Kawahara方程：行波解与周期波的结合
The Kawahara Equation: Traveling Wave Solutions Joining Periodic Waves
摘要：
Kawahara方程是色散波的弱非线性长波模型，当首阶色散效应与次阶校正平衡时出现。Kawahara方程的行波解满足四阶常微分方程，其中行波速度是参数。该四阶方程具有Hamilton结构，且存在一个双参数的单相周期解族，其速度和Hamilton函数都是变化的。对具有相等速度和哈密顿量的周期解对导出了一组跳跃条件。这些是行波存在的必要条件，渐近周期轨道在$\pm \infty$。利用分歧理论和参数延拓方法构造了跳跃条件的多解分支。对于每一对相容周期解，由周期轨道的稳定流形和不稳定流形的交集构造代表行波的异宿轨道。每个分支终止于平衡到周期解，其中平衡是连接到相关联的周期解的孤立波的背景。
英文摘要：
The Kawahara equation is a weakly nonlinear long-wave model of dispersive waves that emerges when leading order dispersive effects are in balance with the next order correction. Traveling wave solutions of the Kawahara equation satisfy a fourth-order ordinary differential equation in which the traveling wave speed is a parameter. The fourth order equation has Hamiltonian structure and admits a two-parameter family of single-phase periodic solutions with varying speed and Hamiltonian. A set of jump conditions is derived for pairs of periodic solutions with equal speed and Hamiltonian. These are necessary conditions for the existence of traveling waves that asymptote to the periodic orbits at $\pm \infty$. Bifurcation theory and parameter continuation are used to construct multiple solution branches of the jump conditions. For each pair of compatible periodic solutions, the heteroclinic orbit representing the traveling wave is constructed from the intersection of stable and unstable manifolds of the periodic orbits. Each branch terminates at an equilibrium-to-periodic solution in which the equilibrium is the background for a solitary wave that connects to the associated periodic solution.
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