Abstract

The Kundu equation, which can be used to describe many phenomena in physics and mechanics, has crucial theoretical meaning and research value. In previous studies, the single Kundu equation has been investigated by the Riemann-Hilbert method, but few researchers have focused on the coupled Kundu equations. To our knowledge, many phenomena in nature can be only described by coupled equations, such as species competition and signal interactions. In this paper, we discuss N-soliton solutions of the coupled Kundu equations according to the Riemann-Hilbert method. Starting from the spectral problem, the coupled Kundu equations are generated, and the Riemann-Hilbert problem is presented. When the jump matrix of the Riemann-Hilbert problem is the identity matrix, the N-soliton solutions of the coupled Kundu equations can be expressed explicitly.

1. Introduction

It is well known that Ginzburg-Landau equation [1], a very important physical model, has been widely used in many scientific research fields, such as superconductivity, superfluidity, fluid dynamics, the diffusion reaction equation, nonlinear optics, and quantum field theory. Kundu equation [2], as a special case of the complex Ginzburg-Landau equation, has also attracted increasingly more attention. Many researchers have discussed the integrable properties and exact solutions of the Kundu equation through various methods [3, 4], but we note that the coupled Kundu equations have been rarely studied [5, 6]. To our knowledge, many significant natural science and engineering problems can be reduced to the study of coupled equations, such as species competition and signal interactions. Thus, it is necessary to discuss the solution of the coupled Kundu equations by advanced methods.

In the research of nonlinear partial differential equations [711] and the study of the properties of nonlinear equations [1216], constructing a precise solution of a nonlinear equation is an important subject in soliton research [1720], and many methods can be used to solve nonlinear equation [2123]. In recent years, with the development of soliton theory [2429], nonlinear partial differential equations relevant to engineering [30, 31] are playing an important role, with more researchers concentrating on Riemann-Hilbert method [32, 33] to study these important equations; for example, the Sasa-Satsuma equation was discussed by Geng [34], and the Ostrovsky-Vakhnenko equation was researched by Fan [35]. The Riemann-Hilbert method is a general term for a method of studying an integrable equation by using the Riemann-Hilbert problem as a tool [36]. The Riemann-Hilbert method is one of the most useful approaches for investigating N-soliton solutions of integrable equations [37], initial boundary value problems [38], and long-time asymptotic behavior [39, 40].

In this paper, based on the Riemann-Hilbert method, we discuss the coupled Kundu equationswhereandare arbitrary complex functions ofand,is a constant, and. Supposing, (1) can be read aswheredenotes the complex conjugate of. Equation (2) is known as the Kundu equation. Whentakes different values, we can obtain the following crucial equations.(i)When, we can deduce the Kaup-Newell equation(ii)When, we can deduce the Chen-Lee-Liu equation(iii)When, we can deduce the Gerdjikov -Ivanov equation

We can easily find that (2)-(5) are special cases for (1). These equations are commonly used in physics and mechanics, such as plasma physics, nonlinear fluids, and quantum physics. Moreover, based on the Riemann-Hilbert method, the coupled Kundu equations are not discussed by other published papers.

The structure of this paper is as follows. Based on the Riemann-Hilbert method, we investigate N-soliton solutions of the coupled Kundu equations. In Section 2, the coupled Kundu equations and their Hamiltonian structures are generated. With the help of the Lax pair, the spectral problem is analyzed, and the Riemann-Hilbert problem is constructed in Section 3. The solution of the Riemann-Hilbert problem is presented in Section 4. In Section 5, the long-time behavior of the solution and the N-soliton solutions is studied. In addition, we also discuss some special cases of the coupled Kundu equations in Section 6.

2. Generation of the Coupled Kundu Equations

In the section, we consider the spectral problemBy solving the adjoint equation where we can obtain the following recursive relations:Based on (9), we conclude

In the following, we consider the auxiliary spectral problemwhere andis a modification. According to the compatibility conditions of (6) and (11) we obtainBased on (14), we deriveand which can be rewritten aswhere through the recursive relations of (9).

Define and wheredenotes the complex conjugate ofand. Then, (16) can be rewritten aswhere.

Taking the Killing-Cartan formas, we can easily obtain In addition, we have relying on the trace identity. The equation can be read as Letting, we obtain. Thus, where Combining (20) with (24), we acquire the Hamilton structure Supposing, we obtain the coupled Kundu equations

3. Riemann-Hilbert Problem

Based on the above discussion and (6) and (11), the Lax pair of the coupled Kundu equations can be read aswhere

Denotewhere and we havewhere Letting the equivalent Lax pair of (32) can be given bywhich implies thatwhererepresents the Hermitian of a matrix and. In addition, we can easily deduce thatIn terms of a scattering problem, the asymptotic condition ofcan be given bywheredenote theunit matrix and the subscript inrefers to the positive and negative side of the-axis. According to (37) and Abel’s formula, we obtainSupposing are both solutions of (32), we know thatandare linearly related.

Denoteand we have which implies thatwhere is the scattering matrix. By means of using (39) and (43), we deduce

Based on above discussion, we focus onsatisfyingAssuming that theterm of (46) is an inhomogeneous term, we know that the solution to the homogeneous equation on the left hand side of (46) is. We can turn (46) into Volterra integral equations foraccording to (38) and the method of variation of parameters. Thus, we conclude

When the integrals on the right sides of (47)-(48) converge, we can find thatare analytical continuations off the real axis. In addition, we can infer that the first column ofand the second column ofcan be analytically continued to the lower half plane. Similarly, the second column ofand the first column ofcan be analytically continued to the upper half plane

Defining the following notation: then the Jost solutionsare analytic inand are analytic in, where

It is easy to find thatand by (47) and (48). With the help of the adjoint scattering equation of (35),we can acquire the analytic counterpart ofin. It is easy to check thatmeets (43). In addition, depending on and (24), we obtainTherefore, we find thatmeets adjoint equation (55).

In the following, we expressandin the following form: Similarly, it is easy to see that the adjoint Jost solutionsare analytic inand are analytic in. In addition,and

From the above equations, we obtain where the superscriptsrefer to the half plane of analyticity. In fact, these analytic properties of the Jost solutions have a direct influence on the scattering matrixIt is easy to see that the analytic properties ofandcan be read as where the superscriptsindicate which scattering elements are analytic in the half ofplane. In addition, we find where the analytic properties ofcould be obtained by.

On the real line, based on (43), (50), and (38), we can obtain the matrix Riemann-Hilbert problemwhere We derive the normalization condition for this Riemann-Hilbert problem by using (53) and (61). Based on (50), (59), and (43), we have

Supposing that we can solve this Riemann-Hilbert problem based on, we can reconstructaccording to the asymptotic expansion of its solution. In addition,andare solutions of (35) and (57), respectively.

By definingand substituting (71) into (35), we have According to a comparison of the coefficient of, we obtainand the solutioncan be given by We haveby taking the Hermitian conjugate of the spectral equation of (35) and (36).

It is not difficult to notice thatormeets (55). In addition,meets (57). Hence,) andmust be linearly dependent. Resorting to (38), in, we infer that) andhave the same boundary conditions. In addition, they have the same solutions as (55). Thus,

We obtain thathas the following property: by using (75), (50), and (59). In addition, resorting to (43), we obtain

4. The Solution of the Riemann-Hilbert Problem

In order to obtain soliton solutions, we assumein (67). From Riemann-Hilbert problem (67), it is easily noticed thatrelies onandin the asymptotic conditions. Thus, we can regardas a constant that is independent of the solutions of the Riemann-Hilbert problem. We assume thathas zerosand thathas zeros. In short, all the zeros ofare simple zeros of. The kernels ofandinclude only a single column vectorand row vector, respectively. We haveBased on (78), we haveTaking the Hermitian conjugate of (79) and using (80), we obtainWe infer thatsatisfyby comparing (81) with the first equation of (79).

The solutions of this special Riemann-Hilbert problem can be written aswhere the matrixcan be defined asThe zerosandare independent of time and space. For the sake of seeking spatial and temporal evolutions for the vectors, we take the derivative of (79) with respect to. We havedepending on (35).

Hence, we can writewhereis a scalar function. The solution of (86) iswhere. Substituting (87) into (83)-(84), we find that theterm cancels out and does not affect the results. Hence, letting, we have Similarly, we obtainIn the same way, the time dependencies ofandare derived. As a result, we derive

5. Long-Time Behavior of the Solution and N-Soliton Solutions

Because the regular matrix Riemann-Hilbert problemwhereis not explicitly solvable, the expression for the solution of the coupled Kundu equations at a later time is not available in a general localized initial condition. However, we can obtain the long-time asymptotic state of the coupled Kundu equations. Then, we can express the solution of (92) as follows:whereAs, we haveandAs, we can easy find thatby using (94). Thus, relying on (93), (98), and (99), we findin the expansionDepending onand (96) can be written as Thus, it is easy to infer thatandcan be expressed by

In the following, we discuss the N-soliton solutions of (28). When, based on (100) and (73), we obtainandwhereis given by (90), andandare given by (84).

In addition, we introduce the notationThen, the solutions ofandcan be written out explicitly aswhere the elements of thematrixare given byandwithout loss of generality.

6. Discussion

In the following, supposingin (108), we obtain the single-soliton solutionSuppose whereandare the real and imaginary parts of, respectively. We obtain

Thus, the single-soliton solutions can be rewritten aswhich are shown in Figure 1.

Assumingin (108), we obtain the two-soliton solutionwhere

In essence, we obtain the expression of the N-soliton solutions in (108) based on the Riemann-Hilbert method. In addition, as an example, the single-soliton solution is precisely given in (113), and the two-soliton solution is expressed in (114). Similarly, according to (108), we can obtain three-soliton solutions, four-soliton solutions, and so on.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Huanhe Dong was supported by a Project of the National Natural Science Foundation of China (No. 11701334), Graduate Innovation Foundation from Shandong University of Science and Technology (No. SDKDYC180346), and the Nature Science Foundation of Shandong Province of China (No. ZR201709180230).