A Equations in the R and \varphi parametrization

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

H. Blas and M. Zambrano


Instituto de Física

Universidade Federal de Mato Grosso

Av. Fernando Correa,   2367

Bairro Boa Esperança, Cep 78060-900, Cuiabá - MT - Brazil

Instituto de Ciências Matemáticas e de Computação; ICMC/USP;

Universidade de São Paulo,

Caixa Postal 668, CEP 13560-970, São Carlos-SP, Brazil

The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schrödinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sine-Gordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential and the saturable type potential satisfying , with a deformation parameter and . The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of . Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

1 Introduction

The theory of soliton collisions in the integrable models is well known in the current literature. In particular, the solitons emerge with their velocities and shapes completely unaltered after collision between them, the only noticed outcome being their phase shifts. However, certain non-linear field theory models with relevant physical applications possess soliton-like solutions and it is difficult to know a priori if they are in fact true solitons. The so-called quasi-integrability concept has recently been put forward in the context of certain deformations of the integrable sine-Gordon (SG), Bullough-Dodd (BD) and focusing non-linear Schrödinger (NLS) models [1, 2]. The idea is that many non-integrable theories posses solitary wave solutions that behave similar to solitons, i.e. the scattering of such solitons basically preserve their shapes and velocities. So, certain deformations of the SG, BD and focusing NLS models were considered as quasi-integrable theories possessing infinite number of charges that are asymptotically conserved. This means that during the collision of two soliton-like solutions such charges do vary in time and sometimes significantly, however their values remain the same in the past (before the collision) and in the future (after the collision). The deformations of the topological SG and BD models studied in [1] furnish kink-type solitary waves presenting soliton-like behaviour. Whereas, the deformed focusing NLS model considered in [2] supports bright solitary waves presenting soliton-like behaviour. The both numerical and analytical techniques were used to describe the concept of quasi-integrability.

Regarding the analytical calculations of solitary wave collisions in deformed NLS models some results have been obtained for the cases of small perturbations of the integrable NLS model [3, 4, 5]. For nearly integrable models it has been implemented a perturbation theory allied to the the inverse scattering transform (IST) method [6]. These methods have been applied mainly to the focusing NLS solitons that decay at infinity, i.e. the bright solitons. On the other hand, the integrable defocusing NLS model supports dark solitons which are intensity dips sitting on a continuous wave background (cwb) with a phase change across their intensity minimum. In order to apply to the dark soliton perturbation and the non-vanishing boundary conditions (nvbc), inherent to this type of solution, various improvements have been made to the calculations and methods [7, 8, 9] based on the so-called complete set of squared Jost solution (eigenstates of the linearised NLS operator). The implementation of direct methods, however, consider small perturbations around the integrable NLS model [5, 10]. Moreover, in [5] it has been argued that the squared Jost functions associated with the dark soliton might be an insufficient basis, so rendering problematic the issue of the completeness of the basis for the solution space of a deformed NLS equation. The solitary wave collisions of certain deformed NLS models, beyond small perturbations, have been considered in the recent literature; e.g. in [11] it has been computed the spatial shifts of fast dark-dark soliton collisions and in [12] some properties of slow dark soliton collisions have been studied.

It would be interesting to understand the issue of quasi-integrability in the other variant of the NLS model, i.e. the deformed defocusing NLS model, since such theories also appear in many areas of non-linear science, condensed matter physics, plasma physics and, in particular, in the study of Bose-Einstein condensates. In this context, our aim is to predict the results of solitary wave collisions and test the quasi-integrability concept in deformed defocusing NLS models with nvbc. However, the non-vanishing boundary condition inherent to the dark solitons, which may change when the deformations are present, introduces some complications when applying the techniques developed for bright solitons in order to study the quasi-integrability concept. In the analytic treatments the nvbc introduces the issue of the renormalization of the charges and anomalies in the expressions of the quasi-conservation laws which must be properly addressed. This is closely related to the fact that the dark soliton solution is a composite object, i.e. it incorporates the continuous wave background (cwb) plus the dark soliton itself. In addition, to simulate the time dependence of field configurations for computing dark solitons properties the choice of the efficient and accurate numerical methods, the so-called time-splitting cosine pseudo-spectral finite difference (TSCP) and the time-splitting finite difference with transformation (TSFD-T) methods [13] will be made in order to control the highly oscillatory phase background. In fact, these methods allowed us to improve in several orders of magnitude the accuracy in the computation of the charges and anomalies presented in [2].

The deformed focusing NLS with bright soliton solutions and the structures responsible for the phenomenon of quasi-integrability have been discussed in [2]. In this context it has been shown that this model possesses an infinite number of asymptotically conserved charges. An explanation found so far for this behaviour of the charges is that some special soliton type solutions are eigenstates of a space-time parity transformation. On the other hand, the deformed defocusing NLS with dark soliton solutions also presents the above remarkable properties, however in this case we can say even more. As we will show, there are some soliton like solutions which present a special space-reflection symmetry for any time, so this property implies the exact conservation of the sequence of the even order charges, which in the absence of this special symmetry would be conserved only asymptotically, as in the models studied in [1, 2]. Here we will show the connection between the space-reflection parity and the exactly conserved charges which involves an interplay between the space-reflection parity and internal transformations in the affine Kac-Moody algebra underlying the anomalous Lax equation. However, it seems to be that such parity property is a sufficient but not a necessary condition in order to have the sequence of exactly conserved charges. In fact, as we will show by numerical simulations, there are certain soliton like configurations without this symmetry which also exhibit such conserved charges. So, these properties constitute the distinguishing new features associated to the deformed defocusing NLS with dark soliton solutions, as compared to the previous quasi-integrable models [1, 2].

The paper is organized as follows. In the next section we discuss the deformed defocusing NLS model with non-vanishing boundary conditions. In section 3, we discuss the concept of quasi-integrability for deformed defocusing NLS, based on an anomalous zero curvature condition. In 3.1 and 3.2 we discuss the relationships between the space-time parity and asymptotically conserved charges, and the space-reflection parity and the exactly conserved charges, respectively. In section 4 we perform the renormalizations of the charges and anomalies relevant for dark soliton solutions. In sections 5.1 and 5.2 we discuss the space-time and space-reflection symmetries of dark soliton solutions of the integrable NLS model, respectively. In 6 we show the vanishing of the anomalies associated to solitary waves. In 6.1, by using general symmetry considerations, we show that the anomalies vanish when evaluated on a general solitary wave of the deformed NLS model. In 6.2 we show, by direct computation, the vanishing of the first non-trivial anomaly when evaluated on a general solitary wave of the deformed NLS. In section 7 we present the results of our numerical simulations which allowed us to compute and study various properties of dark soliton scattering of the full equations of motion for two deformations of the NLS model and several values of the deformation parameter . In 7.1 we present the results of the simulations of the first deformation with , and in 7.2 the second deformation with . In 8 we present some conclusions and discussions. The appendix A presents the eqs. of motion in the and parametrizations, as well as some useful identities and B presents relevant expressions corresponding to the gauge transformation.

2 Deformations of defocusing NLS

We will consider non-relativistic models of a complex scalar field in dimensions with Lagrangian density given by

(2.1)

where stands for the complex conjugation of and . The Euler-Lagrange equation following from (2.1) becomes

(2.2)

The model (2.2) defines the deformed NLS model and it supports dark soliton type solutions in analytical form for some special functions . The potential , corresponds to the integrable NLS model and supports N-dark soliton solutions. The potential defines the non-integrable cubic-quintic NLS model (CQNLS) which possesses analytical bright and dark type solitons [14, 15]. In [16, 14] the bright solitary waves of the cubic-quintic focusing NLS have been regarded as quasi-solitons presenting partially inelastic collisions in certain region of parameter space. Among the models with saturable non-linearities [6], the case also exhibits analytical dark solitons [17]. The deformed NLS model with will deserve a careful consideration below. It passes the Painlevé test for arbitrary positive integers and [18]. However, its Lax pair formulation and analytical solutions for a general set , to the best of our knowledge, are not known in the literature.

The qualitative properties of travelling waves of the NLS model for general non-linearities and non-vanishing boundary conditions have been studied in [19]. Exact analytical dark-soliton solutions of the deformed NLS model with arbitrary potential are not available, and one can resort to numerical simulations to obtain such solutions. However, dark-soliton solutions can be presented in an implicit form as a curve in a complex plane (for the complex variable , where ) (see [6] and references therein) or in a small-amplitude approximation where the deformed NLS model is reduced to the Korteweg-de Vries equation [20].

Among the possible deformations of the NLS model the case , for has recently been considered in [2] in order to study the concept of quasi-integrability for bright soliton collisions. An analytical solution with vanishing boundary condition (bright soliton) for this potential is well known in the literature. However, an analytical dark soliton solution, as far as we know, is not known in the literature.

In this paper we will study analytically and numerically some deformations of the NLS model of the type (2.2). The first deformation we will consider in our study is defined by the non-integrable cubic-quintic NLS model

(2.3)

where we have considered in (2.2) (here and in the next sections ). This deformation possesses analytical solitary waves as we will present below.

The second deformation considers a saturable non-linearity and it is defined by the equation

(2.4)

We do not know any analytical solutions for a general set of parameters of this model, however dark solitary wave solutions will be obtained numerically.

The both deformations reproduce the integrable defocusing NLS model in the limit .

3 Quasi-integrability of deformed NLS

Next we undertake an analytical study of the properties of the deformed NLS model (2.2), in order to do this we will use the well known techniques from the integrable field theories. We follow the developments and notations put forward in [1, 2] on quasi-integrability. Then, one considers an anomalous zero curvature representation of the deformed NLS model (2.2) with the connection given by

(3.1)

where the above Lax potentials are based on a loop algebra with commutation relations

(3.2)

One can verify that the curvature of the connection (3.1) is given by

(3.3)
(3.4)

with

(3.5)

Note that when the equation of motion (2.2) and its complex conjugate, i.e , are satisfied the terms proportional to the Lie algebra generators vanish. In addition, the quantity vanishes for the usual non-linear Schrödinger potential

(3.6)

Then, the curvature vanishes for the NLS model making this theory an integrable field theory. For the generalizations of this theory, i.e. deformations of the potential (3.6), the above curvature does not vanish and the model is regarded as non-integrable [2]. Next, we use some algebraic techniques borrowed from the theory of integrable models in order to analyse the equation (3.4) corresponding to a model with non-vanishing anomaly (3.5), or equivalently in (3.3). Let us denote

(3.7)

and parametrize . Substituting the parametrization (3.7) into (2.2) one gets the system of eqs. of motion

(3.8)
(3.9)

We specialize the parametrizations for the case of the defocusing NLS, i.e. the case . In addition, define a new basis of the loop algebra as

(3.10)

satisfying

(3.11)

The connection in the new basis is obtained through the gauge transformation

(3.12)

So, inserting the connection (3.1) into the above expression one gets

(3.13)

So, considering the fields which satisfy the equation (2.2) and its complex conjugate, i.e. the terms in the directions of vanish in (3.4), the curvature becomes

(3.14)

Next, in order to construct the quasi-conservation laws we perform a further gauge transformation

(3.15)

with

(3.16)

The parameters of the transformation will be determined below so that the component of the new connection lies in the abelian sub-algebra generated by the elements . Under the gauge transformation (3.15) the curvature (3.14) transforms as

(3.17)

The loop algebra is furnished with an integer gradation associated to the grading operator defined by

(3.18)

Since the group element in (3.16) is generated by exponentiating negative grade elements of the algebra and the component of the connection (3.13) lies in the directions of generators of grades and , the component of the new connection has generators with grades . Then, expanding as a linear combination of the eigen-subspaces of the grading operator , one has

(3.19)

where

(3.20)

As usual we will use the generator as a semi simple element in the sense that its adjoint action decomposes the loop algebra into its kernel and image as follows

(3.21)

such that Ker and Im have no elements in common. According to the commutation relations in (3.11) one has that the ’s form a basis of Ker, and a basis of Im. Next, one can make all the components of in the subspace Im to vanish by choosing conveniently the parameters . This can be done recursively, for example , and so on. So, the component get rotated onto the subspace Ker

(3.22)

Notice that in this process it has not been used neither the equation of motion (2.2) nor its complex conjugate. The components will depend on the real fields and , and their -derivatives, but not on the potential . In the Appendix B we provide the first components of .

Next, the component of the connection does not get rotated onto the subspace Ker even when the equation of motion (2.2) and its complex conjugate are used. Then, one has

(3.23)

Since lies in Ker one has that the term in (3.17) has components only in Im. Let us denote

(3.24)

then, we find that

(3.25)

In the appendix C we present the first quantities .

Next, let us discuss the nvbc associated to dark soliton solutions. These solutions present a difficulty in the sense that the field does not vanish at . In general, the behaviour of the solutions are such that

(3.26)

where is a constant phase. The connection (3.1) is not suitable to construct the charges because

(3.27)

However, making the gauge transformation (3.12) and the reparametrization (3.7) the new connection depends upon the modulus and its derivatives, and only on the derivatives of the phase . Therefore, one has that for dark solitons

(3.28)

In addition the abelianized potential in (3.15) also satisfy that condition, i.e.

(3.29)

because the group element also satisfies

(3.30)

Indeed, look at the expressions of the first terms of  and in the appendix B and one notes that depends on and its derivatives and only on the derivatives of the phase .

Then, in the defocusing NLS case with dark soliton solutions one has that the component of the connection satisfies a non-vanishing boundary condition such that , then from (3.25) we have the anomalous conservation laws

(3.31)

Thus, the non-vanishing of the quantity given in (3.14) and the anomalies above imply the non-conservation of the charges. Therefore, the charges and anomalies in (3.31) are valid for the deformed NLS model with nvbc and dark soliton solutions.

3.1 Space-time parity and asymptotically conserved charges

Next, let us discuss a special symmetry which plays an important role in the study of quasi-integrable theories. For certain solutions of the theory (2.1) the charges satisfy the so-called mirror symmetry. This symmetry is realized on a special subset of solutions of the deformed NLS model (2.2). For every solution belonging to this subset one can find a point in space-time and define a parity transformation around this point as

(3.32)

such that the fields and transform as

(3.33)

In order to realize this symmetry at the level of the anomalous zero-curvature equation let us combine the above parity transformation with the order two automorphism of the loop algebra given by

(3.34)

So, consider the composition of the internal and space time transformations as

(3.35)

Then, the component of the connection (3.13) is odd, i.e.

(3.36)

This property can be used to show that the group element which enter in the gauge transformation (3.15) is even, i.e.

(3.37)

Then, one can use the last property to show that the factor in the integrand of the anomaly in (3.31) is even under the space-time parity, i.e.

(3.38)

For more details of this demonstration we refer to [2].

Next, let us note that the in the integrand of the anomaly in (3.31) is a derivative of a functional of . Since is assumed to satisfy (3.33) one has that the time-integrated anomaly vanishes

(3.39)

where the integration is meant to be performed on any rectangle with center in , and the parameters which define the integration intervals are any given fixed values of the shifted space-time coordinates and , respectively, defined in (3.32).

A remarkable consequence of the vanishing of the integrated anomaly is that the charges (3.31) satisfy a mirror time-symmetry around the point

(3.40)

So, this property holds for the special subsets of solutions satisfying (3.33) and belonging to the deformed NLS model (2.2) with an even potential under the parity (3.33). This defines the quasi-integrable deformed NLS model. Then, by taking in (3.40) one has

(3.41)

This relationship shows that the scattering of two-soliton solutions may present an infinite number of charges which are asymptotically conserved. For some very special two-bright soliton configurations in the context of focusing deformed NLS it has been verified that the charges are asymptotically conserved, i.e. even though they vary in time during collision their values in the far past and the far future are the same [2]. As we will show below, for the case of defocusing NLS it is a remarkable fact that for any two-dark soliton configuration it is possible to find a point in space-time , such that the fields and always satisfy the space-time parity symmetry (3.32)-(3.33) without any additional restriction on the soliton parameters.

Next we will summarize the main steps in the construction of solutions in perturbation theory, for a detailed account see [2]. One can choose as a zero order solution a solution of the NLS equation, such that it satisfies (3.33), i.e. it exhibits only the pair and nothing of the pair . So, let us expand the fields in powers of the deformation parameter

(3.42)
(3.43)

Then the equations of motion for the components of the first order solution satisfy non-homogeneous equations and so they can never vanish. However, the pair satisfy homogeneous equations and so they can vanish. In addition, if the pair of components without definite parity is a solution, so is the pair . Then, one can always choose the components of the first order solution whith definite parity. For this choice one can show that the second order solution has also similar properties, i.e. the components satisfy non-homogeneous equations and the components satisfy homogeneous equations. This enables one to choose the second order solution to be the pair with definite parity, and this process repeats in all orders.

Summarizing, we may conclude that

1. If one has a two-dark-soliton-like solution of the equation of motion (2.2), transforming under the space-time parity (3.32) as in (3.33), i.e.

(3.44)

and

2. If the potential in (2.2) evaluated on such a solution is even under the parity , i.e.

(3.45)

such that

(3.46)

3. Then, one has an infinite set of asymptotically conserved charges, i.e.

(3.47)

Therefore, the values of the charges in the infinite past, before the collision of the solitons, are the same as in the infinite future, after the collision. Theories possessing such properties are dubbed as quasi-integrable theories. In particular, the deformed NLS models (2.3) and (2.4) can be shown to satisfy the requirements (3.44) and (3.45) for some field configurations, and then they belong to the class of quasi-integrable theories, as we will discuss in this work.

3.2 Space-reflection symmetry and conserved charges

Next, we consider some special solutions which exhibit a space-reflection symmetry

(3.48)

such that the fields and transform as

(3.49)

As we will see below, in the special case of 2-dark solitons moving in opposite directions and equal velocities, such that they undergo a head-on collision, we have a space-reflection symmetry. The implication of this additional symmetry of the fields under space-reflection (for any shifted time), i.e. and being even fields, on the behaviour of the quantities deserves a further analysis. We will show that some ’s possess an even parity under for any shifted time. Since is odd the anomaly in (3.31) will vanish identically providing an exact conservation equation for the charges . Therefore, these type of charges will not vary during collision, implying their exact conservation in the whole process of scattering.

In order to realize the symmetry (3.48)-(3.49) at the level of the anomalous zero curvature equations (3.14) and (3.17) we introduce a transformation in the internal space along with the space-reflection symmetry . So, let us consider another order two automorphism of the loop algebra given by

(3.50)

Note that the connection component in (3.13) does not have a definite space-reflection parity. In addition, it doest not lie in an eigenspace of the automorphism of the loop algebra. So, consider the joint action of the space and internal transformations as

(3.51)

Then, the component of the connection (3.13) transforms as

(3.52)

Actually, each term of in (3.13) possesses this property. Next, we analyse the transformation property of the new connection in (3.19). Thus, one can have , and so

(3.53)

The term in the third eq. of (3.19) as defined in (3.20) is odd under , then it follows from the third equation in (3.19) that

(3.54)

The have been selected to rotate into the kernel of the adjoint action of and the r.h.s. of (3.54) is in the image of the adjoint action, so for consistency the both sides of (3.54) must vanish

(3.55)

Therefore, is even under . Taking into account this result we see from the fourth equation in (3.19) that

(3.56)

Then, following similar arguments we conclude that

(3.57)

and so that is even under as well. Furthermore, from the fifth equation in (3.19) one has that , and following similar arguments as above we find that

(3.58)

By repeating the same arguments as above one can show that all are even under . This property can be used to show that the group element which enter in the gauge transformation (3.15) is even, i.e.

(3.59)

Since and are odd under , and since is even one can show that in (3.15) is odd under . In addition, since the are even under and the generators satisfy (3.50), it follows from (3.16) that . One can verify these results by inspecting the explicit expressions for the fields provided in the appendix B.

In order to find the parity of the factor in the integrand of the anomaly in (3.31) let us consider some properties of the loop algebra. The Killing form is given by

(3.60)

It is realized by , where tr is the usual finite matrix trace, and . So, from (3.24) one has that

(3.61)

where the invariance property of the Killing form under has been used, and the fact that the transform as in (3.50). Then, taking into account (3.59) one has that

(3.62)

and so one can see that the ’s with are even under . On the other hand, since we have assumed that is even under and that given in (3.14) is a derivative of a functional of , one has that is odd under , i.e. and so that

(3.63)

Therefore, the anomalous conservation laws (3.31) imply

(3.64)

Consequently, the even order charges are exactly conserved. Notice that it holds for any potential which depends only on the modulus . The only requirement is that the field components satisfy (3.49).

Next let us analyse the solutions which admit the system of equations (3.8)-(3.9) such that the field components satisfy the space-reflection symmetry (3.48)-(3.49). We perform this construction in perturbation theory around solutions of the NLS model, so we expand the equations of motion and the solutions into power series in , as

(3.65)
(3.66)

The deformed potential and its gradient can been expanded as

(3.67)

and

(3.69)

An interesting solution is the one satisfying , i.e. the odd components vanish . We will show that this property is maintained in all orders of perturbation theory. So, substituting the solution into the equations (3.8)-(3.9) one gets the zeroth order system of equations for and