关于几类二分量非线性波方程的行波解的研究

Investigating the exact solution and dynamic behaviors of nonlinear wave equations plays an important role in the mathematical physics. In this project，we mainly focuses on several types of two-component nonlinear wave equations，and their traveling wave solutions and their dynamic behavior are comprehensively studied and comparatively analyze. Based on the bifurcation theory of dynamical systems, combined with the first integral and phase portraits of integrable traveling wave systems，the bifurcation and exact solutions of several classes of two-component nonlinear wave equations are studied. Meanwhile, by means of Bäcklund transformation, some multi-soliton solutions of two-component nonlinear wave equation are constructed and the dynamic properties of the solutions are analyzed. Finally, the representations and properties of the exact solutions in several cases, such as the same type of exact solutions of different systems and the same exact solutions of the same system with different expressions and waveforms, are compared and analyzed. The algebraic properties, geometric structures and dynamic behavior of these traveling wave solutions are discussed, and the internal relations between the exact solutions of several kinds of equations and the dynamic behavior, as well as the causes and physical significance of the solutions are clarified.

非线性波方程的精确解及其动力学性质是数学物理中一个重要的研究课题。本项目将围绕几类二分量非线性波方程，对其行波解及其动力学行为进行综合研究与比较分析。应用动力系统分支理论，结合可积行波系统的首次积分和相图，研究几类二分量非线性波方程的分支和精确解及其动力学性质；借助Bäcklund变换，构造某些二分量非线性波方程的多孤子解，并分析其动力学性质。最后，比较分析几种情形下精确解的性质和形式，如不同系统的同一类精确解，以及同一系统具有不同表达方式和波形图的同一类精确解等，探讨研究这几类行波解的代数性质、几何结构和动力学行为，理清几类方程精确解和动力学行为的内在联系，以及解产生的原因和物理意义。

非线性波方程的精确解及其动力学性质是数学物理中一个重要的研究课题。本项目通过动力系统分岔理论和奇行波方程理论，对一些非线性波方程的行波解及其动力学行为进行综合研究与比较分析，包括旋转Camassa-Holm方程、一类具有Conformable分数阶导数的三阶mKdV方程、一类非局域流体动力学方程、非局部Fokas-Lenells方程、复杂Ginzburg-Landau方程和具有反立方非线性项的Raman孤子模型。将奇行波方程理论和奇异摄动理论相结合，研究了Degasperis-Procesi方程的精确行波解，并通过Melnikov方法研究方程对应的扰动系统的孤立波解的存在性。