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 -soliton solutions. When and , the characteristics of soliton dynamics of one-soliton solutions and two-soliton solutions are analyzed with the help of figures.
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 initial-value problem of the Korteweg–de Vries (KdV) equation was exactly solved by the IST method , finding soliton solutions of nonlinear PDEs has become extremely active and some effective methods were proposed such as Hirota’s bilinear method , Painlevé expansion , homogeneous balance method , and function expansion methods [5–10]. Among these methods, the IST  is a systematic method which has achieved considerable development and received a wide range of applications like those in [11–21] 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  and by Hirota and Satsuma to the KdV equation in nonuniform media . Serkin et al. [24–28] 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  successfully constructed a hierarchy of isospectral nonlinear PDEs; here it is written asfrom the compatibility condition, that is, the zero curvature equationof the following spectral problemand its evolution equationwhere , , 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 , the isospectral AKNS hierarchy (1) giveswhich includes the famous KdV equation as a special case.
Subsequently, in the case of spectral parameter being dependent on time , Calogero and Degasperis [29–31] and Li  proposed effective methods to derive different hierarchies of nonisospectral equations. For example, the nonisospectral AKNS hierarchy can be derived from (3)–(5) equipped with . It is easy to see that when , 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 formfrom which we can see that the nonisospectral integrodifferential system (9) with time-dependent coefficient terms is different from that in In order to construct system (9), in this paper we shall employ a new and more general spectral parameter which satisfiesIt is easy to see that the nonisospectral parameter in  is a special case of (12). Here in (12) is equivalent to in . On the other hand, we shall generalize the matrix in to the following form:
In the very recent work , we let the parameter satisfyand employedthen a general nonisospectral integrodifferential system of the formis constructed. Equation (17) can be rewritten aswhich has the expansion in partObviously, there is substantial difference between system (9) and (17) in . It is because, except for the termthe 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 time-dependent 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.
2. Lax Integrability
Proof. Firstly, by virtue of (3) equipped with the new spectral parameter satisfying (12) we haveSupposing thatfrom (24) and (25), we haveby the use of (2).
We next suppose thatand substitute (28) into (27). Then comparing the coefficients of in (27) yieldsfrom which we derive (10). Finally, the substitution of (2) into (10), we arrive at the nonisospectral integrodifferential system (9). Thus, the proof is completed.
3. 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.
3.1. The Time Dependence of Scattering Data
Theorem 2. The scattering data , , , , , and for the generalized spectral problem (4) possess the following time dependence:where , , , and are the scattering data of the generalized spectral problem (4) in the case of .
Proof. It is easy to see that if is a solution of the generalized spectral problem (4) then is also a solution of generalized spectral problem (4). Therefore, can be represented by and which also satisfies the generalized spectral problem (4) but is independent of ; that is, there exist two functions and such thatFirstly, we consider the discrete spectral . Since decays exponentially while must increase exponentially as , we then have . Thus, (31) is simplified asLeft-multiplying (32) by the inner product yieldsPresuming to be the normalization eigenfunction and noting that , we haveFor convenience, we rewrite (34) aswhere the following inner product had been usedfor arbitrary two vectors and .
Using (4), we haveand hence obtainIntegrating (38) with respect to from to yieldsOn the other hand, we rewrite (28) asand then obtainThen (32) becomesNoting thatas , then from (42)-(43) we haveIn a similar way, we obtainSecondly, we consider as a real continuous spectral and take a solution of the generalized spectral problem (4), then the solution of the generalized spectral problem (4)can be represented linearly by and which also satisfies the generalized spectral problem (4) but is independent of , that is, there exist two functions and such thatUsing the asymptotical propertiesas , from (47) we obtainSubstituting the Jost relationship into (47) yieldsLetting and usingfrom (50) we deriveSimilarly, we haveFinally, solving (44), (45), (52), and (53) yields (30). We therefore finish the proof.
3.2. Exact Solutions and Soliton Solutions
Theorem 3. Given the scattering data for the generalized spectral problem (4), the nonisospectral integrodifferential system (9) has exact solutions as follows:where satisfies the Gel’fand-Levitan-Marchenko (GLM) integral equation:with
In order to give explicit form of solutions (54), we consider . In this reflectionless potentials case, the GLM integral equation (55) can be solved exactly. For convenience, we use to rewrite (55) as
Introducing the vectorswe can write (60) in the matrix formswhereand is a unit matrix.
Substituting (65) into (54), we obtain the following -soliton solutions of the nonisospectral integrodifferential system (9):Particularly, when , (66) give the one-soliton solutions:where and are determined by the Riccati equations
3.3. Soliton Dynamics
In this part, we further investigate the soliton dynamics of system (9) by means of one-soliton solutions and two-soliton solutions. To determine and of (69), we employ Zhang et al.’s direct algorithm  of exp-function method and gain two special solutions of (69):In Figures 1 and 2, two local spatial structures of one-soliton solutions (67) and (68) are shown by selecting the parameters as , , , and . We can see from Figures 1 and 2 that the local spatial structures of one-soliton solutions (67) and (68) possess the bell-shaped characteristics. The dynamical evolutions of two-soliton solutions determined by (66) are shown in Figures 3 and 4, where the parameters are selected as , , , , , , , and , respectively. Figures 3 and 4 show that the inelastic scatterings can happen between two-soliton solutions determined by (66).
4. 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 and , 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
This work was supported by the Natural Science Foundation of China (11547005), the Natural Science Foundation of Liaoning Province of China (201705040007), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002), and the Liaoning BaiQianWan Talents Program of Liaoning Province of China (2013921055).
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