1 Introduction


The concept of quasi-integrability for modified non-linear Schrödinger models

L. A. Ferreira , G. Luchini  and Wojtek J. Zakrzewski 


Instituto de Física de São Carlos; IFSC/USP;

Universidade de São Paulo

Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil

email: laf@ifsc.usp.br

email: gabriel.luchini@gmail.com


 Department of Mathematical Sciences,

University of Durham, Durham DH1 3LE, U.K.

email: W.J.Zakrzewski@durham.ac.uk

We consider modifications of the nonlinear Schrödinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an infinite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form , with being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model.

1 Introduction

The concept of a soliton, introduced half a century ago by Zabusky and Kruskal [1], was based by the seminal work of Fermi, Pasta and Ulam [2]. Solitons are special solutions of non-linear evolution equations that propagate without changing their shapes and without dissipating their energies. The solitons interact among themselves but the special property they possess is, that after a long time after their scattering, the only effect of it is a shift in their position (the so-called time delay or time advance) w.r.t. the values they would have had had the scattering have not taken place. There is no emission of radiation during their interaction and, well after the scattering process, their shape and other physical properties like energy, are preserved. For the case of dimensional theories this behaviour of solitons has been understood in the context of integrable field theories. Indeed, it has been observed that (practically) all models possessing soliton solutions admit a representation of their equations of motion in terms of the so-called Lax-Zakharov-Shabat (LZS) equation or zero curvature condition [3], where the Lax potential or connection lives in an infinite dimensional Kac-Moody algebra. The LZS equation has led to the development of many exact and non-perturbative methods to study such dimensional theories, including the construction of exact solutions and of an infinite number of conservation laws [4, 5]. In the context of such a soliton theory, the above mentioned striking properties of solitons can be credited to the constraints on their dynamics imposed by the infinite number of exactly conserved charges coming from the LZS equation.

Of course, the class of dimensional integrable field theories, admitting the LZS equation, is not very large. Indeed, most of the two dimensional physical non-linear phenomena are described by theories that do not belong to that class. In many cases however, integrable models can be used as approximations to more realistic theories, and many interesting developments have been done in that direction. In fact, the literature on applications of perturbations around integrable theories is quite vast and diverse, and we shall not attempt to quote the many interesting and important results obtained. We shall concentrate, however, on the fact that many non-integrable theories possess solutions that behave much like solitons despite the lack of a large number of conservation laws.

In this context, two of us [6] have recently looked at a class of models which generalizes the integrable sine-Gordon model and used it to introduce the concept of quasi-integrability. According to [6] a dimensional field theory is quasi-integrable if although it does not admit a representation of its equations of motion in terms of the LZS equation, it does possess soliton like solutions that which, when they undergo a scattering process, preserve their basic physical properties like mass, topological charges, etc. It is also required that the theory should possess an infinity number of quasi-conservation laws with the property that the corresponding charges are conserved when evaluated on the one-soliton solutions, and are asymptotically conserved in the scattering of these solitons. In other words, during the scattering of the solitons the charges do vary in time, but they return to their original values (in the far past), when the solitons are well separated after the collision (in the far future). Essentially, the theory possesses anomalous conservation laws of the form

(1.1)

with the label being an integer. For instance, in the scattering of two solitons one has

(1.2)

For breather like solutions it was shown in [6] that in many special cases the vanishing in (1.2) occurs when the time integral is performed over a period determined by the breather, i.e. the charges are periodic in time, .

Reference [6] has also considered particular modifications [7] of the sine-Gordon model that admit topological soliton like solutions (kinks), and a representation of their equations of motion in terms of an anomalous (non-zero) LZS equation. Adapting techniques of integrable field theories to this anomalous equation, an infinite set of quasi-conserved charges was constructed. Employing both analytical and numerical techniques the scattering of solitons was studied and it was verified that for some special solutions the asymptotic conservation of charges does take place. The key observation of [6] was based on the fact that the two-soliton solutions satisfying (1.2) had the property that their fields were eigenstates of a very special space-time parity transformation

(1.3)

where the point in space-time, depends upon the parameters of the solution. Since the charges are obtained from some densities, i.e. , so are the anomalies . Therefore, the vanishing of , follows from the properties of under (1.3). Note that the solutions for which the fields are eigenstates of the parity (1.3) cannot be selected by choosing appropriate initial boundary conditions. The reason for this is simple: the boundary conditions are set at a given initial time and the transformation (1.3) relates the past and the future of the solutions. In other words, boundary conditions are kinematical statements, and the fact that a field is an eigenstate under (1.3) is a dynamical statement. For these reasons, the physical mechanism that guarantees that such special solutions have the required parity properties is not clear yet.

The models studied in [6] were perturbed sine-Gordon models; i.e. Lorentz covariant models with topological solitons. Thus it would be interesting to see whether similar phenomena hold in other models, with other symmetries. Hence in this paper we look at the nonlinear Schrödinger (NLS) model and its perturbations. Even though this model is also integrable, it differs from the sine-Gordon in the sense that it possesses solitons which are not topological, their dynamics is governed by a first order (in time derivatives) equation and it does not possess any breather like structures. However, this model is probably even more important than the sine-Gordon model in its applications, which are abundant in all areas of nonlinear science. Hence the understanding of quasi-integrability in this context would have very important implications. The modifications of the NLS model we consider in this paper have equations of motion of the form

(1.4)

where is a complex scalar field and is a potential dependent only on the modulus of . The unperturbed NLS equation corresponds to . We start our analysis of such models by writing the equations of motion (1.4) as an anomalous LZS equation of the form

(1.5)

where the connection is a functional of and its derivatives, and takes values in the loop algebra (Kac-Moody algebra with vanishing central element), and is the anomaly that vanishes when is the NLS potential.

We construct the infinite set of quasi-conserved charges by employing the standard techniques of integrable field theories known as Drinfeld-Sokolov reduction [8], or abelianization procedure [9, 10, 11]. Using these techniques we gauge transform the component of the connection into an infinite dimensional abelian subalgebra of the loop algebra, generated by . Even though the anomaly prevents the gauge transformation to rotate the component into the same abelian subalgebra, the component of the transformed curvature (1.5) in that subalgebra, leads to an infinite set of quasi-conservation laws, , or equivalently leads to (1.1) with and .

Next we employ a more refined algebraic technique, involving two transformations, to understand the conditions for the vanishing of the integrated anomalies. The first is an order two automorphism of the loop algebra and the second is the parity transformation (1.3). For the solutions for which the field transforms under (1.3) as

(1.6)

we show that , where and are any given fixed values of the space-time coordinates and , respectively, introduced in (1.3). This shows that

(1.7)

which is a type of a mirror symmetry for the charges. Therefore, for a two-soliton solution satisfying (1.6), the asymptotic conservation of the charges (1.2) follows from such stronger result.

Such results certainly unravel important structures responsible for the phenomena that we have called quasi-integrability. They involve an anomalous LZS equation, internal and external symmetries, and algebraic techniques borrowed from integrable field theories. However, they rely on the assumption (1.6) which is, as we have argued above, a dynamical statement since it relates the past and the future of the solutions. In order to shed more light on this issue we study the relation between (1.6) and the dynamics defined by (1.4).

It is easier to work with the modulus and phase of , and so we parametrize the fields as , with and being real scalars fields. We split them into their eigen-components under the parity (1.3), as , and . The assumption (1.6) implies that the solution should contain only the components , and nothing of the pair . By splitting the equations of motion (1.4) into their even and odd components under (1.3), we show that there cannot exist non-trivial solutions carrying only the pair . In addition, if the potential in (1.4) is a deformation of the NLS potential, in the sense that we can expand it as

(1.8)

with being a deformation parameter, then we can make even stronger statements. In such a case we expand the equations of motion and the solutions into power series in , as

(1.9)

If we select a zero order solution, i.e. a solution of the NLS equation, satisfying (1.6), i.e. carrying only the pair , then the equations for the first order fields, which are obviously linear in them, are such that the pair satisfies inhomogeneous equations, while the pair , satisfies homogeneous ones. Therefore, , is a solution of the equations of motion, but , is not. By selecting the first order solution such that the pair is absent, we see that the same happens in second order, i.e. that the pair also satisfies inhomogeneous equations, and the pair the homogeneous ones. By repeating this procedure, order by order, one can build a perturbative solution which satisfies (1.6), and so has charges satisfying (1.7). Note that the converse could not be done, i.e. we cannot construct a solution involving only the pair . So, the dynamics dictated by (1.4) favours solutions of the type (1.6).

Finally we show that the one-bright-soliton and the one-dark-soliton solutions of the NLS equation satisfy the condition (1.6), and that not all two-bright-soliton solutions satisfy it. However, one can choose the parameters of the general solution so that the corresponding two-bright-soliton solutions do satisfy (1.6). This involves a choice of the relative phase between the two one-bright-solitons forming the two-soliton solution. We do not analyze in this paper the two-dark soliton solutions of the NLS equation. Therefore, our perturbative expansion explained above can be used to build a sub-sector of two-bright-soliton solutions of (1.4) that obeys (1.6) and so has charges satisfying (1.7). This would constitute our quasi-integrable sub-model of (1.4).

Despite the fact that the equations of motion satisfied by the -order fields are linear, the coefficients are highly non-linear in the lower order fields and so, unfortunately, these equations are not easy to solve. We then use numerical methods to study the properties of our solutions. In addition, such numerical analysis can clarify possible convergence issues of our perturbative expansions. We chose to perform our numerical simulations for a potential of the form

(1.10)

We performed several simulations using the 4th order Runge Kutta method of simulating the time evolution. These simulations involved the NLS case with the two bright solitons sent towards each other with different values of velocity (including ) and for various values of the relative phase. We then repeated that for the modified models. We looked at various values of and have found that the numerical results were reliable for only a small range of around 0. For very small values we saw no difference from the results for the NLS model but for or the results of the simulations became less reliable. Hence, we are quite confident of our results for and in the numerical section we present the results for .

We also present the results for the first anomaly as seen in our simulations. We find that our results confirm our expectations.

The paper is organized as follows: in section 2 we describe in detail the models to be studied, construct the anomalous LZS equation, the quasi-conserved charges and establish the conditions, that have to be satisfied by the solutions, for the integrated anomalies to vanish. We also give an argument, valid in a space-time of any dimension, for a field theory to possess charges satisfying symmetries of the type given in (1.7). In section 3 we discuss how the dynamics of the model favours solutions satisfying (1.6). We also discuss further the relation between the dynamics and parity for the case when the potential is a deformation of the NLS potential. In section 4 we discuss the parity properties of the one and two-soliton solutions of the NLS theory and show how to select those that satisfy (1.6). We then present, in section 5, the results of our numerical simulations which support the analytical results discussed in the previous sections. The conclusions are given in section 6, and in the appendix A we present the details of the calculation used in section 2, and in appendix B we use the Hirota method to construct one and two-bright-soliton solutions of the NLS theory.

2 The model

We consider a non-relativistic complex scalar field in () dimensions with the Lagrangian given by

(2.1)

where is the complex conjugate of . The equations of motion are

(2.2)

together with its complex conjugate. The corresponding Hamiltonian is given by

(2.3)

We shall consider solutions of (2.2) satisfying the following boundary conditions

(2.4)

It is easy to check that the energy , momentum and normalization of the solutions of the equations of motion(2.2) satisfying (2.4), as defined below, are conserved in time.

(2.5)
(2.6)
(2.7)

In fact, these conserved quantities correspond to the Noether charges of the model. The energy is connected with the invariance of (2.1) under time translations, the momentum under the space translations, and is related to the following internal symmetry of the Lagrangian (2.1)

(2.8)

The integrable Non-Linear Schrödinger theory (NLS) corresponds to the potential

(2.9)

which leads to the NLS equation

(2.10)

The sign of the parameter plays an important role in the properties of the solutions. Indeed, for we have the so-called bright soliton solutions given by

(2.11)

with , and being real parameters of the solution. For we have the dark soliton solution given by

(2.12)

Note, that the solutions are defined up to an overall constant phase due to the symmetry (2.8).

The equation (2.2) admits an anomalous zero curvature representation (Lax-Zakharov-Shabat equation) with the connection given by

(2.13)

where the generators , , and integer, satisfy the so-called loop algebra commutation relations

(2.14)

which can be realized in terms of the finite algebra generators as , with an arbitrary complex parameter. The curvature of the connection (2.13) is given by

(2.15)

with

(2.16)

In consequence, when the equations of motion (2.2) are imposed, the terms, on the r.h.s. of (2.15), proportional to and vanish. Note also that, by taking

(2.17)

the anomaly , given in (2.16), vanishes for the NLS potential (2.9). In fact, this vanishing of the curvature (2.15) for the NLS equation makes this classical field theory integrable.

In this paper we will consider the generalisation of this theory (i.e. the deformations of the NLS potential) which make the resultant theory nonintegrable, i.e. those for which the anomaly (2.16) does not vanish. However, as we will show, the corresponding theories exhibit properties very similar to the integrable ones, like the solitons preserving their shapes after the scattering etc. In addition, we will show, using the algebraic technioques borrowed from integrable field theories, that the anomalous Lax-Zakharov-Shabat equation (2.15) leads to an infinite number of quasi-conservation laws. And, we will find that, under some special circumstances, the corresponding charges are conserved asymptotically in the scattering of soliton type solutions of these (non-integrable) theories.

In order, to employ the algebraic techniques mentioned above it is more convenient to work with a new basis of the loop algebra and a new parameterization of the fields. In our work we will use the modulus of and its phase , defined as

(2.18)

In addition, we will parameterize the complex parameters and , appearing in the connection (2.13), as

(2.19)

The new basis of the loop algebraic is then defined as

(2.20)

which satisfy

(2.21)

As usual we perform the gauge transformation

(2.22)

and find that the connection (2.13) has now become

(2.23)

For the fields which satisfy the equations of motion (2.2) the curvature becomes

(2.24)

To go further we carry out the usual abelianization technique of the integrable field theories [8, 9, 10, 11]; i.e. we perform a further gauge transformation

(2.25)

with

(2.26)

The parameters are chosen, as we will explain below, so that the component of the transformed connection lies in the infinite abelian subalgebra spanned by the generators .

An important role in our construction is played by the grading operator defined as

(2.27)

The component of the connection (2.23) has generators of grade and . Since the group element (2.26) is an exponentiation of negative grade generators, the component of the transformed connection has generators of grades ranging from to . Splitting the transformed potential (2.25) into its eigen-subspaces under the grading operator (2.27), i.e. , we find that

(2.28)

where we have denoted (see (2.25)).

An important ingredient of this construction is the observation that the generator is a semi simple element (in fact any is) in the sense that it splits the loop algebra into the kernel and image of its adjoint action, i.e.

(2.29)

The Ker and Im subspaces do not have common elements, i.e. any element of commuting with cannot be written as a commutator of with some other element of . One notes from (2.21) that constitute a basis of Ker, and , , a basis of Im. In addition, one notes from (2.28) that the first time that appears in the expansion of , is in the component of grade , and it appears in the form . Therefore, one can choose the parameters in so that they cancel the image component of . This can be done recursively starting at the component of grade and working downwards. It is then clear that the gauge transformation (2.26) can rotate the component of the connection into the abelian subalgebra generated by the ’s, i.e.

(2.30)

Note from (2.23) that depends on the real fields and . Thus, the components are polynomials in these fields and their -derivatives, and they do not depend on the potential . In consequence, the component of the connection is the same for any choice of the potential. In the appendix A we give explicit expressions for the first few components of .

On the other hand the component of the connection (2.23) depends on the choice of the potential . In fact, for the case of the NLS potential (2.9) we note that the gauge transformation (2.25), with the group element (2.26) fixed as above, does rotate into an abelian subalgebra generated by the ’s, when the equations of motion (2.10) are satisfied. For other choices of potentials this does not take place even when the equations of motion (2.2) are imposed. Thus, we find that

(2.31)

Next we note that does not have the grade component due to the fact that the coefficient of in , and the coefficient of in , are the same up to a sign (see (2.23)). Under the gauge transformation (2.25) the curvature transforms to , and so from (2.24) we see that

(2.32)

Since lies in the kernel of it follows that has components only in the image of . Thus, denoting

(2.33)

we find that

(2.34)

The explicit expressions for the first few , , are given in appendix A. Note that if the time component of the connection satisfies the boundary condition , which is the case in the example we consider, then we have anomalous conservation laws

(2.35)

Of course, in the case of the NLS theory we get an infinite number of conserved quantities since the anomaly , given in (2.16) or (2.24), vanishes for the NLS potential (2.9).

We now use a more refined algebraic technique to explore the structure of the anomalies . The key ingredients are the two transformations, one in the internal space of the loop algebra and the other in space-time. The first transformation is an order automorphism of the loop algebra (2.21) given by

(2.36)

The second transformation is a space-time reflection around a given point , i.e.

(2.37)

Consider now solutions of the equations of motion (2.2) of the theory (2.1) such that, in addition, they satisfy the following property under the parity (2.37) (see (2.18))

(2.38)

Then, the -component of the connection (2.23) transforms as

(2.39)

and so it is odd under the joint action of the two transformations:

(2.40)

In fact, this property is valid for every individual component of . Thus we see that we have , and so

(2.41)

Since is odd under , it follows from the second equation of (2.28) that

(2.42)

The r.h.s. of (2.42) is clearly in the image of the adjoint action, and we have chosen the to rotate into the kernel of that same adjoint action. Therefore, the only possibility for (2.42) to hold is that both sides vanish, i.e. that

(2.43)

and so that is even under . Using this fact we see from the third equation in (2.28) that

(2.44)

Furthermore, using same arguments we conclude also that

(2.45)

and so that is even under as well. Again, from the fourth equation in (2.28) we see that , and so by the same arguments as before we conclude that

(2.46)

Repeating this reasoning we reach the conclusion that all are even under . So, the group element , given in (2.26), is even under

(2.47)

To go further we note that since and are odd under , and since is even (2.25) demonstrates that has to be odd under . One can verify all these claims by inspecting the explicit expressions for the parameters given in appendix A. Since the are even under , and since the generators satisfy (2.36), it follows from (2.26) that and .

Next we use the Killing form of the loop algebra given by

(2.48)

which can be realized by , with being the ordinary finite matrix trace, and , . In this case we see from (2.33) that

(2.49)

where in the last equality we have used the fact that the Killing form is invariant under , and that all the ’s are odd under it. Thus, using (2.47) we have that

(2.50)

and so we see that all the ’s are even under . Note that , given in (2.24), is an -derivative of a functional of . Since we have assumed that is even under , we see from (2.38) that is odd, i.e. that and so that

(2.51)

where and are given fixed values of the shifted time and space coordinate respectively, introduced in (2.37). Therefore, by taking , we conclude that the non-conserved charges (2.35) satisfy the following mirror time-symmetry around the point: .

(2.52)

In consequence, even though the charges vary in time, they are symmetric w.r.t. to . Note that we have derived this property for any potential which depends only on the modulus of . The only assumption we have made is that we are considering fields which satisfy (2.38).

In the next sections we will show that such solutions are very plausible and that, in fact, the one and two-soliton solutions of the theories (2.1) can always be chosen to satisfy (2.38). This fact has far reaching consequences for the properties of the theories (2.1). For instance, by taking one concludes that the scattering of two-soliton solutions presents an infinite number of charges which are asymptotically conserved. Since the -matrix relies only on asymptotic states, it is quite plausible that the theories (2.1) share a lot of interesting properties with integrable theories (but which have been believed to be only true for integrable field theories).

2.1 Another way of understanding it

The properties leading to charges satisfying (2.52) can be realized, in fact, in a much wider context. Indeed, consider a field theory in a space-time of dimensions with fields labelled by , . These fields can be scalars, vectors, spinors, etc, and the indices just label their components. Consider a fixed point in space-time, and a reflection around it, i.e.

(2.53)

Suppose that a such field theory possesses a classical solution such that the fields evaluated on it are eigenvectors of up to constants, i.e. that

(2.54)

Consider now a functional of the fields and of their derivatives , that is even under when evaluated on a particular solution, i.e.

(2.55)

Next, look at a rectangular spatial volume bounded by hyperplanes crossing the axes of the space coordinates at the points , , corresponding to fixed values of the shifted space coordinates introduced in (2.53), i.e. such that the point lies in the very center of . The integral of this functional over

(2.56)

satisfies

(2.57)

When evaluated on the solution each term in the integrand in (2.57) is odd under . The reasons for this are simple: any derivative of the form , when evaluated on , has an eigenvalue of equal to . Since evaluated on is even under , it follows that any derivative of the form has an eigenvalue of equal to , when evaluated on . Therefore, when evaluated on each term of the integrand on the r.h.s. of (2.57) is odd under . Consequently, one finds that

where the superscript denotes that is evaluated on the solution , and is a given fixed value of the shifted time introduced in (2.53).

Summarizing our results: if one has a solution of the theory such that all the fields evaluated on this solution are eigenstates of , i.e. they satisfy (2.54), then any even functional of these fields and their derivatives leads to charges that satisfy a mirror time-symmetry like (2.1).

In the case studied in this paper we have shown that the -component of the connection, , is odd under the transformation , i.e. . Since lies in the abelian subalgebra generated by the ’s (see (2.30)), which are odd under (see (2.36)), it follows that the charge densities are even under . Therefore, the charges introduced in (2.35) are in the class of charges (2.56) discussed in this subsection. So, the assumption of the existence of a solution satisfying (2.38) has much deeper consequences. It implies not only that the charges (2.35) satisfy the mirror time-symmetry (2.52), but also that any charge built out of a density that is even under when evaluated on this solution, also satisfies (2.52). The fact that a solution satisfies (2.38) implies that its past and future w.r.t. to the point in time