Deformed Sine-Gordon Models, Solitons and Anomalous 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 N−soliton 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 N−soliton 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

where wis a real scalar field w, Vwis the scalar potential and V1w≡ddwVw. 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].

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

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].

under the special space–time reflection

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

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

In addition, consider an even potential Vunder Px

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).

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.

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,±3t−qa,±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 α−3t(middle figures). The bottom figures show ∫dt′α+3t′, vanishing in Figure 3 and ∫dt′α−3t′asymptotically 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.

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 α−3t(middle figures). The bottom figures show ∫dt′α+3t′, vanishing in Figure 5 and ∫dt′α−3t′asymptotically 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.

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) α−3t(middle figures). The bottom figures of Figures 7 and 8 show the vanishing ∫dt′α+3t′and periodic ∫dt′α−3t′expressions. 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.

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+=∫dx2∂t2w+∂x2wV,E37

a−=∫dx4∂t∂xwV.E38

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

γ+=∫dx∂tw2−∂xw2∂tV,E39

γ−=−∫dx∂tw2−∂xw2∂xV.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 9–12 show the anomalies a±and γ±and their corresponding densities. The anomalies a−and γ−vanish 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 γ−vanish in the Figures 11 and 12, respectively, since their densities are odd under space reflection.

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

and the next linear first order equation for ψ

The compatibility condition ∂η∂ξu−∂ξ∂ηu=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

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.

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.