\r\n\t
\r\n\tAdditionally, authors are expected to cover the medicinal effect of tea polyphenols viz. catechins, theaflavins and thearubigins which are the major chemical compounds of tea. Green tea contains a higher quantity of catechins as compared to black tea which becomes transformed into more complex compounds (theaflavins and therubigins) during the manufacturing of black tea. Out of many therapeutic uses these bioactive compounds, protection against cardiovascular diseases, atherosclerosis, cancer, gene mutation, and diabetics have been reported and are more promising. This book will also highlight the nutraceuticals of tea. The catechins along with other various bioactive compounds present in tea have many therapeutic properties which attribute to the development of various food products where tea constitutes as an active ingredient. That is why it is important to stress the potential use of tea and their bioactive constituents (catechins, polysaccharides, vitamins, amino acids etc.) in food products with added nutraceutical values. Finally, the contamination of tea and its effect on our health will also be covered. Tea and its food products may contain various types of contaminations which may include toxic heavy metals, pesticides, microorganisms, and environmental pollutants etc. which are present from tea infusion and from residues. These contaminations which are above the regulatory limit may pose a serious threat to public health. Submitted chapters are expected to contain novel information, be informative as well as thought-provoking.
Optical solitons arise as localized optical fields that preserve their shapes during propagation when the medium’s diffraction and self-phase modulation are balanced [1]. Since they were first observed experimentally in 2003 [2], spatial solitons [3] in nonlinear optical systems with additional optical lattices have received a significant interest. Currently, fundamental solitons and vortices on a variety of lattices including real and complex structures in the media with cubic (Kerr) [4, 5], saturable [6] and with competing nonlinearities [7] have been studied. Solitons may also exist in aperiodic or quasicrystal lattices [8, 9, 10, 11, 12] and in lattices with defects [13] and dislocations [14, 15].
In the studies above, the governing equations are nonlinear cubic Schrödinger (NLS) equation and their variants, e.g., NLS with additional terms and/or with external lattices. On the other hand, numerous optical materials, such as potassium niobate (KNbO3) [16] or lithium niobate (LiNbO3) [17], acquire quadratic nonlinear responses as well [18, 19, 20, 21, 22, 23, 24]. NLS equation with coupling to a mean term (NLSM) system governs the pulse dynamics in quadratically polarized media. The general NLSM system is given by [18, 25, 26].
Here
NLSM equations were derived by Benney and Roskes [27] for a finite water depth, neglecting the surface tension. In 1974, Davey and Stewartson [28] investigated the evolution of a three-dimensional wave packet for a finite water depth and obtained an equivalent form of these equations. The integrability of the NLSM equations that were derived for the shallow water limit was studied by Ablowitz and Haberman [29]. Djordevic and Reddekopp [30] improved the study of Benney and Roskes by including the surface tension in 1977. Later, Ablowitz et al. [18, 25, 26] derived from first principles that NLSM type equations model the evolution of the electromagnetic field in a quadratically polarized media.
One major drawback of the dynamics of solitons under the NLS equation in 2D is that they exhibit collapse. As a matter of fact, in [31], it was revealed that the collapse dynamics of NLSM solitons are similar to the collapse of NLS solitons. Possible collapse arrest mechanisms have been studied extensively in nonlinear optics, e.g., nonlinear saturation [32, 33]. Merle and Raphael [34] investigated the collapse of the NLS solitons and its variants in depth. Furthermore, Gaeta and coworkers [35] carried out detailed experiments in order to expose the nature of the singularity formation in cubic optical media and demonstrated experimentally that collapse occurs with a self-similar profile.
Recently, wave collapse in the NLSM system was studied in [36] and it is shown that in both water waves and optics, collapse occurs with a quasi self-similar profile. By evolving the NLSM solutions that are computed by the Spectral Renormalization method which is essentially a fixed point iteration scheme. In this study, it is also revealed that the NLSM solitons have astigmatic profiles and their collapse can be arrested by adding nonlinear saturation into the system. Wave self-rectification and beam ellipticity as a collapse arrest mechanism for NLSM solitons was put forward by in [16]. In the aforementioned work, for simplicity, the authors considered Gaussian profiles with various input powers and astigmatism and then evolve these profiles for long distances. More recently, wave collapse in the NLSM system was arrested by adding a periodic external lattice in [23] and it has been shown that deeper lattices may serve as a collapse arrest mechanism.
In this chapter, the NLSM model (1) is extended by adding a quintic term and an external lattice. Using this new model, the dynamics of the fundamental lattice solitons in a nonlocal nonlinear medium with quintic nonlinearity are explored. The study is concentrated on effects formed by the imposed lattice depth and the strength of quintic nonlinearity. The stability of the obtained lattice solitons is examined by the nonlinear evolution and linear spectra. In the light of the conducted numerical analysis, it is confirmed that fundamental lattice solitons can exist in nonlocal nonlinear media with both self-focusing and self-defocusing quintic nonlinearities and stability of these solitons is achieved for a broad range of system parameters. Since many optical materials such as chalcogenide glasses are engineered to reveal fifth and seventh-order effects in addition to cubic nonlinear effects [37], and high-order nonlinearities can arise even with pure Kerr materials [38, 39, 40], it is crucial to consider the soliton dynamics consists of higher-order nonlinearities.
The outline of the chapter is as follows: In Section 2, we present the model equations and compute the periodic lattice solitons numerically. In Section 3, we explore the linear spectra and nonlinear evolution of the solitons. Results are outlined in Section 4.
It is known that, the steady-state solutions of the NLSM system (1) collapse [36], and the collapse of solitons can be arrested by adding external lattices [23]. The NLSM model (1) is extended as follows to describe the nonlocal qubic-quintic nonlinear medium with an external lattice,
where
where
The top view and diagonal cross-section of the lattice
The squared operator (SOM) method (that was proposed by Yang and Lakoba [41]) is utilized to calculate the soliton solutions of the (2 + 1)D NLSM model (2). The SOM method is based on integrating squared-operators of evolution equations. Derivation of these operators and the scheme of the method are explained below.
A real-valued nonlinear evolution equation can be presented in the following form
where
where
here
It was shown that the SOM method converges to a solitary wave solution for a broad-range of nonlinear evolution equations when a convenient initial condition is given and time step
To calculate a solitary wave solution of the model Eq. (2) by the SOM, the following scheme is constructed. Inserting the solution suggestion
where
where
After obtaining
while the mean term
In order to avoid division by zero error, the first element of
It should be noted that
The fundamental solitons of the NLSM model (2) are obtained by the SOM method for the considered parameters (14) and are shown in Figure 2.
3D profiles (first column), top views (second column) of fundamental solitons that are obtained (a) for
As shown in previous studies [23, 24, 36], fundamental solitons of the NLSM system (2) (with or without lattice) are not radially symmetric due to the anisotropy in the medium. To investigate the effect of the quintic nonlinearity on the level of astigmatism in the solitons, the following formulation is defined as a measure of astigmatism
Here,
Contour plots of solitons are depicted in Figure 3 for
Contour image of fundamental solitons that are obtained for
To see the impact of the anisotropy parameter
The on-axes amplitudes of fundamental solitons along x-axis (first column), along y-axis (second column) and contour image of solitons (third row) (a) for
Comparing the contour images in Figures 3 and 4, it is seen that the astigmatism of the fundamental solitons changes significantly with
The stability properties of fundamental solitons (that are obtained by the SOM method) are examined by the linear eigenvalue spectra and the direct simulation of the nonlinear model (2).
The linear stability of solitons are studied by the linear spectra of fundamental solitons. To obtain the linear spectrum, the fundamental soliton is perturbed as follows
where
where
and the matrix coefficients of
We can solve this eigenvalue problem by the Fourier collocation method [42]. Any eigenvalue with a positive real part in the spectrum indicates the linear instability of the fundamental soliton considered.
The power (or energy is some other contexts) of solitons can be computed by P=
Moreover, in [44, 45], it was shown that the solitons can be stable nonlinearly only if the slope condition (19) is satisfied.
In the light of these results, prior to the nonlinear stability analysis, the power and linear stability of solitons are studied in Figure 5. These power-eigenvalue diagrams are calculated for the parameters that are given in Eq. (14), and the linear (in)stability intervals of gap solitons are determined via the computation of eigenvalue spectra of the solitons at each point of the power curves (see Figure 5(a1)–(b1)). Furthermore, the maximum real part in the eigenvalue spectra (spectral radius) is examined in Figure 5(a2)–(b2) for the gap solitons. The solitons are found to be linearly stable for self-defocusing quintic nonlinearity (
The power-eigenvalue (
In addition to the power-eigenvalue diagram, the power and linear stability of solitons are studied in Figure 6 for varied values of
(a) Soliton power for varying quintic nonlinearity coefficient
Figure 6 shows that the power of solitons is decreased as
To confirm obtained linear stability results, we study nonlinear evolution of solitons by means of direct simulation of the governing Eq. (2). To this end, using the fourth-order Runge–Kutta (RK4) scheme fundamental soliton is advanced in
In Figure 7, linear stability spectrum (first column), nonlinear evolution of the peak amplitude (second column), the evolution of soliton power (third column), and 3D view of the evolved soliton (fourth column) are plotted for the fundamental solitons that correspond to “a”, “b”, “c”, “d” and “e” points in the right panel of Figure 6(a). Here, the impact of quintic nonlinearity is examined by increasing
Linear spectra (first column), nonlinear evolution of peak amplitudes (second column), the evolution of soliton power (third column), and 3D view of the soliton after evolution (fourth column) for (a)
In Figure 7(b)–(d), it is shown that the linear spectrum of the fundamental soliton is purely-imaginary (none of their eigenvalues have a real part), and the peak amplitudes oscillate during the propagation, thus stable evolution can be achieved for the considered solitons in the quadratic-cubic-quintic nonlinear medium when
From previous studies, it is known that increasing potential depth
Peak amplitude of the evolved solitons for varying lattice depths when the fundamental soliton is obtained for (a)
Furthermore, it is known that increasing anisotropy parameter
Peak amplitude of the evolved solitons for varying anisotropy coefficients (
It must be noted that even though soliton solutions of the NLSM system (2) can be obtained when
Fundamental lattice solitons were obtained in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity. The steady-state solutions are calculated by the SOM algorithm and the stability properties of solitons have been explored by linear spectra and nonlinear evolution of the amplitude by direct simulation of the nonlinear model. The band-gap boundaries and stability intervals of solitons were found and the power of solitons was investigated for varied
The stability analysis showed that fundamental lattice solitons can stay stable for the weak self-focusing and self-defocusing quintic nonlinearity. Strong self-focusing (
Furthermore, it has been observed that when the quintic nonlinearity is strong in the medium, solitons collapse, and the collapse of solitons can be arrested by a deeper lattice for self-focusing quintic nonlinearity (
In conclusion, the numerical existence and stability of fundamental lattice solitons have been presented in a nonlocal nonlinear medium with the quintic nonlinear response, and it has been demonstrated that stability of unstable solitons can be improved by modification of potential depth and strong anisotropy coefficient.
The authors declare no conflict of interest.
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A viral disease can be defined as an infectious disease that has recently appeared within a population or exists in nature with the rapid expansion of incident or geographic range. 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Biochemistry examines macromolecules - proteins, nucleic acids, carbohydrates, and lipids – and their building blocks, structures, functions, and interactions. Much of biochemistry is devoted to enzymes, proteins that catalyze chemical reactions, enzyme structures, mechanisms of action and their roles within cells. Biochemistry also studies small signaling molecules, coenzymes, inhibitors, vitamins, and hormones, which play roles in life processes. Biochemical experimentation, besides coopting classical chemistry methods, e.g., chromatography, adopted new techniques, e.g., X-ray diffraction, electron microscopy, NMR, radioisotopes, and developed sophisticated microbial genetic tools, e.g., auxotroph mutants and their revertants, fermentation, etc. More recently, biochemistry embraced the ‘big data’ omics systems. Initial biochemical studies have been exclusively analytic: dissecting, purifying, and examining individual components of a biological system; in the apt words of Efraim Racker (1913 –1991), “Don’t waste clean thinking on dirty enzymes.” Today, however, biochemistry is becoming more agglomerative and comprehensive, setting out to integrate and describe entirely particular biological systems. The ‘big data’ metabolomics can define the complement of small molecules, e.g., in a soil or biofilm sample; proteomics can distinguish all the comprising proteins, e.g., serum; metagenomics can identify all the genes in a complex environment, e.g., the bovine rumen. 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Dr. Blumenberg’s research is focused on the epidermis, expression of keratin genes, transcription profiling, keratinocyte differentiation, inflammatory diseases and cancers, and most recently the effects of the microbiome on the skin. He has published more than 100 peer-reviewed research articles and graduated numerous Ph.D. and postdoctoral students.",institutionString:null,institution:{name:"New York University Langone Medical Center",institutionURL:null,country:{name:"United States of America"}}},subseries:[{id:"14",title:"Cell and Molecular Biology",keywords:"Omics (Transcriptomics; Proteomics; Metabolomics), Molecular Biology, Cell Biology, Signal Transduction and Regulation, Cell Growth and Differentiation, Apoptosis, Necroptosis, Ferroptosis, Autophagy, Cell Cycle, Macromolecules and Complexes, Gene Expression",scope:"The Cell and Molecular Biology topic within the IntechOpen Biochemistry Series aims to rapidly publish contributions on all aspects of cell and molecular biology, including aspects related to biochemical and genetic research (not only in humans but all living beings). We encourage the submission of manuscripts that provide novel and mechanistic insights that report significant advances in the fields. Topics include, but are not limited to: Advanced techniques of cellular and molecular biology (Molecular methodologies, imaging techniques, and bioinformatics); Biological activities at the molecular level; Biological processes of cell functions, cell division, senescence, maintenance, and cell death; Biomolecules interactions; Cancer; Cell biology; Chemical biology; Computational biology; Cytochemistry; Developmental biology; Disease mechanisms and therapeutics; DNA, and RNA metabolism; Gene functions, genetics, and genomics; Genetics; Immunology; Medical microbiology; Molecular biology; Molecular genetics; Molecular processes of cell and organelle dynamics; Neuroscience; Protein biosynthesis, degradation, and functions; Regulation of molecular interactions in a cell; Signalling networks and system biology; Structural biology; Virology and microbiology.",annualVolume:11410,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/14.jpg",editor:{id:"165627",title:"Dr.",name:"Rosa María",middleName:null,surname:"Martínez-Espinosa",fullName:"Rosa María Martínez-Espinosa",profilePictureURL:"https://mts.intechopen.com/storage/users/165627/images/system/165627.jpeg",institutionString:null,institution:{name:"University of Alicante",institutionURL:null,country:{name:"Spain"}}},editorTwo:null,editorThree:null,editorialBoard:[{id:"79367",title:"Dr.",name:"Ana Isabel",middleName:null,surname:"Flores",fullName:"Ana Isabel Flores",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRpIOQA0/Profile_Picture_1632418099564",institutionString:null,institution:{name:"Hospital Universitario 12 De Octubre",institutionURL:null,country:{name:"Spain"}}},{id:"328234",title:"Ph.D.",name:"Christian",middleName:null,surname:"Palavecino",fullName:"Christian Palavecino",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0033Y000030DhEhQAK/Profile_Picture_1628835318625",institutionString:null,institution:{name:"Central University of Chile",institutionURL:null,country:{name:"Chile"}}},{id:"186585",title:"Dr.",name:"Francisco Javier",middleName:null,surname:"Martin-Romero",fullName:"Francisco Javier Martin-Romero",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSB3HQAW/Profile_Picture_1631258137641",institutionString:null,institution:{name:"University of Extremadura",institutionURL:null,country:{name:"Spain"}}}]},{id:"15",title:"Chemical Biology",keywords:"Phenolic Compounds, Essential Oils, Modification of Biomolecules, Glycobiology, Combinatorial Chemistry, Therapeutic peptides, Enzyme Inhibitors",scope:"Chemical biology spans the fields of chemistry and biology involving the application of biological and chemical molecules and techniques. In recent years, the application of chemistry to biological molecules has gained significant interest in medicinal and pharmacological studies. This topic will be devoted to understanding the interplay between biomolecules and chemical compounds, their structure and function, and their potential applications in related fields. Being a part of the biochemistry discipline, the ideas and concepts that have emerged from Chemical Biology have affected other related areas. 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Behind these definitions are hidden all the aspects of normal and pathological functioning of all processes that the topic ‘Metabolism’ will cover within the Biochemistry Series. 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Thus proteomics, an area of research that detects all protein forms expressed in an organism, including splice isoforms and post-translational modifications, is more suitable than genomics for a comprehensive understanding of the biochemical processes that govern life. The most common proteomics applications are currently in the clinical field for the identification, in a variety of biological matrices, of biomarkers for diagnosis and therapeutic intervention of disorders. From the comparison of proteomic profiles of control and disease or different physiological states, which may emerge, changes in protein expression can provide new insights into the roles played by some proteins in human pathologies. Understanding how proteins function and interact with each other is another goal of proteomics that makes this approach even more intriguing. Specialized technology and expertise are required to assess the proteome of any biological sample. Currently, proteomics relies mainly on mass spectrometry (MS) combined with electrophoretic (1 or 2-DE-MS) and/or chromatographic techniques (LC-MS/MS). MS is an excellent tool that has gained popularity in proteomics because of its ability to gather a complex body of information such as cataloging protein expression, identifying protein modification sites, and defining protein interactions. The Proteomics topic aims to attract contributions on all aspects of MS-based proteomics that, by pushing the boundaries of MS capabilities, may address biological problems that have not been resolved yet.",annualVolume:11414,isOpenForSubmission:!0,coverUrl:"https://cdn.intechopen.com/series_topics/covers/18.jpg",editor:{id:"200689",title:"Prof.",name:"Paolo",middleName:null,surname:"Iadarola",fullName:"Paolo Iadarola",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSCl8QAG/Profile_Picture_1623568118342",institutionString:null,institution:{name:"University of Pavia",institutionURL:null,country:{name:"Italy"}}},editorTwo:{id:"201414",title:"Dr.",name:"Simona",middleName:null,surname:"Viglio",fullName:"Simona Viglio",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRKDHQA4/Profile_Picture_1630402531487",institutionString:null,institution:{name:"University of Pavia",institutionURL:null,country:{name:"Italy"}}},editorThree:null,editorialBoard:[{id:"72288",title:"Dr.",name:"Arli Aditya",middleName:null,surname:"Parikesit",fullName:"Arli Aditya Parikesit",profilePictureURL:"https://mts.intechopen.com/storage/users/72288/images/system/72288.jpg",institutionString:null,institution:{name:"Indonesia International Institute for Life Sciences",institutionURL:null,country:{name:"Indonesia"}}},{id:"40928",title:"Dr.",name:"Cesar",middleName:null,surname:"Lopez-Camarillo",fullName:"Cesar Lopez-Camarillo",profilePictureURL:"https://mts.intechopen.com/storage/users/40928/images/3884_n.png",institutionString:null,institution:{name:"Universidad Autónoma de la Ciudad de México",institutionURL:null,country:{name:"Mexico"}}},{id:"81926",title:"Dr.",name:"Shymaa",middleName:null,surname:"Enany",fullName:"Shymaa Enany",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bRqB9QAK/Profile_Picture_1626163237970",institutionString:null,institution:{name:"Suez Canal University",institutionURL:null,country:{name:"Egypt"}}}]}]}},libraryRecommendation:{success:null,errors:{},institutions:[]},route:{name:"chapter.detail",path:"/chapters/28693",hash:"",query:{},params:{id:"28693"},fullPath:"/chapters/28693",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)}()