The coupled mKdV equation

is an important member of the AKNS hierarchy [1] . Moreover, it has various applications in mathematical and physical fields. In [2] , Prof. Geng has given its quasi-periodic solution by using algebra-geometric methods. The equation can be solved by the method of the inverse scattering transformation, Hirota direct method, Lax pairs nonlinearization approach and others [3] - [6] . There are a lot of references for the topic [7] - [14] .

In this paper, we study the Equation (1) with the help of the Riemann-Hilbert method following [15] [16] . The present paper is organized as follows. In section 2, we give the Jost solution of the spectral equation. In section 3, we discuss the analytic property of the Jost solution. In section 4, we give the Matrix Riemann-Hilbert Problem. In section 5, we obtain the soliton-solution of the coupled KdV Equation (2), and we drop the curve of the solutions with the aid of the Matlab.

First, we consider the coupled KdV equation

As is well known [2] , the Equation (2) can be derived as the compatibility of the system

where the 2 × 2 matrices U and V of the form

where k is an arbitrary constant spectral parameter.

When, we obtain the special solution of Equation (3). For convenience, we denote the special solution as. Then, the spectral Equation (3) is transformed into

where,.

In what follows, we study the Jost solutions of the Equation (6) satisfying the asymptotic conditions, at. Since, these boundary conditions guarantee that for all x.

In fact, the Jost functions are not mutually independent. They are interconnected by the scattering matrix:

Let us rewrite the spectral Equation (6) with the boundary conditions in the integral form:

for the first column entries of the Jost matrix.

It is easy to know that the exponent in (8) decreases for. The first column of the matrix is analytic in the upper half plane and continuous on the real axis. Similarly, we know that the second column of the matrix is analytic as well in the same domain. Then, we give a solution of Equation (6):

It can see that it is analytic as a whole in the upper half plane.

The analytic solution can be expressed in terms of the Jost function. In view of (7), we derive

with

In the same way,

It follows from the above formal as that

In what follows, we define a function. It is obvious that

Then, is a solution of the adjoint spectral problem. On the real axis

and, has an asymptotic expansion as follows:

and substitute it into the spectral Equation (6). Comparing with powers of k, we derive

In order to solve the coupled KdV Equation (2), we should find the analytic solution.

Through tedious calculation, we obtain RH problem

with,.

It is easy to know that only depends on k, the x-dependence being given by the simple exponential function F. Moreover, it is obvious that for, in view of (12).

In order to obtain the soliton solution of the coupled KdV equation, we suppose that the zeros of and are simple and finite number. We know that determinants of the matrices and are given by and. We assume that

In this case, the RH problem (15) with zeros can be solved in view of its regulation.

To obtain the relevant regular problem, let us introduce a rational matrix function

where the eigenvector solves.

Here is the rank 1 projector, and.

In view of (11), we know that near the point. We obtain at the point. The matrix function will be regularized by the rational function

it is easy to know that the matrix has no zeros in.

The regularization of all the other zeros is performed similarly, and eventually we obtain the following representation for the analytic solutions:

where the rational matrix function accumulates all zeros of the RH problem, while the matrix functions solve the regular RH problem (without zeros)

with, thus.

The matrix will be called the dressing factor. It follows from (16) that the asymptotic expansion for the dressing factor is written as

We note that the dress matrix can be written as

Thus, we derived vectors and instead of N vectors. It is obvious that at the point. To avoid divergence at, we should pose, that is

We note that the matrix can be decomposed into the following form:

where. Similarly,

where. In what follows, we rewrite (13) as

Let us differentiate the equation in x, and in view of (6), we derive

thus, we have

In the same way, we obtain the evolutionary equation

In this end, we establish explicitly the vector as

where is a vector integration constant.

Similarly, according to, we obtain the solution

where is a vector integration constant.

We consider the case and pose,. Then, we have

where, are components of the constant vector.

where, are components of the constant vector.

The dress formula (19) reduced to

At the same time, we have, from which, we obtain

Denoting, , thus

In the same way, defining, , thus

Substituting (31) and (32) into (30), we have

Moreover, , hence,

From which, we have the solutions of the coupled KdV Equation (2)

Here, , , and determine the soliton velocity and amplitude, respectively, while, , and give the initial position and phase of the soliton. In what follows, we plot the graph for in order to analyze the solutions (35). Figure 1 and Figure 2 are the imaginary part and real part of, respectively. From the two solution curves, we can see that the difference between the real and imaginary part.

In the same way, we drop the solution curves of v for Figure 3 and Figure 4.

From the graphs, it is shown that u and v have the similar solution form. The difference exists between the real and imaginary part. In fact, we chose different parameters, and the solution curves between the real part and imaginary part had corresponding changes.

The authors acknowledge the support by National Natural Science Foundation of China (Project No: 11301149), Henan Natural Science Foundation For Basic Research under Grant No: 162300410072, doctor Foundation (D2015001) and Young backbone teachers in Henan province.

Su, T., Ding, G.H. and Wang, Z.W. (2016) Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem. Applied Mathematics, 7, 1789-1797. http://dx.doi.org/10.4236/am.2016.715150