Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System

Searching for integrable systems and constructing their exact solutions are of both theoretical and practical value. In this paper, Ablowitz–Kaup–Newell–Segur (AKNS) spectral problem and its time evolution equation are first generalized by embedding a new spectral parameter. Based on the generalized AKNS spectral problem and its time evolution equation, Lax integrability of a nonisospectral integrodifferential system is then verified. Furthermore, exact solutions of the nonisospectral integrodifferential system are formulated through the inverse scattering transform (IST) method. Finally, in the case of reflectionless potentials, the obtained exact solutions are reduced to n-soliton solutions. When n = 1 and n = 2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.


Introduction
Nonlinear phenomena involved in many fields such as physics, biology, chemistry, and mechanics are often related to nonlinear partial differential equations (PDEs).The investigation of exact solutions of nonlinear PDEs plays an important role because of its direct connection with dynamical processes in these nonlinear phenomena.Since the initialvalue problem of the Korteweg-de Vries (KdV) equation was exactly solved by the IST method [1], finding soliton solutions of nonlinear PDEs has become extremely active and some effective methods were proposed such as Hirota's bilinear method [2], Painlevé expansion [3], homogeneous balance method [4], and function expansion methods [5][6][7][8][9][10].Among these methods, the IST [1] is a systematic method which has achieved considerable development and received a wide range of applications like those in [11][12][13][14][15][16][17][18][19][20][21] since it is put forward by Gardner, Greene, Kruskal, and Miura in 1967.One of the advantages of the IST is that it can solve a whole hierarchy of nonlinear PDEs associated with a certain spectral problem.As early as in 1976, the framework of IST with varying spectral parameter was introduced for the first time by Chen and Liu to the nonlinear Schrödinger (NLS) equation with a linear external potential [22] and by Hirota and Satsuma to the KdV equation in nonuniform media [23].Serkin et al. [24][25][26][27][28] pointed out that the soliton dynamics of nonautonomous ones which interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations can be described in the framework of the IST theory with varying in time spectral parameter.
In soliton theory, nonlinear PDEs associated with some linear spectral problems can be generally classified as the isospectral equations which often describe solitary waves in lossless and uniform media and the nonisospectral equations describing the solitary waves in a certain type of nonuniform media.Specifically, when the spectral parameter of the associated linear spectral problem is independent of time, one could construct isospectral equations.While starting from the spectral problem with a time-dependent spectral parameter, nonisospectral equations are usually derived.In 1974, Ablowitz, Kaup, Newell, and Segur [21] successfully constructed a hierarchy of isospectral nonlinear PDEs; here it is written as Complexity from the compatibility condition, that is, the zero curvature equation of the following spectral problem and its evolution equation where  = (, ),  = (, ), and their derivatives of any order with respect to  and  are smooth functions which vanish as  tends to infinity, the spectral parameter  is independent with  and , and , , and  are undetermined functions of , , , , and .
When  = 2, the isospectral AKNS hierarchy (1) gives which includes the famous KdV equation   =   + 6  as a special case.Subsequently, in the case of spectral parameter  being dependent on time , Calogero and Degasperis [29][30][31] and Li [32] proposed effective methods to derive different hierarchies of nonisospectral equations.For example, the nonisospectral AKNS hierarchy [20] ( can be derived from (3)-( 5) equipped with   = (2)  /2.It is easy to see that when  = 0, 1, 2, the nonisospectral AKNS hierarchy (1) gives the following nonisospectral systems: The aim of this paper is to generalize AKNS spectral problem (4) and its evolution equation (5) for testing the integrability of the following new and more general nonisospectral integrodifferential system: and extending the IST to system (9).With the help of (2), we can rewrite system (9) in the form from which we can see that the nonisospectral integrodifferential system (9) with time-dependent coefficient terms is different from that in [33] ( In order to construct system (9), in this paper we shall employ a new and more general spectral parameter  which satisfies It is easy to see that the nonisospectral parameter   = (2)  /2 in [33] is a special case of (12).Here  in ( 12) is equivalent to  in [33].On the other hand, we shall generalize the matrix  in [33] | (,)=(0,0) = ( to the following form: ) . ( In the very recent work [34], we let the parameter  satisfy and employed Complexity 3 then a general nonisospectral integrodifferential system of the form is constructed.Equation ( 17) can be rewritten as which has the expansion in part Obviously, there is substantial difference between system ( 9) and ( 17) in [34].It is because, except for the term the other three terms of ( 10) cannot be contained in (18).Due to appearance of the third term of (21), system ( 9) is a variable-coefficient system with not only space-dependent coefficients but also timedependent coefficients.However, (17) has not such a term with time-dependent coefficients.In fact, the different selections for ( 12) and ( 14) lead to the difference between system (9) and ( 17).The rest of the paper is organized as follows.In Section 2, we prove the Lax integrability of system (9) by generalizing AKNS spectral problem (4) and its evolution equation (5).In Section 3, system ( 9) is solved via the IST.As a result, the uniform formulae of exact solutions are obtained.In the special case of reflectionless potentials, the obtained exact solutions are reduced to -soliton solutions.In Section 4, we conclude this paper.

Lax Integrability
Theorem 1. Suppose that the function  in (5) has the form then the nonisospectral integrodifferential system ( 9) can be derived from (3) and thus system ( 9) is Lax integrable.
Proof.Firstly, by virtue of (3) equipped with the new spectral parameter  satisfying (12) we have Supposing that from ( 24) and ( 25), we have by the use of (2).We next suppose that and substitute ( 28) into (27).Then comparing the coefficients of 2 in ( 27) yields from which we derive (10).Finally, the substitution of ( 2) into (10), we arrive at the nonisospectral integrodifferential system (9).Thus, the proof is completed.

Soliton Solutions
In this section, we first determine the time dependence of scattering data for the AKNS spectral problem (4) with the generalized time evolution equation ( 5) caused by (22).Based on the determined scattering data, we then construct exact solutions of nonisospectral integrodifferential system (9).We finally reduce the obtained exact solutions to soliton solutions and analyze the soliton dynamics.

Exact Solutions and Soliton Solutions. According to
Theorem 1 and the results in [20], we have the following Theorem 3.

Conclusions and Discussions
In summary, we have verified Lax integrability of the new and more general nonisospectral integrodifferential system (9).This is due to the generalizations on AKNS spectral problem (4) and its time evolution equation ( 5) by embedding a new spectral parameter.To exactly solve the nonisospectral integrodifferential system (9), the IST is employed.As a result, exact solutions (54) are obtained.In the case of reflectionless potentials, the obtained exact solutions (54) are reduced to -soliton solutions (66).When  = 1 and  = 2, the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.To the best of our knowledge, the nonisospectral integrodifferential system (9), the exact solutions (54), and the -soliton solutions (66) have not been reported in literatures.How to construct other nonisospectral integrodifferential systems and their soliton solutions in the framework of IST method is worthy of study.This is our task in the future.