Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem

We study the coupled mKdV equation by the dressing method via local RiemannHilbert problem. With the help of the Lax pairs, we obtain the matrix RiemannHilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.


Introduction
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.

Jost Solution
First, we consider the coupled KdV equation 6 ,
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.
, we obtain the special solution of Equation ( 3).For convenience, we denote the special solution as . Then, the spectral Equation ( 3) where, In what follows, we study the Jost solutions ( ) , H x k ± of the Equation (6) satisfying the asymptotic conditions H I ± → , at x → ±∞ .Since 0 trU = , these boundary con- ditions guarantee that det 1 H ± = for all x.In fact, the Jost functions H ± are not mutually independent.They are intercon- nected by the scattering matrix ( )

Analysis Solutions
Let us rewrite the spectral Equation ( 6) with the boundary conditions in the integral form: for the first column entries of the Jost matrix H − .It is easy to know that the exponent in (8) decreases for 0 Imk > .The first column 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 is a solution of the adjoint spectral problem.On the real axis ( ) ( ) 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 + Ω .

Matrix RH Problem
Through tedious calculation, we obtain RH problem ( ) ( ) ( ) with It is easy to know that ( ) ( ) Ω Ω only depends on k, the x-dependence being given by the simple exponential function F.Moreover, it is obvious that ( ) (12).
In order to obtain the soliton solution of the coupled KdV equation, we suppose that the zeros of ( ) a k and ( ) a k are simple and finite number.We know that determi- nants of the matrices + Ω and Ω are given by ( ) a k and ( ) a k .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 1 , , Here j ϒ is the rank 1 projector 2 j j ϒ =ϒ , and In view of ( 11), we know that Ω will be regularized by the rational function it is easy to know that the matrix 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 ω ± solve the regular RH problem (without zeros) ( ) ( ) ( ) ( ) ( ) with ( ) The matrix Γ 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 ( ) k Γ can be written as ( ) Thus, we derived 2N vectors j x and j y instead of N vectors j Y .It is ob- vious that ( ) ( ) To avoid divergence at j k k → , we should pose ( ) We note that the matrix ( ) can be decomposed into the following form: where where . In what follows, we rewrite ( 13) as Let us differentiate the equation ( ) , and in view of ( In the same way, we obtain the evolutionary equation In this end, we establish explicitly the vector j as where 0 j is a vector integration constant.
Similarly, according to , we obtain the solution where 0 j  is a vector integration constant.

One Soliton Solution
We consider the case 1 N = and pose 1 k i where, 1 2 , p p are components of the constant vector 0 1 .
The dress formula (19) reduced to At the same time, we have , from which, we obtain Denoting ( ) In the same way, defining ( ) Substituting (31) and (32) into (30), we have From which, we have the solutions of the coupled KdV Equation ( 2 Here, ξ , ξ  , η and η determine the soliton velocity and amplitude, respectively, while α , β , α and β  give the initial position and phase of the soliton.In what follows, we plot the graph for ( ) , u x t in order to analyze the solutions (35).Figure 1 and Figure 2 are the imaginary part and real part of ( ) , u x t , 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.

[ ] 1 H
− of the matrix H − is analytic in the upper half plane and continuous on the real axis 0 Imk = .Similarly, we know that the second column

[ ] 2 H
+ of the matrix H + is analytic as well in the same domain.Then, we give a solution of Equation (6):