一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解

A New Matrix Modified Korteweg-de Vries Equation:Riemann-Hilbert Approach and Exact Solutions

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.

A new matrix modified Korteweg-de Vries(mmKdV for short)equation with a p×q complex-valued potential matrix function is first studied via Riemann-Hilbert approach,which can be reduced to the well-known coupled modified Korteweg-de Vries equations by selecting special potential matrix.Starting from the special analysis for the Lax pair of this equation,the authors successfully establish a Riemann-Hilbert problem of the equation.By introducing the special conditions of irregularity and reflectionless case,some interesting exact solutions,including the N-soliton solution formula,of the mmKdV equation are derived through solving the corresponding Riemann-Hilbert problem.Moreover,due to the special symmetry of special potential matrices and the N-soliton solution formula,the authors make further efforts to classify the original exact solutions to obtain some other interesting solutions which are all displayed graphically.It is interesting that the local structures and dynamic behaviors of soliton solutions,breather-type solutions and bell-type soliton solutions are all analyzed via taking different types of potential matrices.