Asymptotics of regular solutions to the Camassa–Holm problem

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.