the source node, and the identification of all relay nodes in the path is embedded in the packets. Relay nodes can forward packets according to the planned path instead of selecting next hop by themselves [1]. During the route construction phase, a source node establishes a route by flooding route request packets (RREQs) in the network. Upon receiving an RREQ, the destination node responds by sending a route reply packet (RREP), which carries the route attached in the RREQ back to the source node. It supports asymmetric channel and reduces the cost of route maintenance. However, it increases the packet overhead and the routing cost. In addition, the performance degrades rapidly with the increase of mobility.
Hop‐by‐hop routing allows each relay node to select its next hop by itself. The selection of next hop is usually based on its local view of the network. It can make the network more scalable and flexible, but the final path may not be optimal.
Clustered routing divides the sensor nodes into geographical or logical groups, and the selection of cluster head considers the position distribution of nodes and residual energy. In each group, a cluster head is used as a gateway to communicate with other groups, and sensor nodes in each group only need to transmit packets to the cluster head. This strategy has a good adaptability and reduces data redundancy.
Although the hierarchical protocol stack structure is clear, scalable, robust, and easy to implement, the inter‐layer information is difficult to be shared and not conducive to the global optimization of network performance. Cross‐layer routing strategies take into account the functions and information available on other layers. Through the power control and frame collision control, it can achieve the relevant hierarchical interaction and minimize energy costs while maximizing the performance of the entire network [7].
Reinforcement learning‐based routing aims to find the most suitable route by using Q‐learning algorithm to learn the network states and adapts to topology changes at runtime intelligently without any off‐line training. During the routing process, a node analyzes its residual energy and energy distribution of a group of nodes, establishes a reinforcement function, and selects the appropriate node to forward packets [8]. This strategy can extend the lifetime of the network as much as possible.
Opportunistic routing takes advantages of the packet transmission opportunities due to mobility and broadcast of nodes, not only determining the next hop but also selecting suitableneighbor nodes as a set of candidate forwarding nodes based on certain routing metrics. All nodes in the set receive packets and then the node with the highest priority completes the following transmission. It makes full use of channel broadcast characteristics to improve throughput and transmission reliability [9].
A multi‐path grid‐based geographic routing protocol (MGGR) is proposed in Ref. [10]. It assumes that there is a sink fixed at the top in the middle of the grid, and the other sensor nodes are mobile and equipped with locator. Routing is performed in a grid‐by‐grid manner through gateways that use disjoint paths to relay data packets [10]. The disjoint paths and gateway election algorithm adopted in the protocol are helpful for balancing energy while achieving high delivery ratio and small end‐to‐end delay.
In Ref. [11], a location‐based adaptive routing protocol (CARP) selects different paths for different levels of data packets, relying on the dynamic characteristics of underwater environments. CARP can adapt dynamic underwater environments, improve the quality of the network communication, and have better performance in transmission delay, energy consumption, and packet acceptance rate.
The hop‐by‐hop vector‐based forwarding routing protocol (HH‐VBF) is homogeneous to VBF which uses the notion of a “routing vector” to act as the axis of the “virtual pipe” [12]. However, it constructs different virtual pipes for the per‐hop vector from each individual forwarder to the sink, instead of a single virtual pipe from the source to the sink as VBF. HH‐VBF is less sensitive to the routing pipe radius threshold and has much better performance such as energy consumption and successful delivery rate than VBF in sparse networks.
A solution for routing joint control and node replacement decisions is proposed in Ref. [13] to minimize the node replacement costs and develop effective methods for practical implementation. Sensor nodes in the network are laid on the ocean floor to collectively gather and transmit seismic information. As the results indicate, it provides a lower average node replacement cost and meets higher service‐level requirements, while it has a higher degree of simplicity.
A fault tolerant routing protocol (FTR) [14] assumes that the topology of the network is static and only a small fraction of the nodes is involved in routing. It detects substitutive links to revise the main path and build backup path. Simulations show that FTR achieves higher packet delivery ratio, small end‐to‐end delay, higher network throughput, and lower energy consumption than VBF and epidemic routing (ER) protocols.
In Ref. [15], the authors proposed a mobicast routing protocol. All sensor nodes are randomly distributed in a 3‐D environment, and an AUV travels along a user‐defined route as a mobile sink to collect data from sensor nodes within a 3‐D zone of relevance (ZOR) and wakes up sensors in the next 3‐D ZOR [15]. In addition, only sensor nodes in the 3‐D ZOR can be notified to enter the active mode to transmit collected data. For this reason, it has better performance such as a successful delivery rate, power consumption, and message overhead.
In Ref. [16], the authors proposed an integration method between localization and routing by using an AUV as an anchor node and “iterative localization”. Each sensor node establishes multiple routes to the sink by extending control packets exchanged for localization and creates routing table to the sink in parallel with the localization process [16]. This method provides robustness to communication failure and decreases energy consumption between each localization.
A reliable and energy efficient vector‐based forwarding protocol (REE‐VBF) was proposed in Ref. [17]. The transmission way of data packets is local flooding, and an optimal node with the smallest desire coefficient is selected by establishing a set of adaptive selection mechanisms [17]. The optimal node forwards the packet at first to prohibit others transmitting repeatedly. Therefore, REE‐VBF has a better performance on energy efficiency and reliability than others. In addition, it is suitable for the communication in shallow water areas.
In the directional flooding‐based routing protocol (DFR) [18], a node knows its own location, its one‐hop neighbors’ location, and the location of a sink. It relies on a packet flooding scheme, with which the flooding zone is determined dynamically by the link quality among neighbors to increase the successful delivery ratio instead of relying on a path establishment between the source and sink node. The results show that DFR has a higher packet delivery ratio, less communication overhead, and shorter end‐to‐end delay than VBF.
A geographical optimized reflection‐enabled routing protocol immune to link ambiguity (GORRILA) is proposed in Ref. [19], aiming to establish the best stable route from a source to a destination. It utilizes directional antennas to incorporate surface‐reflected non‐line‐of‐sight (NLOS) links in the routing process, instead of relying on the LOS link between one‐hop neighbors, which adds delay to the data delivery time in establishing routes. In addition, a physical and medium access cross‐layer scheme was also designed to optimize the route selection for maximum network throughput.
A sector‐based routing with destination location prediction (SBR‐DLP) protocol [20] was designed for a fully mobile topology network, in which each node is mobile and the destination node moves along a pre‐planned route. In the routing process, the area around the current forwarder is divided into a plurality of sectors, wherein the first sector is halved by the vector from the transponder to the destination vector and the other sectors are tagged according to their angular differences from that vector. The SBR‐DLP is highly adaptive to network dynamics and can improve the packet delivery ratio.
A reliable and energy‐efficient routing protocol (RER) was proposed in Ref. [21]. In the route discovery phase, a simple next‐hop selection is introduced to select a node with the minimum packet transmission delay to the current node as its next‐hop node. After a route is set up, multiple nodes in the network can also send data packets to the sink through the same established route instead of building a new route. As a result, it allows the real‐time data to arrive at the destination earlier and extends the network lifetime. The simulations show that RER performs better than the traditional routing protocols in terms of packet delivery ratio, average end‐to‐end delay, and average energy consumption.
Another source routing protocol called local area source routing (LASR) [22] is proposed for a small area of UASN. In this case, a source node does not need to care about the outside area of its transmission range. For this reason, each node just needs to share information of its neighboring nodes and finds a minimum cost path up to a determined sub‐destination within its transmission range. Simulation results show that the proposed solution can reduce energy consumption, in comparison with other 3‐D on‐demand routing protocols.
In channel‐aware depth‐adaptive routing protocol (CDRP) [23], the source node constructs a virtual ideal path to the sink node when it has data to send. Besides, the noise and the speed of sound under different water depths are taken into account in the selection of forwarders to reduce end‐to‐end delay and improve packet delivery ratio.
In view of the impact of the beam width and 3‐D direction of underwater sensors in UASNs, beam width and direction concerned routing (BDCR) proposed in Ref. [24] can achieve relatively high packet delivery rate and ensure reasonable energy consumption. On the selection of forwarder, the sensor node makes its preliminary decision not only on its own distance to the sink node but also on the distance of the previous sender to the sink node. Then, a correction mechanism based on beam width and direction is designed to make the final decision.
Chen et al. proposed a per‐hop‐based routing protocol called depth adaptive routing protocol (DARP) in Ref. [25], which takes not only the sound speed in different water depths but also the depth and the distance to the sink into consideration. In the routing process, a source node broadcasts the packets and then the neighbors wait for some time to decide whether they are eligible to forward the packets or not. The simulation results verify that DARP outperforms other routing protocols in terms of end‐to‐end delay.
UASNs-MIMO (UMIMO)‐routing [26] utilizes multiple‐input multiple‐output‐orthogonal frequency division multiplexing (MIMO‐OFDM) links to adaptively leverage the trade‐off between multiplexing and diversity gain. With cross‐layer design, it adapts its behavior to the noise and interference in underwater channels to choose a suitable transmission mode and allocate transmit power on subcarriers [26]. Moreover, the energy consumption is minimized according to the cooperation of transmitter and receiver to achieve the desired QoS (quality of service) according to application needs and channel condition.
SEANAR [27] is an energy‐efficient and topology‐aware routing protocol. It assigns larger weights to nodes with higher connectivity to the sink and adopts a simple yet effective greedy approach for making routing decisions. Although this is a simple greedy approach that uses the degree information (i.e., the number of neighbors) as the criteria to choose the successor, simulation results show that SEANAR achieves higher packet delivery ratio and a lower energy consumption in comparison with greedy forwarding and VBF‐based routing protocol.
Redundancy‐based adaptive routing (RBAR) [28] is a routing protocol designed for underwater delay tolerant networks (DTNs). It adopts a binary tree‐based forwarding procedure for the packet replication process, which allows a node to hold a packet as long as possible until it has to make another copy to satisfy its delay requirements [28]. Simulations show that RBAR can meet different delay requirements and achieve a good balance between delivery ratio, delay, and energy consumption.
A location‐based clustering algorithm for data gathering (LCAD) [29] is a cluster‐based routing protocol. The whole network is divided into 3‐D grids and the sensor nodes are fixed at different depths. The sensor nodes at each tier are organized in clusters with multiple cluster heads. Data gathered from the sensor nodes are sent to their respective cluster heads and then cluster heads deliver the data to the sink via AUVs. Its performance depends on the positioning of the cluster heads. The simulation shows that it improves the network lifetime by at least five times as against a scenario, which does involve clustering.
A distributed underwater clustering scheme (DUCS) without GPS support was proposed in Ref. [30]. The nodes organize themselves into local clusters. A non‐cluster head node forwards the packets to cluster heads in a single hop and then the cluster head transmits the packets to a sink via the relay of other cluster heads in a multi‐hop mode. In order to solve a battery draining problem for the cluster head, the routing protocol performs random rotation of the cluster head. The simulation shows that DUCS can achieve a very high packet delivery ratio and minimize the proactive routing message exchange.
An energy optimized path unaware layered routing protocol (E‐PULRP) [31] puts the sensor nodes into different layers in the form of concentric shells around a sink node. In each layer, an intermediate relay node is selected to deliver packets from the source node to the sink node. A mathematical framework is developed for energy consumption optimization. In comparison with other routing protocols for UASNs, E‐PULRP is simpler and more topology independent.
The location unaware multi‐hop routing protocol based on a hybrid, energy‐efficient, distributed clustering approach (LUM‐HEED) [32] is a new, homogeneous, multi‐hop routing protocol. It can be adaptive to a hierarchal structure network model in which each node is initialized with a certain degree according to its distance to the sink. The difference between LUM‐HEED and HEED protocols is that the nodes in HEED protocol must be location aware for communicating with the sink node. Note that the sensor nodes nearer to the sink have a higher degree. Simulation results show that LUM‐HEED has better performance than HEED in terms of network lifetime and network traffic.
A mobility aware routing protocol, called temporary cluster‐based routing (TCBR) [33], divides the sensor nodes in different clusters according to their locations. Ordinary nodes collect and forward the packets to a nearer cluster head and then courier nodes can move vertically and deliver them to the multiple sinks deployed on the water surface with a piston module. TCBR takes the advantage of multiple‐sink architecture with requiring any location information of sensor nodes. This makes it suitable not only for stationary and mobile networks but also for the hybrid networks [33].
A staggered time division multiple access (TDMA) underwater medium access control (MAC) protocol with routing (STUMP‐WR) [34] is a distributed and channel‐scheduling routing protocol and designed for heavily loaded underwater networks. The sensor nodes select and schedule links to overlap communications by using a distributed algorithm for leveraging the long propagation delays [34]. The simulation shows that STUMP‐WR outperforms several protocols proposed for underwater networks in terms of bits delivered per unit of energy and throughput.
The depth‐controlled routing protocol (DCR) in Ref. [35] uses the distance of the nodes to its nearest sink in a greedy forwarding strategy. In the network initialization phase, a sensor node moves to get a good and stable topology to improve the connectivity and routing performance. DCR shows better results than the current depth‐based routing protocol (DBR) in terms of data delivery ratio, delay, and average number of redundant packets with the same energy per packet consumption.
Focused on energy balancing, the dual‐sink efficient and balanced energy consumption technique (DSEBET) in Ref. [36] first establishes links between nodes based on their optimum distance value and then picked relay nodes based on their minimum distance value for the transmission of data. Long‐distance nodes from one sink will share their data to another sink if they come in range of sink; otherwise, they will establish a multi‐hop path for transmission of data to the respective sink [36]. In this way, energy is balanced to improve network lifetime and throughput.
A new routing protocol called MobiSink was proposed in Ref. [37] to tackle the problem of high energy consumption and reduce the instability period based on the deficiencies of depth‐based routing and energy‐efficient depth‐based routing. Each of the four mobile sinks moves in its own region, following linear motion in the horizontal region to collect data from the nodes [37]. If the sink enters the transmission range of these nodes, the packet is forwarded directly from the high‐level node to the sink. In this way, the data forwarding load is reduced on the middle node.
In the dual‐sink (DS) VBF protocol [38], all nodes are dynamic, and dual‐sink architecture is deployed to increase the number of nodes that are participating in the data‐forwarding procedure. In comparison with VBF, in DS‐VBF, each node calculates its distance from the nearest sink and transmits the packets to it. It considers both residual energy and location information instead of just location information to discover an optimized path to save energy [38]. Thus, it can improve the packet delivery ratio, reduce end‐to‐end delay, and balance total energy consumption to prolong network lifetime.
The hop–by‐hop dynamic addressing‐based (H2‐DAB) routing protocol in Ref. [2] chooses the next hop node in a greedy strategy for broadcasting based on the hop count, called HopID, from source nodes to sink nodes on the surface water. H2‐DAB can easily handle the node movements and support multiple sink architecture. However, the problem of multi‐hop routing still exists with which nodes near the sink drain more energy because they are used frequently.
Similar to the geo‐based routing in single‐sink–based UASNs, the geo‐based routing in multi‐sink UASNs uses the position information too. The difference is that in multi‐sink–based UASNs, packets are sent to one of the sink nodes [1].
In Ref. [35], the DCR is a centralized and distributed geographic routing protocol, with depth adjustment‐based topology control for recovery of invalid communication area. A greedy forwarding strategy is proposed for geographic routing.
The DS‐VBF in Ref. [38] is a dynamic and geographical routing protocol, considering both residual energy and location information as priority factors to discover an optimized routing path to save energy. Every sensor node is aware of its own location, and each data packet contains the location information of the source, forwarder, and destination nodes. In addition, a range of fields used for node mobility notion is also known by each sensor node. Based on the simulation results in comparison with VBF, average end‐to‐end delay is reduced but remaining energy and packet reception ratio are increased.
A trajectory‐aware communication scheme based on statistical inference to model position uncertainty, combined with a practical cross‐layer optimization for a WHOI acoustic micro‐modem, is adopted in a paradigm‐changing geographic routing protocol [39] to minimize energy consumption. Acoustic communications are used to transfer information between gliders and finally to a surface station. Implemented and tested in the proposed underwater communication test bed, it shows improvement over other routing protocols with only statistical approach or cross‐layer approach in terms of end‐to‐end reliability, throughput, and energy consumption [39].
An AUV‐aided routing method integrated path planning (AA‐RP) protocol [40] uses AUVs to collect data from sensor nodes following a dynamic path, which is planned by AUVs. It utilizes the cooperation of multi‐tasks to reduce energy consumption and avoid hot spot and zone problem with a dynamic gateway node scheme. The simulation shows that, although the AA‐RP does not require location information in the routing process, it can balance energy consumption, avoid hot point and hot zone problem, and save energy by combining multi‐tasks with a good delivery ratio.
The energy‐efficient interference‐aware routing protocol in Ref. [41] is a centralized cross‐layer heuristic solution for an efficient use of the scarce resources of UASNs. It provides a class of scheduling, power control, and routing policies and selects the next transmission node by considering different delays in packet delivery, maximal node buffer size, distance to the sink, and channel usage. Such routing can increase the overall network throughput and outperforms others in terms of energy consumption and throughput.
A new geographical and distributed routing algorithm was tailored for the characteristics of 3‐D underwater environment in Ref. [42]. A model characterizing the acoustic channel utilization efficiency allows setting the optimal packet size for underwater communication. Moreover, the problem of data gathering was investigated at the network layer by considering the cross‐layer interactions with MAC layers, forward error correction schemes between the routing functions, and the characteristics of underwater acoustic channels [42]. In the light of different application requirements, two distributed routing algorithms were introduced to minimize energy consumption.
The gossiping in underwater acoustic mobile ad‐hoc networks (GUWMANET) scheme in Ref. [43] realizes medium access and routing functionalities in a cross‐layer design. It is based on impulse communication as a physical layer method. GUWMANET needs only 10 bits of additional overhead in combination with the generic underwater application language (GUWAL), which has a 16‐bit header with a multi‐cast source and destination address.
A multi‐path power‐control transmission (MPT) scheme [44] smartly combines power control with multi‐path routing and packets at the destination. MPT is divided into three parts: multi‐path routing, source‐initiated power‐control transmission, and destination packet combining. With carefully designed power‐control strategies, MPT consumes lesser energy than the conventional one‐path transmission scheme without retransmission [44]. Besides, since no hop‐by‐hop retransmission is allowed, MPT introduces much shorter delays than the traditional one‐path scheme with retransmission.
The CARP in Ref. [45] is a distributed cross‐layer solution for multi‐hop delivery of data to a sink in underwater networks. Next‐hop selection takes explicitly into account the history of data packet delivery, the link quality, and how successful a neighbor has been in forwarding data toward the sink [45]. The results show that CARP can achieve throughput efficiency that is up to twice the throughput of focused beam routing (FBR) and almost three times that of DBR. It also obtains remarkable performance improvements over FBR and DBR in terms of end‐to‐end packet latency and energy consumption.
A novel Q‐learning‐based delay tolerant routing (QDTR) protocol [8] with predictions empowered by adaptive filters is adaptive and energy efficient. The adaptive filters are used to predict future neighbor contact. With the Q‐learning agent, the routing protocol can adapt to changes in the network. Since the routing problem is formulated as a Markov decision process (MDP), in which the state space is composed by all the nodes in the network, QDTR is fully distributed without any central control. The simulation results have shown that QDTR yields significantly better network performance in energy consumption, end‐to‐end delay, and delivery ratio in comparison with most of the existing DTN routing protocols.
A multi‐level routing protocol for acoustic‐optical hybrid underwater wireless sensor networks (MURAO) [46] is a multi‐level Q‐learning‐based routing protocol for a novel acoustic‐optical hybrid UASN. The network is physically partitioned into several groups and logically divided into two layers. Taking advantage of the long range but slow acoustic transmission and fast optical communications with multi‐level Q‐learning, MURAO performs better than the flat Q‐learning‐based routing.
In Ref. [47], a Q‐learning‐based tracking scheme based on the buffer size and residual energy of the individual node was used to find the next forwarder. It aims to reduce the dropping on the packets, the number of forwarders, and energy consumption of the sensor nodes. The lifetime of the network is expected to increase.
Another Q‐learning‐based energy‐efficient and lifetime‐aware routing protocol (QELAR) was proposed in Ref. [48] to prolong the lifetime of networks. The residual energy of each sensor node as well as the energy distribution among a group of nodes is factored in the throughput routing process to calculate the reward function, which aids in selecting the adequate forwarders for packets. Compared with VBF, QELAR has a longer lifetime.
The geographic and opportunistic routing with depth adjustment‐based topology control for communication recovery over void regions (GEDAR) in Ref. [49] adjusts the topology by moving void nodes to new depths and using greedy opportunistic forwarding mechanisms to transmit packets. The communication void region occurs whenever the data is transferred to a node that is not closer to the destination than the node; the node located in a communication void region is called void node. Compared with the baseline routing protocols, GEDAR outperforms in data packet delivery ratio.
The void‐aware pressure routing protocol (VAPR) in Ref. [50] exploits periodic beaconing to build directional trails toward the closest sonobuoy and features greedy opportunistic directional forwarding mechanisms for packet delivery. It can be efficiently performed even in the presence of voids [50]. The simulations show that VAPR outperforms existing schemes by significantly lowering the frequency of recovery fallbacks and effectively handling node mobility.
With the opportunistic‐based DARP in Ref. [51], forwarding node selection is dynamic and independent for each node. DARP takes different acoustic signal speed, depth, and distance to sink into account to find the minimum end‐to‐end delay path, which may not be the shortest path directly from the source to the sink. Furthermore, it does not need to continuously maintain neighbors’ information or to exchange control packets.
The HydroCast in Ref. [52] is a hydraulic pressure‐based anycast routing protocol, which exploits measured pressure levels to route a packet upward to lower depths. The opportunistic routing mechanism can limit co‐channel interference by selecting the subset of forwarder. The dead‐end recovery method can guarantee the delivery. Because HydroCast uses adaptive timer setting at each hop, it is mainly used for depth‐based communication with sparse network and performs better than DBR in terms of delivery ratio and end‐to‐end delay.
We summarize the routing strategies mentioned above in Table 2. Most routing strategies are suitable for a static UASN, and just a small scale of them can be applied in a mobile UASN. However, one of the most important characteristics of UASNs is mobility. From the adaptability of the application scenarios, geo‐based routing protocols cannot work well in mobile UASNs due to frequent localization. In addition, the characteristics of acoustic channels are also the limitation of the design of geo‐based routing protocols. Clustered routing protocols also do not perform well in mobile UASNs because of the grouping cost.
Routing strategy\\ | Scenarios | ||||
---|---|---|---|---|---|
Single‐sink–based UASNs | Multi‐sink–based UASNs | Zero‐sink–based UASNs | |||
Static sink | Mobile sink | Static topology | Mobile topology | ||
Geo‐based routing | ✓ | ✓ | ✓ | ||
Source routing | ✓ | ||||
Hop‐by‐hop routing | ✓ | ✓ | ✓ | ✓ | |
Clustered network | ✓ | ✓ | |||
Cross‐layer routing | ✓ | ✓ | |||
Reinforcement learning‐based routing | ✓ | ||||
Opportunistic routing | ✓ |
Categories of routing strategies.
Cross‐layer routing protocols may have good performance in static UASNs. Source‐routing protocols are based on the location of sensor nodes and usually adapted to single‐sink–based UASNs. On the opposite, hop‐by‐hop routing protocols can be applied in both single‐sink and multi‐sink–based UASNs.
Reinforcement learning‐based routing protocols have good adaptability. It may be applied to dynamic networks by the design of reward functions.
The above discussion shows that it is impossible to design just one or two routing protocols that can cost‐effectively support all underwater application scenarios. Many routing proposals for UASNs are only in the simulation phase and have not been testified in the actual environments. Researchers still follow the design philosophy of routing protocols for terrestrial wireless networks, which is not enough for UASNs. An optimal design must take into account each different favorable feature available in different scenarios.
This work is supported (in part) by National Natural Science Foundation of China (NSFC) under Grant 61472237.
Solitons can be regarded as isolated waves that travel without loss of energy. The solitons emerge with their velocities and shapes completely unchanged after collision to each other, the only outcome being their phase shifts. The soliton solution is the main feature of the integrable models [1, 2, 3]. However, certain non-linear models in physics, with solitary wave solutions, are not integrable. Recently, certain deformations of integrable models such as the sine-Gordon (SG), nonlinear Schrödinger (NLS), Korteweg-de Vries (KdV) and Toda models have been introduced, such that they exhibit soliton-type solutions with some properties resembling to their counterparts of the truly integrable ones. In this context the so-called quasi-integrability concept has been put forward [4]. These properties have been examined in the frameworks of the anomalous zero-curvature [4, 5, 6, 7] and the Riccati-type pseudo-potential approaches [8, 9, 10], respectively.
The main developments have been focused on the construction of infinite number of quasi-conservation laws which give rise to asymptotically conserved charges, i.e. conserved charges, such that their values vary during the scattering of the solitons only. The main observation in the both approaches to quasi-integrability is that, in general, the conserved charges of the standard integrable systems turn out to be the so-called asymptotically conserved charges in the deformed models. In fact, the exact conservation laws of the usual integrable systems become quasi-conservation laws of the deformed integrable models. The non-homogeneous terms of the quasi-conservation laws are dubbed as anomalies such that they vanish when integrated on the space–time plane, provided that the fields satisfy a special space–time symmetry.
The properties of the soliton-like configurations in the quasi-integrable models are, so far, largely unknown. We summarize the main results. First, the one-soliton sectors exhibit infinite conserved charges. Second, the space–time integration of the anomalies vanish when one-soliton like solutions are located far away from each other. The anomalies are significant around the space–time regions of their interaction. Third, a sufficient condition for the vanishing of the space–time integrated anomalies is that the
In the context of the Riccati-type method there have been shown that the deformed SG, KdV and NLS models [8, 9, 10], respectively, possess linear system formulations and that they exhibit infinite towers of exact non-local conservation laws. The NLS-type, KdV-type and SG-type models share the same importance due to their potential applications, since they are ubiquitous in all areas of nonlinear physics, such as Bose-Einsten condensation and superconductivity [12, 13, 14], soliton gas and soliton turbulence in fluid dynamics [15, 16, 17, 18, 19, 20], the Alice-Bob physics [21, 22] and the understanding of a kind of triality among the gauge theories, integrable models and gravity theories [23].
Here, we discuss the previous results in the field by utilizing a deformed sine-Gordon model. We will introduce the relationship between the space–time parity and asymptotically conserved charges. Next, we clarified on the space-reflection parity related to the linear combination of the dual sets of anomalous quantities. In addition, it is focused on the space-reflection symmetry of some two-soliton solutions of deformed sine-Gordon models. Then one proceeds to construct a tower of exactly conserved charges for each solution possessing a definite space-reflection parity. Lastly, by considering linear combinations of the anomalous conserved charges it is showed, through analytical and numerical methods, that there is a subset of exactly conserved charges.
A modified SG model and the space–time symmetries are presented in the next section. In Section 3, the towers of quasi-conservation laws are presented. In Section 4 our numerical simulations are described. The linear formulation and the non-local conservation laws are discussed in the Riccati-type pseudo-potential approach in Section 5. Finally, in Section 6 we present some conclusions.
Let us consider the relativistic field theories in
where
where
So, we introduce the deformation parameter
The model (1) possesses several towers of anomalous charges associated to quasi-conservation laws [4, 8, 11]. In [11] it has been introduced a subset of exactly conserved charges associated to space-reflection eigenstates as kink-antikink, kink-kink and breather configurations, respectively. New types of two sets of dual towers of asymptotically conserved charges have been uncovered [8]. Remarkably, even the usual sine-Gordon models possesses anomalous charges. So far, it is attributed to the space–time symmetry properties of the solitons. Those charges can be relevant in the study of soliton gases and formation of certain structures, such as soliton turbulence, soliton gas dynamics and rogue waves [16].
The quasi-integrability has been introduced for deformed sine-Gordon models such that the field
under the special space–time reflection
defined around a given point
and assume that the scalar field is an eigenstate of the operator
In addition, consider an even potential
Several towers of quasi-conservation laws, with anomaly terms possessing odd parities under (6)–(8), have been found [8, 11]. Next, we consider those quasi-conservation laws and examine their anomalies in view of the symmetries (4)–(5) and (6)–(8), respectively.
We will discuss some of the infinite towers of quasi-conservation laws of the deformed SG model (1).
The usual SG charges turn out to be the anomalous charges of the DSG. So, one has the infinite set of quasi-conservation laws [4, 11].
where the quantities
In
where the quantities
These towers of quasi-conservation laws reproduce the same polynomial form as in the usual sine-Gordon charge densities. In fact, the anomalies
The importance and the relevance of such a dual construction will become clear below when the linear combinations of the charges in (9) and (10) give rise to infinite towers of exactly conserved charges, provided that the space-integral of the linear combination of the anomaly densities
The above dual sets of quasi-conservation laws are used to construct a sequence of conserved charges and vanishing anomalies. The space-reflection symmetry of some soliton solutions of the deformed SG model will imply the existence of an infinite tower of conserved charges. So, let us examine a linear combination, at each order
with the charges
in which the quantities
Since the theory (1) is invariant under space–time translations one has that the energy momentum tensor is conserved. In fact, one has
where
The first non-trivial anomalies become [11].
Notice that for the SG potential (3) the factor
The properties of the quantities
Let us write the anomalies in terms of the
where we have defined the anomaly density
Following analogous procedure as above one has
where we have defined the anomaly density
The anomalies
By direct construction it has been found new towers of anomalous charges in [8]. In the next subsections we will discuss those charges and anomalies in relation to the symmetry (4) and (5).
The quasi-conservation laws [8].
define the asymptotically conserved charges
The dual quasi-conservation laws become
where we have introduced the dual asymptotically conserved charges
The densities of the anomalies
Let us define the quasi-conservation laws [8].
where we have introduced the asymptotically conserved charges
The interchange
where we have defined the dual asymptotically conserved charges
Similarly, the densities of the anomalies
The relevant anomalies of the lowest order quasi-conservation laws of the above towers will be simulated below for 2-soliton interactions.
Remarkably, the above charges turn out to be anomalous even for the standard sine-Gordon model. In fact, the relevant 2-soliton solutions have been constructed analytically [4, 11] which possess a definite parity under (4)–(5), such that the odd anomaly densities vanish upon space–time integration. The usual explanation for the appearance of novel anomalous charges in the standard sine-Gordon model is the symmetry argument. The anomalous charges also appear in the standard KdV and its deformations [9].
These charges have been computed for soliton collisions in the treatment of soliton gases and formation of some structures in integrable systems, such as integrable turbulence and rogue waves. In the context of the usual KdV model it has been analyzed the behavior of the statistical moments defined by (see e.g. [16, 17])
Here we will check numerically the lowest order expressions of the various towers of quasi-conservation laws presented above. For this purpose we will numerically solve the Eq. (1) with the particular deformed potential (2). In the Figures 1 and 2 we plot the kink-kink and kink-antikink collisions, respectively. Moreover, we show the first conserved charges, i.e. the energy and momentum for these field configurations.
Kink-kink with velocities
Kink-antikink with velocities
We have checked our results by numerical simulation of the anomalies
So, let us write (11) in the form
where
The simulations of the kink-antikink, kink-kink and breather systems of the deformed SG model will consider, as the initial condition, two analytic solitary wave solutions presented in Eq. (1.2) of [4], located some distance apart and stitched together at the middle point.
In the Figures 3 and 4 we show the results for kink-antikink system with velocities
In the Figures 5 and 6 we show the results for kink-kink system with velocities
So, one can conclude that for kink-antikink (kink-kink) solution the definite parity related to the space-reflection symmetry is a necessary condition in order to achieve a conserved
The both kink-antikink and kink-kink solitons of the SG model with opposite and different velocities do not possess the required parity symmetry. However, it has been shown that in the center-of-mass reference frame (
Figures 7 and 8 show the results for breather (kink-antikink bound state) with
We will compute the linear combinations of the lowest order anomalies of the second and third types of towers in (22)–(27) and (28)–(33), respectively,
The two anomalies in (35) can be written as
Similarly, the two anomalies in (36) can be written as
Notice that under the space–time reflection transformation (4) and (5), the densities of the above anomalies
Under the space-reflection symmetry (6) and (8), some of the densities of the above anomalies will present odd parities; therefore, they must vanish upon space integration. So, in such cases one can have exact conserved charges. These results will be verified for certain solutions as we will see below in the numerical simulations for the kink-kink and kink-antikink solutions.
Figures 9–12 show the anomalies
Top: The anomaly densities
Top: Anomaly densities
Top: Anomaly densities of
Top: Anomaly densities of
These results suggest that the quasi-integrable models set forward in the literature [4, 6, 7], and in particular the model (1), would possess more specific integrability structures, such as an infinite set of exactly conserved charges, and some type of linear formulations for certain deformed potentials. So, in the next section we will tackle the problem of extending the Riccati-type pseudo-potential formalism to the deformed sine-Gordon model (1).
The Lax equations and Backlund transformations, as well as the conservation laws for the well-known non-linear evolution equations can be generated from the pseudo-potentials and the properties of the Riccati Equation [25, 26, 27, 28, 29].
So, in the next steps we consider a convenient deformation of the usual pseudo-potential approach to integrable field theories. Let us consider the system of Riccati-type equations
and the next linear first order equation for
The compatibility condition
From the system (41) and (42) one can get a quasi-conservation law
This equation has been used to construct a tower of infinite number of quasi-conservation laws [8]. For the standard SG one has
In this section we search for a linear system formulation of the DSG model. It is achieved by taking into account the Riccati Eq. (41) and the conservation law (44), as well as the Eq. (43). So, the following system of equations has been proposed as a linear formulation of the deformed SG model [8].
where the auxiliary non-local field
In fact, taking into account the expression for the auxiliary field
with
In (49) the coefficient of the linear term in
For non-linear equations, not necessarily integrable, which can be derived from a compatibility condition of an associated linear system with spectral parameter, explicit expressions of local and non-local currents can be obtained (see e.g. [30, 31]). In the non-linear
So, one can construct an infinite set of non-local conserved currents through an inductive procedure. Let us define the currents
where
where
Then one can show by an inductive procedure that the (non-local) currents
The first current conservation law
The next conservation law
The construction of analogous linear systems have been performed for deformations of the KdV and NLS models [9, 10]. The construction of the classical Yangian as a Poisson-Hopf type algebra [34] for those non-local currents is worth to pursue in a future work.
Our work presents an in-depth demonstration of the quasi-integrability property of the modified sine-Gordon models and the presence of several towers of infinite number of asymptotically conserved charges for soliton configurations satisfying the space–time symmetry (4) and (5). In addition, it is observed that there exist a subset of towers of infinite number of exactly conserved charges, provided that some two-soliton configurations are eigenstates (even or odd) of the space-reflection symmetry (6)–(8).
Moreover, we have uncovered a linear system formulation (45) of the modified SG model, and an infinite set of exact non-local conservation laws (58) associated to that linear formulation.
The space–time and internal symmetries related to quasi-integrability deserve further investigations, due to their applications in several areas of non-linear science, but we hope that the results reported here have opened new lines of research in the context of the quasi-integrability phenomena.
JPRC acknowledges brazilian CAPES for financial support. HB thanks FC-UNI (Lima-Perú) and FC-UNASAM (Huaraz-Perú) for partial support and kind hospitality. BMC and CR thank UNASAM for partial financial support.
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\\n\\nWe aim at improving the quality and availability of scholarly communication by promoting and practicing:
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The Open Access publishing movement started in the early 2000s when academic leaders from around the world participated in the formation of the Budapest Initiative. They developed recommendations for an Open Access publishing process, “which has worked for the past decade to provide the public with unrestricted, free access to scholarly research—much of which is publicly funded. Making the research publicly available to everyone—free of charge and without most copyright and licensing restrictions—will accelerate scientific research efforts and allow authors to reach a larger number of readers” (reference: http://www.budapestopenaccessinitiative.org)
\n\nIntechOpen’s co-founders, both scientists themselves, created the company while undertaking research in robotics at Vienna University. Their goal was to spread research freely “for scientists, by scientists’ to the rest of the world via the Open Access publishing model. The company soon became a signatory of the Budapest Initiative, which currently has more than 1000 supporting organizations worldwide, ranging from universities to funders.
\n\nAt IntechOpen today, we are still as committed to working with organizations and people who care about scientific discovery, to putting the academic needs of the scientific community first, and to providing an Open Access environment where scientists can maximize their contribution to scientific advancement. By opening up access to the world’s scientific research articles and book chapters, we aim to facilitate greater opportunity for collaboration, scientific discovery and progress. We subscribe wholeheartedly to the Open Access definition:
\n\n“By “open access” to [peer-reviewed research literature], we mean its free availability on the public internet, permitting any users to read, download, copy, distribute, print, search, or link to the full texts of these articles, crawl them for indexing, pass them as data to software, or use them for any other lawful purpose, without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself. The only constraint on reproduction and distribution, and the only role for copyright in this domain, should be to give authors control over the integrity of their work and the right to be properly acknowledged and cited” (reference: http://www.budapestopenaccessinitiative.org)
\n\nOAI-PMH
\n\nAs a firm believer in the wider dissemination of knowledge, IntechOpen supports the Open Access Initiative Protocol for Metadata Harvesting (OAI-PMH Version 2.0). Read more
\n\nLicense
\n\nBook chapters published in edited volumes are distributed under the Creative Commons Attribution 3.0 Unported License (CC BY 3.0). IntechOpen upholds a very flexible Copyright Policy. There is no copyright transfer to the publisher and Authors retain exclusive copyright to their work. All Monographs/Compacts are distributed under the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0). Read more
\n\nPeer Review Policies
\n\nAll scientific works are Peer Reviewed prior to publishing. Read more
\n\nOA Publishing Fees
\n\nThe Open Access publishing model employed by IntechOpen eliminates subscription charges and pay-per-view fees, enabling readers to access research at no cost. In order to sustain operations and keep our publications freely accessible we levy an Open Access Publishing Fee for manuscripts, which helps us cover the costs of editorial work and the production of books. Read more
\n\nDigital Archiving Policy
\n\nIntechOpen is committed to ensuring the long-term preservation and the availability of all scholarly research we publish. We employ a variety of means to enable us to deliver on our commitments to the scientific community. Apart from preservation by the Croatian National Library (for publications prior to April 18, 2018) and the British Library (for publications after April 18, 2018), our entire catalogue is preserved in the CLOCKSS archive.
\n\nOpen Science is transparent and accessible knowledge that is shared and developed through collaborative networks.
\n\nOpen Science is about increased rigour, accountability, and reproducibility for research. It is based on the principles of inclusion, fairness, equity, and sharing, and ultimately seeks to change the way research is done, who is involved and how it is valued. It aims to make research more open to participation, review/refutation, improvement and (re)use for the world to benefit.
\n\nOpen Science refers to doing traditional science with more transparency involved at various stages, for example by openly sharing code and data. It implies a growing set of practices - within different disciplines - aiming at:
\n\nWe aim at improving the quality and availability of scholarly communication by promoting and practicing:
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Her research interests include archaea metabolism, enzymes purification and characterization, gene regulation, carotenoids and bioplastics production, antioxidant\ncompounds, waste water treatments, and brines bioremediation.\nRosa María’s other roles include editorial board member for several journals related\nto biochemistry, reviewer for more than 60 journals (biochemistry, molecular biology, biotechnology, chemistry and microbiology) and president of several organizing committees in international meetings related to the N-cycle or respiratory processes.",institutionString:null,institution:{name:"University of Alicante",institutionURL:null,country:{name:"Spain"}}},editorTwo:null,editorThree:null},{id:"15",title:"Chemical Biology",coverUrl:"https://cdn.intechopen.com/series_topics/covers/15.jpg",isOpenForSubmission:!0,editor:{id:"441442",title:"Dr.",name:"Şükrü",middleName:null,surname:"Beydemir",slug:"sukru-beydemir",fullName:"Şükrü Beydemir",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0033Y00003GsUoIQAV/Profile_Picture_1634557147521",biography:"Dr. Şükrü Beydemir obtained a BSc in Chemistry in 1995 from Yüzüncü Yıl University, MSc in Biochemistry in 1998, and PhD in Biochemistry in 2002 from Atatürk University, Turkey. 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He is a member of the Turkish Biochemical Society, American Chemical Society, and German Genetics society. Dr. Ekinci published around ninety scientific papers, reviews and book chapters, and presented several conferences to scientists. He has received numerous publication awards from several scientific councils. 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He worked on the structure-function relationships of glycoconjugates and his main project was the investigations on the biological roles of the de-N-glycosylation enzymes (Endo-N-acetyl-β-D-glucosaminidase and peptide-N4-(N-acetyl-β-glucosaminyl) asparagine amidase). From 2002 he contributes to the understanding of the Blood-brain barrier functioning using proteomics approaches. He has published more than 70 papers. 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Since then, he has been working as an Adjunct Professor in the same Department at the University of Pavia. His research activity during the first years was primarily focused on the purification and structural characterization of enzymes from animal and plant sources. During this period, Prof. Iadarola familiarized himself with the conventional techniques used in column chromatography, spectrophotometry, manual Edman degradation, and electrophoresis). Since 1995, he has been working on: i) the determination in biological fluids (serum, urine, bronchoalveolar lavage, sputum) of proteolytic activities involved in the degradation processes of connective tissue matrix, and ii) on the identification of biological markers of lung diseases. 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David Pan",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSEI9QAO/Profile_Picture_1623656213532",institutionString:null,institution:{name:"University of Alabama in Huntsville",institutionURL:null,country:{name:"United States of America"}}},{id:"72920",title:"Prof.",name:"Yves",middleName:"Philippe",surname:"Rybarczyk",fullName:"Yves Rybarczyk",profilePictureURL:"https://mts.intechopen.com/storage/users/72920/images/system/72920.jpeg",institutionString:"Dalarna University, Faculty of Data and Information Sciences",institution:{name:"Dalarna University",institutionURL:null,country:{name:"Sweden"}}}]},{id:"27",title:"Multi-Agent Systems",keywords:"Collaborative Intelligence, Learning, Distributed Control System, Swarm Robotics, Decision Science, Software Engineering",scope:"Multi-agent systems are recognised as a state of the art field in Artificial Intelligence studies, which is popular due to the usefulness in facilitation capabilities to handle real-world problem-solving in a distributed fashion. The area covers many techniques that offer solutions to emerging problems in robotics and enterprise-level software systems. Collaborative intelligence is highly and effectively achieved with multi-agent systems. Areas of application include swarms of robots, flocks of UAVs, collaborative software management. Given the level of technological enhancements, the popularity of machine learning in use has opened a new chapter in multi-agent studies alongside the practical challenges and long-lasting collaboration issues in the field. It has increased the urgency and the need for further studies in this field. We welcome chapters presenting research on the many applications of multi-agent studies including, but not limited to, the following key areas: machine learning for multi-agent systems; modeling swarms robots and flocks of UAVs with multi-agent systems; decision science and multi-agent systems; software engineering for and with multi-agent systems; tools and technologies of multi-agent systems.",annualVolume:11423,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/27.jpg",editor:{id:"148497",title:"Dr.",name:"Mehmet",middleName:"Emin",surname:"Aydin",fullName:"Mehmet Aydin",profilePictureURL:"https://mts.intechopen.com/storage/users/148497/images/system/148497.jpg",institutionString:null,institution:{name:"University of the West of England",institutionURL:null,country:{name:"United Kingdom"}}},editorTwo:null,editorThree:null,editorialBoard:[{id:"275140",title:"Dr.",name:"Dinh Hoa",middleName:null,surname:"Nguyen",fullName:"Dinh Hoa Nguyen",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRbnKQAS/Profile_Picture_1622204093453",institutionString:null,institution:{name:"Kyushu University",institutionURL:null,country:{name:"Japan"}}},{id:"20259",title:"Dr.",name:"Hongbin",middleName:null,surname:"Ma",fullName:"Hongbin Ma",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRhDJQA0/Profile_Picture_2022-05-02T08:25:21.jpg",institutionString:null,institution:{name:"Beijing Institute of Technology",institutionURL:null,country:{name:"China"}}},{id:"28640",title:"Prof.",name:"Yasushi",middleName:null,surname:"Kambayashi",fullName:"Yasushi Kambayashi",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002aYOQxQAO/Profile_Picture_1625660525470",institutionString:null,institution:{name:"Nippon Institute of Technology",institutionURL:null,country:{name:"Japan"}}}]}]}},libraryRecommendation:{success:null,errors:{},institutions:[]},route:{name:"profile.detail",path:"/profiles/264876",hash:"",query:{},params:{id:"264876"},fullPath:"/profiles/264876",meta:{},from:{name:null,path:"/",hash:"",query:{},params:{},fullPath:"/",meta:{}}}},function(){var e;(e=document.currentScript||document.scripts[document.scripts.length-1]).parentNode.removeChild(e)}()