Abstract

A periodic boundary value problem is considered for a modified Camassa–Holm equation, which differs from the well-known classical equation by several additional quadratic terms. Three important conditions on the coefficients of the equation are formulated under which the original equation has the Camassa–Holm type. The dynamic properties of regular solutions in neighborhoods of all equilibrium states are investigated. Special nonlinear boundary value problems are constructed to determine the “leading” components of solutions. Asymptotic formulas for the set of periodic solutions and finite-dimensional tori are obtained. The problem of infinite-dimensional tori is studied. It is shown that the normalized equation in this problem can be compactly written in the form of a partial differential equation only for the classical Camassa–Holm equation. An asymptotic analysis is presented in the cases when one of the coefficients in the linear part of the equation is sufficiently small, while the period in the boundary conditions is sufficiently large.

中文翻译：

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Camassa-Holm问题正则解的渐近性

摘要

对于改进的Camassa-Holm方程，考虑了周期边值问题，该方程与众所周知的经典方程的区别在于几个附加的二次项。在方程的系数上提出了三个重要条件，根据这些条件，原始方程具有Camassa–Holm类型。研究了所有平衡态邻域中正则解的动力学性质。构建特殊的非线性边值问题，以确定解决方案的“主导”组成部分。获得了周期解集和有限维花托的渐近公式。研究了无穷维花托的问题。结果表明，该问题的归一化方程可以用偏微分方程的形式紧凑地编写，仅适用于经典的Camassa–Holm方程。