Open access peer-reviewed chapter - ONLINE FIRST

Deformed Sine-Gordon Models, Solitons and Anomalous Charges

By Harold Blas, Hector F. Callisaya, João P.R. Campos, Bibiano M. Cerna and Carlos Reyes

Submitted: September 10th 2020Reviewed: December 8th 2020Published: January 25th 2021

DOI: 10.5772/intechopen.95432

Downloaded: 28

Abstract

We study certain deformations of the integrable sine-Gordon model (DSG). It is found analytically and numerically several towers of infinite number of anomalous charges for soliton solutions possessing a special space–time symmetry. Moreover, it is uncovered exact conserved charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings with equal and opposite velocities. Moreover, we provide a linear formulation of the modified SG model and a related tower of infinite number of exact non-local conservation laws. We back up our results with extensive numerical simulations for kink-kink, kink-antikink and breather configurations of the Bazeia et al. potential V q w = 64 q 2 tan 2 w 2 1 − sin w 2 q 2 , q ∈ R , which contains the usual SG potential V 2 w = 2 1 − cos 2 w .

Keywords

  • quasi-integrability
  • solitons
  • deformed sine-Gordon
  • anomalous charges
  • non-local charges

1. Introduction

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 Nsoliton possesses definite parity under a shifted parity and delayed time reversion (PsTd) symmetry. When the anomaly densities possess odd parities the space–time integration of them vanish, which imply the existence of anomalous charges. Fourth, the conserved charges of the usual integrable systems turn out to be the anomalous charges upon deformation. Fifth, there exist infinite towers of infinitely many anomalous charges, different in form from the ones of the usual integrable models. New towers of anomalous charges have been uncovered in [8, 9, 10]. Remarkably, even the usual integrable models possess quasi-conservation laws with anomalous charges for analytical Nsoliton with CPsTdsymmetry [9, 10]. For the standard SG theory it has been discussed for the 2-soliton sector of the theory [8]. Sixth, there is a subset of exact conserved charges for soliton eigenstates simply of the shifted space-reflection Ps. The deformed NLS model for two-soliton solutions [6, 7] and the deformed sine-Gordon model [11] for two-kink and breather solutions exhibit this property.

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.

2. A deformation of the sine-Gordon model

Let us consider the relativistic field theories in 1+1-dimensions with equation of motion1

ξηw+V1w=0,E1

where wis a real scalar field w, Vwis the scalar potential and V1wddwVw. The family of potentials Vwwill represent certain deformations of the usual SG model. The theory (1) has been studied using the techniques of integrable field theories, such as the anomalous zero-curvature [4, 11] and deformed Riccati-type pseudo-potential formulations [8], respectively. In our simulations we will consider [4, 24].

Vwq=2q2tan2w1sinwq2,E2

where qis a real parameter such that for q=2the potential reduces to the SG potential

Vw2=1161cos4w.E3

So, we introduce the deformation parameter εas q=2+ε, such that in the limit ε=0one reproduces the SG model.

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 wand the potential Vsatisfy the symmetry [4, 8, 11].

P:ww+const.;VwVw,E4

under the special space–time reflection

PPsTd,Ps:x˜x˜,Td:t˜t˜˜,x˜xxΔ,t˜=ttΔ,E5

defined around a given point xΔtΔ. Moreover, let us consider the space-reflection transformation

Px:xx,E6

and assume that the scalar field is an eigenstate of the operator Px

Px:wϱw,ϱ=±1.E7

In addition, consider an even potential Vunder Px

PxV=V.E8

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.

3. Quasi-conservation laws of the deformed SG model

We will discuss some of the infinite towers of quasi-conservation laws of the deformed SG model (1).

3.1 First type of towers: The SG-type quasi-conservation laws

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].

ddtqa2n+1=dxβ2n+1,n=1,2,3,E9

where the quantities qa2n+1define the anomalous charges, provided that the time-integrated anomalies dtdxβ2n+1vanish for solitons satisfying (4) and (5). This condition, when combined with Eq. (9), implies qa2n+1t+=qa2n+1t. So, we have that qa2n+1are anomalous for n=1,2,3,.. The charges qa2n+1maintain the same form as the ones of the usual SG.

In 1+1-dimensional Lorentz invariant integrable field theories one has dual integrability conditions or Lax equations. Analogously, for the deformations of the SG model there exist a dual formulation for each equation as in (9) by interchanging ξηin the procedure to obtain the relevant quasi-conservation laws. So, one can get

ddtq˜a2n+1=dxβ˜2n+1,n=1,2,3,E10

where the quantities q˜a2n+1define the dual asymptotically conserved charges, provided that the time-integrated anomalies dtdxβ˜2n+1vanish. Likewise, this result implies q˜a2n+1t+=q˜a2n+1t.

These towers of quasi-conservation laws reproduce the same polynomial form as in the usual sine-Gordon charge densities. In fact, the anomalies β2n+1and β˜2n+1vanish identically provided that the deformed potential Vwrecovers the form of the standard SG potential.

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 β2n+1and β˜2n+1vanish for special two-soliton solutions.

3.1.1 Space-reflection parity and conserved charges

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 n=1,2,, of the above two sets of quasi-conserved charges qa2n+1(9) and q˜a2n+1(10). Consider the new quasi-conservation laws

ddtqa,±2n+1=dxβ±2n+1,n=1,2,,E11

with the charges qa,±2n+1and anomalies β±2n+1, respectively, defined as

qa,±2n+1116qa2n+1±q˜a2n+1,E12
βa,±2n+1116β2n+1±β˜2n+1E13

in which the quantities q2n+1and β2n+1defined in (9) and the quantities q˜a2n+1and β˜2n+1in (10) have been used, respectively.

Since the theory (1) is invariant under space–time translations one has that the energy momentum tensor is conserved. In fact, one has β1=β˜1=0at the zero’th order n=0, and the linear combinations of the charges qa1and q˜a1leads to the energy and momentum, respectively [11].

q+1=+dx12tw2+12xw2+V,E14
q1=+dxxwtw,E15

where E=q+1is the energy and P=q1is the momentum.

The first non-trivial anomalies become [11].

β±3=±12Zξξw2ηηw2,ZV2+16V1.E16
β±5=±12Z24ξw2ξ2w+ξ4wξw±24ηw2η2w+η4wηw.E17

Notice that for the SG potential (3) the factor Zabove vanishes identically; therefore, the anomalies vanish β±3=0, and the relevant charges q±3turn out to be the exactly conserved charges of the standard SG model at this order.

The properties of the quantities q±2n+1and dxβ±2n+1in (11) will depend on the symmetry properties of the solitons, in particular on the space-reflection symmetry of β±2n+1, as we will see below. So, let us examine the space-reflection symmetry of them.

Let us write the anomalies in terms of the xand tderivatives. So, once the eq. of motion (1) is used to substitute t2wx2wVw, as well as, neglecting surface terms one has

α+32dxf+3xt,E18
f+3xtV+16Vxtw2+xxw2,E19

where we have defined the anomaly density f+3. Notice that for even parity potentials (8) and for definite parity (even or odd) fields wthe density f+3is an odd function, and thus the xintegrated anomaly α+3vanishes.

Following analogous procedure as above one has

α3=4dxf3xt,E20
f3xtV+16Vtwx2w+xwxtw,E21

where we have defined the anomaly density f3. Notice that for even parity potentials (8) and for definite parity (even or odd) fields wthe density f3is an even function, and thus the xintegrated anomaly α3will not vanish solely by a space-reflection parity reason.

The anomalies α±3and dtα±3in (18) and (20), will be computed numerically for two-solitons and breather-like solutions below.

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).

3.2 Second type of towers

The quasi-conservation laws [8].

ddtQaN=aN,E22
QaNdx1NξwN+VξwN2,E23
aNdxN2ξwN3ξ2wV,N3,E24

define the asymptotically conserved charges QaNand the corresponding anomalies aN.

The dual quasi-conservation laws become

ddtQ˜aN=a˜N,E25
Q˜aNdx1NηwN+VηwN2,E26
a˜NdxN2ηwN3η2wV,N3,E27

where we have introduced the dual asymptotically conserved charges Q˜aNand the relevant anomalies a˜N.

The densities of the anomalies aNand a˜Nin (24) and (27), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (22) and (25), respectively, allow the construction of asymptotically conserved charges.

3.3 Third type of towers

Let us define the quasi-conservation laws [8].

ddtQaN=γN,E28
QaNdx12VN1ξw2+1NVN,E29
γNdx12ξw2ηVN1,N2,E30

where we have introduced the asymptotically conserved charges Q̂aNand the corresponding anomalies γN.

The interchange ηξallows us to reproduce the dual quasi-conservation laws. So, one has

ddtQ˜aN=γ˜N,E31
Q˜aNdx12VN1ξw2+1NVN,E32
γ˜Ndx12ηw2ξVN1,N2,E33

where we have defined the dual asymptotically conserved charges Q˜aNand the anomalies γ˜N.

Similarly, the densities of the anomalies γNand γ˜Nin (30) and (33), respectively, possess odd parities under (4) and (5), so the quasi-conservation laws (28) and (31), respectively, allow the construction of asymptotically conserved charges.

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]) Mnt=+vndx,n1; where vis the KdV field. The M1,2cases are conserved charges. It is remarkable that the moments, M3,4, respectively, in the interaction region of two-solitons, behave as the anomalous charges of the quasi-integrable KdV models [9]. In fact, in the quasi-integrable KdV models the moments M2,3are in fact anomalous charges [9]. So, since the two-soliton collision is an important ingredient in the formation of soliton turbulence and the dynamics of soliton gases, we can expect they will be important in the quasi-integrable counterparts. In the case of the SG soliton ensemble, to our knowledge, it is needed a further theoretical research.

4. Numerical simulations

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.

Figure 1.

Kink-kink with velocities v 2 = − v 1 = 0.15 and q = 2.01 in (2), for initial (green), collision (blue) and final (red) times. Bottom, the energy ( E ) and momentum ( P ) charges of the kink-kink.

Figure 2.

Kink-antikink with velocities v 2 = − v 1 = 0.15 and q = 2.01 in (2), for initial (green), collision (blue) and final (red) times. Bottom, the energy ( E ) and momentum ( P ) charges of the kink-antikink.

4.1 First non-trivial anomalies of the SG-type quasi-conservation laws

We have checked our results by numerical simulation of the anomalies α±3(18)(21) for kink-antikink, kink-kink and breather solutions of the model (2).

So, let us write (11) in the form

qa,±3tqa,±3t0=t0tdtα±3t,E34

where α±3twere defined in (18) and (20) and t0is the initial time.

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.

4.1.1 Kink-antikink

In the Figures 3 and 4 we show the results for kink-antikink system with velocities v2=v1=0.5and ε=0.06. The plots of (19) and (21) as f±3xtvsxare shown for three successive times (top figures). Their integration in space furnish vanishing α+3tand non-vanishing α3t(middle figures). The bottom figures show dtα+3t, vanishing in Figure 3 and dtα3tasymptotically vanishing in Figure 4, respectively. According to (34) our numerical simulations show the asymptotically conservation of the charge qa,3and the exact conservation of the charge qa,+3, within numerical accuracy.

Figure 3.

f + 3 , α + 3 and ∫ dt α + 3 in (18) and (19) for kink-antikink with velocities v 2 = − v 1 = 0.5 and ε = 0.06 . The density figure shows initial t i (green), collision t c (blue) and final t f (red) times of the kink-antikink.

Figure 4.

f − 3 , α − 3 and ∫ dt α − 3 in (20)–(21) for kink-antikink with velocities v 2 = − v 1 = 0.5 and ε = 0.06 . The density figure shows initial t i (green), collision t c (blue) and final t f (red) times of the kink-antikink.

4.1.2 kink-kink

In the Figures 5 and 6 we show the results for kink-kink system with velocities v2=v1=0.5and ε=0.06. The plots of (19) and (21) as f±3xtvsxare shown for three successive times (top figures). Their integration in space furnish vanishing α+3tand non-vanishing α3t(middle figures). The bottom figures show dtα+3t, vanishing in Figure 5 and dtα3tasymptotically vanishing in Figure 6. According to (34) our numerical results show the asymptotically conservation of the charge qa,3and the exact conservation of the charge qa,+3, within numerical accuracy.

Figure 5.

f + 3 , α + 3 and ∫ dt α + 3 in (18) and (19) for kink-kink with velocities v 2 = − v 1 = 0.5 and ε = 0.06 . The density figure shows initial t i (green), collision t c (blue) and final t f (red) times of the kink-kink.

Figure 6.

f − 3 , α − 3 and ∫ dt α − 3 in (20) and (21) for kink-kink with velocities v 2 = − v 1 = 0.5 and ε = 0.06 . The density figure shows initial t i (green), collision t c (blue) and final t f (red) times of the kink-kink.

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 qa,+3charge, within numerical accuracy.

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 (x,t) the parity symmetries are recovered, as discussed in [11]. So, the simulations performed in these reference frames, in the both kink-antikink and kink-kink cases, will provide vanishing α+3anomalies as shown above.

4.1.3 Breather: kink-antikink bound state

Figures 7 and 8 show the results for breather (kink-antikink bound state) with ε=0.06. The densities f±3xtin (19) and (21), respectively, have been plotted as functions of xfor three successive times (top figures). They show the vanishing α+3tand non-vanishing (periodic in time) α3t(middle figures). The bottom figures of Figures 7 and 8 show the vanishing dtα+3tand periodic dtα3texpressions. According to (34) our numerical results show the oscillation of the charges qa,3around a fixed value and the exact conservation of the charge qa,+3, within numerical accuracy.

Figure 7.

f + 3 , α + 3 and ∫ dt α + 3 in (18) and (19) for breather with ε = 0.06 . The density is shown for three times t ∈ t f − T 0 t f , T 0 = 7.025 . The long-lived breather for t f ≈ 10 5 .

Figure 8.

f − 3 , α − 3 and ∫ dt α − 3 in (20) and (21) for breather with ε = 0.06 . The density is shown for three times t ∈ t f − T 0 t f , T 0 = 7.025 . The long-lived breather for t f ≈ 10 5 .

4.2 Lowest order anomalies of the second and third types of towers

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,

a±a3±a˜3,E35
γ±γ2±γ˜2.E36

4.2.1 Second and third types of towers and lowest order anomalies

The two anomalies in (35) can be written as

a+=dx2t2w+x2wV,E37
a=dx4txwV.E38

Similarly, the two anomalies in (36) can be written as

γ+=dxtw2xw2tV,E39
γ=dxtw2xw2xV.E40

Notice that under the space–time reflection transformation (4) and (5), the densities of the above anomalies a±3and γ±, respectively, are odd; then they must vanish upon space–time integration. Therefore, one has asymptotically conserved charges associated to the relevant quasi-conservation laws.

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 912 show the anomalies a±and γ±and their corresponding densities. The anomalies aand γvanish as shown in the Figures 9 and 10, respectively, for symmetric kink-antikink soliton (see Figure 2), within numerical accuracy, since their densities are odd under space reflection. Similarly, for anti-symmetric kink-kink soliton (see Figure 1) the anomalies a+and γvanish in the Figures 11 and 12, respectively, since their densities are odd under space reflection.

Figure 9.

Top: The anomaly densities (37) and (38), respectively, plotted in x − coordinate for three times t i (green), t c (blue) and t f (red). Bottom: The anomalies a ± vs t , for kink-antikink collision shown in Figure 2.

Figure 10.

Top: Anomaly densities (39) and (40), respectively, plotted in x − coordinate for three times t i (green), t c (blue) and t f (red). Bottom: Anomalies γ ± vs t , for kink-antikink shown in Figure 2.

Figure 11.

Top: Anomaly densities of (37) and (38), respectively, plotted in x − coordinate for three successive times t i (green), t c (blue) and t f (red). Bottom figures show the relevant anomalies a ± vs t , for kink-kink shown in Figure 1.

Figure 12.

Top: Anomaly densities of (39) and (40), respectively, plotted in x − coordinate for three successive times t i (green), t c (blue) and t f (red). Bottom: Anomalies γ ± vs t , for kink-kink shown in Figure 1.

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).

5. Riccati-type pseudo-potentials and non-local conservation laws

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

ξu=2λ1u+ξw+ξwu2,E41
ηu=2λV2u12λV1+12λV1u2+ψ,E42

and the next linear first order equation for ψ

ξψ+2λ1ψ2uξwψ=2λ2uλξuZ,ZV2w+16Vw1.E43

The compatibility condition ηξuξηu=0of the system (41) and (42), taking into account (43), provides the equation of motion of the DSG model (1). Moreover, the ordinary differential equation for ψin the variable ξcan be integrated by quadratures [8]. Its expression will become highly non-local and, once inserted into (42), the system of Eqs. (41) and (42) will provide a non-local Riccati-type representation of the DSG model (1).

From the system (41) and (42) one can get a quasi-conservation law

ηuξw+ξλV212λuV1=λξwuZ+ξ.E44

This equation has been used to construct a tower of infinite number of quasi-conservation laws [8]. For the standard SG one has Z=ψ=0; so the Eq. (44) can generate the well known conservation laws of the usual SG model.

5.1 Pseudo-potentials and a linear system associated to DSG

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].

L1Φ=0,L2Φ=0,E45
L1ξAξ,Aξλ2ξw22ξw3ξ2w,E46
L2ηAη,Aη2λλV+ζ,E47

where the auxiliary non-local field ζis defined as

ζ=dξ6V1ξw2ξ2w2V2ξw4ξ2w2.E48

In fact, taking into account the expression for the auxiliary field ζ, the compatibility condition of the linear problem (45) provides the equation

Δξηλ6ξwξ2wΔξη+2ξw2ξ2w2ξΔξη=0,E49

with

Δξηξηw+V1w.E50

In (49) the coefficient of the linear term in λΔξηmust vanish, providing the DSG equation of motion (1). The other terms in (49) must also vanish provided that Δξη=0is imposed. So, L1and L2in (45) become a pair of linear operators associated to the DSG model (1).

5.2 Non-local conservation laws

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 σmodel the non-local conserved charges imply the absence of particle production and the first non-trivial one alone fixes almost completely the on-shell dynamics of the model (see e.g. [3, 32]). These charges may be constructed through an iterative procedure [33]. Following this method one gets a set of infinite number of non-local conservation laws for the system (45). In fact, this system satisfies the properties: i) AξAηis a “pure gauge”; i.e. Aμ=μΦΦ1,μ=ξ,η; ii) Jμ=AξAηis a conserved current satisfying

ηAξξAη=0.E51

So, one can construct an infinite set of non-local conserved currents through an inductive procedure. Let us define the currents

Jμn=μχn,μξ,η;n=0,1,2,E52
dχ1=Aξ+AηdI0ξη+λdI1ξη;E53
Jμn+1=μχnAμχn;χ0=1,E54

where

dI0ξηa0ξη+b0ξη,dI1ξηa1ξη+b1ξη,E55

where

a02ξw3ξ2w;b0ζ=dξ6V1ξw2ξ2w2V2ξw4ξ2w2;E56
a112ξw2;b12V.E57

Then one can show by an inductive procedure that the (non-local) currents Jμnare conserved

μJnμ=0,n=1,2,3,,+.E58

The first current conservation law μJ1μ=0reduces to the Eq. (51), and then provides the first two conservation laws

ηa0ξb0=0,ηa1ξb1=0.E59

The next conservation law μJ2μ=0, in powers of λ, furnishes

ηa0I0ξb0I0=0,E60
ηa0I1+a1I0ξb0I1+b1I0=0,E61
ηa1I1ξb1I1=0.E62

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.

6. Conclusions

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.

Acknowledgments

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.

Notes

  • In the x and t laboratory coordinates: η = t + x 2 , ξ = t − x 2 , ∂ η = ∂ t + ∂ x , ∂ ξ = ∂ t − ∂ x , ∂ η ∂ ξ = ∂ t 2 − ∂ x 2

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Harold Blas, Hector F. Callisaya, João P.R. Campos, Bibiano M. Cerna and Carlos Reyes (January 25th 2021). Deformed Sine-Gordon Models, Solitons and Anomalous Charges [Online First], IntechOpen, DOI: 10.5772/intechopen.95432. Available from:

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