快速衰减初值的修正Korteweg-de Vries方程的孤子解:Riemann-Hilbert方法

Soliton Solutions of the Modified Korteweg-de Vries Equation with Fast Decay Initial Value:Riemann-Hilbert Approach

本文借助于Riemann-Hilbert (RH)问题研究修正Korteweg-de Vries (mKdV)方程,给出一种有效方法来获得快速衰减初值空间下的孤子解.在正散射过程建立Jost函数和散射矩阵重要性质来构建一个合适的RH问题,进而建立mKdV方程的解和RH问题解之间的关系.在反问题过程中,考虑了两类散射数据,包括简单零点和二阶零点,以及求解相应的RH问题,成功构建在这两种情形下mKdV方程的显示解.最后,结合具体参数,详细分析了几类孤子解的传播行为.

The Riemann-Hilbert(RH)problem is performed to study the modified Korteweg-de Vries(mKdV)equation in this work,and we give an effective method to obtain the soliton solution with the rapidly decaying initial value.The important properties of Jost functions and scattering matrix are obtained by the direct scattering to construct a suitable RH problem,and then the relationship between the solution of the RH problem and the potential function is established.In the inverse problem,two cases of scattering data,including simple zeros and double zeros,are considered,and the corresponding RH problem is solved.Then,the general forms of solutions for mKdV equation in two cases are given successfully.Finally,in combination with specific parameters,the multi-soliton solutions image propagation in two cases are given in detail.