一类耦合KdV型方程的周期波解、孤波解及它们随Hamilton能量的演变关系
Periodic and Solitary Wave Solutions of a Class of Coupled KdV Equation and Their Evolution Relation with Hamilton Energy
摘要: 本文研究了一类非线性耦合KdV型波动方程的精确孤波解、周期波解以及它们随Hamilton能量的演变关系。文中利用平面动力系统的方法对该方程进行了详细的定性分析，通过首次积分法求出了该方程的3种孤波解，其中对有界有理函数波解和扭状孤波解的求解是创新性的解。求出了该方程的7种雅可比椭圆函数周期波解，尤其是利用恰当变换求出了非对称同宿轨所围的闭轨对应的新周期波解，以及包围同宿轨的闭轨对应的新周期波解。文中将所求的孤波解和周期波解与Hamilton能量对应起来，并发现了所研方程为什么能产生孤波解和周期波解，实际上是该方程解的振幅对应的Hamilton系统的能量变化起着关键的作用。
Abstract: In this paper, the exact solitary and periodic wave solutions of a class of nonlinear coupled KdV wave equations and their evolution with Hamilton energy are studied. The equation is qualitative analysis in detail by using the method of planar dynamical system. Three kinds of solitary wave solutions of the equation are obtained by the first integral method, among which the bounded ra-tional function wave solution and the kink-shaped solitary wave solutions are new ones. Seven kinds of Jacobi elliptic function periodic wave solutions of the equation are obtained, especially the new periodic wave solutions corresponding to the closed orbit surrounded by the asymmetric homoclinic orbit and the new periodic wave solutions corresponding to the closed orbit surrounding the homoclinic orbit are obtained by using the appropriate transformation. The soliton and periodic wave solutions are associated with the Hamilton energy, and the reason why the soliton and periodic wave solutions can be produced is found. In fact, the energy change of the Hamilton system corresponding to the amplitude of the solution of the equation plays a key role.
文章引用：张卫国, 张雪, 姚倩. 一类耦合KdV型方程的周期波解、孤波解及它们随Hamilton能量的演变关系[J]. 理论数学, 2023, 13(4): 1090-1121. https://doi.org/10.12677/PM.2023.134116

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