Scientific Research

An Academic Publisher

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

Share and Cite:

*Applied Mathematics*,

**7**, 1789-1797. doi: 10.4236/am.2016.715150.

1. Introduction

The coupled mKdV equation

(1)

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.

2. Jost Solution

First, we consider the coupled KdV equation

(2)

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

(3)

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

(4)

(5)

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

(6)

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:

(7)

3. Analysis Solutions

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

(8)

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

(9)

with

(10)

In the same way,

(11)

It follows from the above formal as that

(12)

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:

(13)

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

(14)

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

4. Matrix RH Problem

Through tedious calculation, we obtain RH problem

(15)

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:

(16)

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

(17)

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

(18)

We note that the dress matrix can be written as

(19)

(20)

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

(21)

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

(22)

where. Similarly,

(23)

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

(24)

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

thus, we have

(25)

In the same way, we obtain the evolutionary equation

(26)

In this end, we establish explicitly the vector as

(27)

where is a vector integration constant.

Similarly, according to, we obtain the solution

(28)

where is a vector integration constant.

5. One Soliton Solution

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

(29)

where, are components of the constant vector.

(30)

where, are components of the constant vector.

The dress formula (19) reduced to

(31)

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

. (32)

Denoting, , thus

(33)

In the same way, defining, , thus

(34)

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

Moreover, , hence,

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

(35)

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.

Figure 1. The soliton solution curve of imaginary part of for, , , , , , , , ,.

Figure 2. The soliton solution curve of real part of for, , , , , , , , ,.

Figure 3. The soliton solution curve of imaginary part of for, , , , , , , , ,.

Figure 4. The soliton solution curve of real part of for, , , , , , , , ,.

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.

Acknowledgements

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.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | Ablowitz, M.J. and Segur, H. (1981) Solitons and the Inverse Scattering Transform. SIMA, Philadelphia. |

[2] |
Geng, X.G., Wu, Y.T. and Cao, C.W. (1999) Quasi-Periodic Solutions of the Modified Kadomtsev-Petviashvili Equation. Journal of Physics A: Mathematical and General, 32, 3733-3742. http://dx.doi.org/10.1088/0305-4470/32/20/306 |

[3] |
Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and the Inverse Scattering. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511623998 |

[4] | Gu, C.H., Hu, H.S. and Zhou, H.S. (1999) Darboux Transformation in Soliton Theory and Its Geometric Applications. Shanghai Scientific and Technical Publishers, China. |

[5] |
Geng, H.S. and Tam, H.W. (1999) Darboux Transformation and Soliton Solutions for Generalized Nonlinear Schrodinger Equations. Journal of the Physical Society of Japan, 68, 1508-1542. http://dx.doi.org/10.1143/JPSJ.68.1508 |

[6] |
Geng, X.G. (1992) A Hierarchy of Nonlinear Evolution Equations and Corresponding Finite Dimensional Completely Integrable System. Physics Letters A, 180, 375-380.
http://dx.doi.org/10.1016/0375-9601(92)90057-S |

[7] | Deife, P. and Zhou, X. (1993) A Steepest Descent Method for Oscilatory Riemann-Hilbert Problems, Asymptotics for the mKdV Equation. Annals of Mathematics, 137, 295-268. |

[8] |
Claeys, T. and Grava, T. (2009) Universality of the Break-Up Profile for the KdV Equation in the Small Dispersion Limit Using the Riemann-Hilbert Approach. Communications in Mathematical Physics, 286, 979-1009. http://dx.doi.org/10.1007/s00220-008-0680-5 |

[9] |
Wang, M.L., Zhou, Y.B. and Li, Z.B. (1996) Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics. Physics Letters A, 216, 67-75. http://dx.doi.org/10.1016/0375-9601(96)00283-6 |

[10] |
Tam, H.W., Hu, X.B. and Wang, D.L. (1999) Two Integrable Coupled Nonlinear Systems. Journal of the Physical Society of Japan, 68, 369-379. http://dx.doi.org/10.1143/JPSJ.68.369 |

[11] |
Fokas, A.S. and Gelfand, I.M. (1994) Integrability of Linear and Nonlinear Evolution Equations and the Associated Nonlinear Fourier Transforms. Letters in Mathematical Physics, 32, 189-210. http://dx.doi.org/10.1007/BF00750662 |

[12] | Calogero, F. and Degasperis, A. (1982) Solitons and Spectral Transform I. North-Holl, Amsterdam. |

[13] |
Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. and Morris, H.C. (1982) Solitons and Nonlinear Wave Equations. Physics Letters A, 89, 168-170.
http://dx.doi.org/10.1016/0375-9601(82)90199-2 |

[14] | Newell, A.C. (1985) Solitons in Mathematics and Physics. SIAM, Philadelphia. |

[15] |
Dotorov, E.V., Matsuka, N.P. and Rothos, V.M. (2003) Perturbation-Induced Radiation by the Ablowitz-Ladik Soliton. Physical Review E, 68, Article ID: 066610.
http://dx.doi.org/10.1103/PhysRevE.68.066610 |

[16] |
Fokas, A.S. (1997) A Unified Transformation Method for Solving Linear and Certain Nonlinear PDES. Proceedings of the Royal Society of London A, 453, 1441-1443.
http://dx.doi.org/10.1098/rspa.1997.0077 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.