On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation
We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schrödinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show that they exhibit stable structures only when we combine the field and parity transformed complex conjugate field. Further, we derive two dark/antidark soliton solution with the help of DT method. Unlike the bright soliton case, dark/antidark soliton solution exhibits stable structure for the field and the parity transformed conjugate field separately. In the dark/antidark soliton solution case we observe a contrasting behaviour between the envelope of the field and parity transformed complex conjugate envelope of the field. For a particular parametric choice, we get dark (antidark) soliton for the field while the parity transformed complex conjugate field exhibits antidark (dark) soliton. Due to this surprising result, both the field and PT transformed complex conjugate field exhibit sixteen different combinations of collision scenario. We classify the parametric regions of dark and antidark solitons in both the field and parity transformed complex conjugate field by carrying out relevant asymptotic analysis. Further we present -dark/antidark soliton solution formula and demonstrate that this solution may have combinations of collisions.
About five years ago, Ablowitz and Musslimani have proposed the following nonlocal nonlinear Schrödinger (NNLS) equation Ablowitz ()
where is a slowly varying pulse envelope of the field, and represent space and time variables respectively and * denotes complex conjugation. The NNLS equation (1) is invariant under the parity-time (PT) transformation. PT symmetric systems, which allow lossless propagation due to their balance of gain and loss, have attracted considerable attention in recent years pt1 (); pt2 (); pt3 (); pt4 (); pt5 (). Equation (1) attracted many researchers to study its physical and mathematical aspects intensively, see for example Refs. Fokas (); AM (); zhang (); AM2 (); Yan (). The integrability of (1) is proved by (i) the existence of a Lax pair, (ii) existence of infinite number of conservation laws and (iii) existence of N-soliton solutions Ablowitz (); Gerdjikov (). The initial value problem was studied by Ablowitz et al.Ablowitz (). Breathers, dark, antidark soliton, algebraic soliton, higher order rational solutions, periodic and hyperbolic solutions of (1) have been derived for this equation in Refs. Liming (); dad (); Ablowitz2 (); chinese1 (); chinese2 (); Khare (). Discrete version of Eq. (1) has also been proposed in discrete1 (); discrete (); Ma (). Recently, Stalin and two of the present authors have constructed more general bright soliton solutions for (1) by developing a nonstandard bilinearization procedure Stalin (). In this procedure, besides Eq. (1) the authors have also considered the parity transformed complex conjugate equation of (1), namely
since they have assumed and evolve independently. Since Eq. (1) is nonlocal, to evaluate the dependent variable at , the other variable has to be evaluated at simultaneously. The authors have obtained more general one and two soliton solutions of Eqs. (1) and (2) by solving them in a combined manner and studied the collision dynamics between two solitons. The approach proposed by the authors is different from the standard one in the literature and produce a more general class of soliton solutions. In particular, the authors have shown that the system can admit both symmetry broken solutions (the solution, , which does not match with the one resulting from after taking complex conjugation and space inversion in it) and symmetry preserving solutions (the solution, , which matches with the one resulting from after taking complex conjugation and space inversion in it). Such broken symmetry solutions are also called spontaneously broken symmetric solutions in the literature, for example in the case of double-well parity symmetric equation. These two categories of solutions can be identified only by augmenting a separate evolution equation (2) for the parity transformed complex conjugate equation in the solution process. The nonstandard bilinearization procedure has also been applied to two coupled NNLS equations and several new localized solutions and collision dynamics have been unearthed Stalin2 ().
As far as the NNLS Eq. (1) is concerned the symmetry broken and symmetry preserving solutions have been analyzed only for the bright soliton case. A natural question arises in this context is what happens to the dark soliton case. These soliton solutions for Eq. (1) have already been reported in the literature AM2 (); dad (). However, as we pointed out above, to bring out a more general dynamical evolution of dark soliton one should consider not only Eq. (1) but also Eq. (2) in the solution process. In this work, we intend to consider both the equations and construct a more general class of dark soliton solution.
As a by-product of this work, we also extend Darboux transformation (DT) method suitable for this class of nonlocal equations. To make our studies a complete one, to begin with, we derive the bright soliton solution using the DT method by considering the nonlocal term as a separate quantity. We then move on to construct dark solitons for this problem. The dark soliton solutions which we report in this paper is a more general one and new to the literature. In the first iteration of DT method we get two dark soliton solutions. A careful analysis of this solution reveals that for a particular parametric choice while exhibits dark (antidark) soliton, surprisingly, exhibits antidark (dark) soliton. This is because, the fields and produce dark and antidark soliton solutions in different parametric regions since they evolve independently. In the two soliton solution, the component has two solitons, they may have either dark or antidark soliton forms. Therefore four types of collision between two solitons can happen in component alone, that is (i) two dark solitons collision, (ii) antidark and dark solitons collision, (iii) dark and antidark solitons collision and (iv) two antidark solitons collision. The component can also have these four types of collisions between the two solitons irrespective of the structures of . Due to this novel behaviour we get combinations of collision scenario in the two soliton solution alone. This novel property can happen only by considering the symmetry broken solutions, that is by considering as a separate quantity in the solution process. If we consider the symmetry preserving solutions they do contain only four types of collisions, because the component would have similar structure as that of . In our studies we plot nine distinct collision structures for the two soliton solution. By carrying out relevant asymptotic analysis of the two soliton solution we classify the parametric regions of dark and antidark solitons in both the components and . We then derive the four soliton solution from the second iteration of the DT method. Since the solution is cumbersome we only give plots of the solution. For the four soliton solution we get combinations of collision behaviour. We plot some of the combinations of collision for illustration purpose. By iterating the DT for times we get dark soliton solution formula. We can also generalize the collision scenario to soliton solution and get combinations of collisions in it.
The plan of the paper as follows. In Sec. II, we present the DT method to construct Nth iterated solution formula for obtaining N-bright, dark and antidark soliton solutions of Eqs. (1) and (2). We present explicit one and two bright soliton solutions and study the collision dynamics between two solitons in Sec. III. In Sec. IV we construct dark and antidark soliton solutions of (1) and (2) and classify the parametric regions of them. Finally we conclude our results in Sec. V.
2 Darboux Transformation of NNLS equation
where the block matrices , , and are given by
2.1 First Iteration of DT
A Darboux transformation (DT) is a special gauge transformation Matveev (),
in which the matrices , , and assume the same forms as that of , , and except that the potentials and have now acquired new expressions, namely and in and . Substituting the transformation (5) into (6) and comparing the resultant expressions with (3), we find
Plugging the expressions , , , and in Eq. (6) and equating the coefficients of various powers of on both sides, we get the following relations between old and new potentials, namely
Substituting the assumed form of in Eq. (8d) and equating the matrix elements on both sides, we find
where is the solution of (3) at . The exact forms of and are given by,
The above equations exactly match with the equations given in (8e) and (8f). Using the relation (8d), together with the expressions given in (11), the Eqs. (8a) - (8c) are all satisfied. Thus the DT (5) preserves the forms of the Lax pair associated with the NNLS equation (1) and (2). Equation (8c) establishes the relationship between new and original potentials.
The first iterated DT is given by (vide Eq.(5)). If is the solution of at then it should satisfy
Expressing Eq. (15) in matrix form and using Cramer’s rule we can determine the exact forms of and which are given by
From (16) it is evident that to determine and one should know the explicit expressions of , , and which are the solutions of the eigenvalue problem (3). Solving (3) with appropriate seed solution and , one can obtain the explicit expressions of , , and . With the known expressions of , , and the matrix elements and can now be fixed. Plugging the latter into (10), we obtain the formula for two parameter symmetry preserving and symmetry broken solutions for Eqs. (1) and (2) in the form
2.2 Second Iteration of DT
Second iteration of DT can be written as Matveev ()
where , , , If , is solution of at then it should satisfy
where and are given by
The second iteration of DT provides us a new solution in the form
Expressing Eq. (19) in matrix elements and using Cramer’s rule we can find the exact forms of and as
2.3 th Iteration of DT
th iteration of DT can be written as
If , is solution of at , it should satisfy
In the above , , , , and , are given by
Nth iteration of DT leads us to a new solution of the form
Expressing Eq. (24) in matrix elements and using Cramer’s rule we can find exact forms of and as
where , and are given by
3 Bright soliton solutions of NNLS equation
3.1 One bright soliton solution
In this subsection, we construct the one bright soliton solution of Eqs. (1) and (2). To construct them, we feed a vacuum solution, that is as seed solution to the focusing NNLS equation (). By solving the Lax pair equations (3) with and by considering the forms of and as given in Eq. (12) one can get the symmetry broken eigenfunctions of the form , , and , with and . Substituting these basic solutions in the first iterated DT formula (17) one can generate two parameter family of symmetry broken one bright soliton solution of (1) and (2) as
where . The solution (31) is the more general one soliton solution of (1) and (2). We call it as symmetry broken solution (except for specific parameter values given below) since and are independent and cannot deduce one from the other. By choosing , , , , and in (31), we can get the solution which is presented in Stalin (). One can also deduce symmetry preserving one soliton solution of NNLS equation from (31) by confining the parametric conditions in the form , , , , and , where , , and are real constants. As a result, we obtain
The above solution coincides with the one given in Ablowitz (). One can get from (32) by taking complex conjugate of it and inversing the space variable in it. The symmetry broken solution (31) for generic parametric conditions can also be written in terms of trigonometric functions as
The above solution (33) develops singularity at , . This one bright soliton solution is plotted in Fig. 1 for the parameter values , , , , , . The absolute value of plotted in Fig. 1(a) shows that the amplitude of the soliton decays at in the direction. In contrast, the absolute value of parity transformed conjugate field grows at in the direction and is illustrated in the Fig.1(b). We can get a stable propagation of soliton for the absolute value of which is demonstrated in Fig.1(c).