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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

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 [

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 [

When

Subsequently, in the case of spectral parameter

The aim of this paper is to generalize AKNS spectral problem (

In the very recent work [

The rest of the paper is organized as follows. In Section

Suppose that the function

Firstly, by virtue of (

Supposing that

We next suppose that

In this section, we first determine the time dependence of scattering data for the AKNS spectral problem (

The scattering data

It is easy to see that if

Firstly, we consider the discrete spectral

Presuming

For convenience, we rewrite (

Using (

On the other hand, we rewrite (

Noting that

In a similar way, we obtain

Secondly, we consider

Using the asymptotical properties

Substituting the Jost relationship

According to Theorem

Given the scattering data for the generalized spectral problem (

In order to give explicit form of solutions (

Using (

Introducing the vectors

Supposing

Substituting (

In this part, we further investigate the soliton dynamics of system (

Local spatial structure of one-soliton solution (

Local spatial structure of one-soliton solution (

Dynamical evolutions of two-soliton solution determined by (

Dynamical evolutions of two-soliton solution determined by (

In summary, we have verified Lax integrability of the new and more general nonisospectral integrodifferential system (

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).