Soliton-like solutions of the Camassa--Holm equation with variable coefficients and a small dispersion
The paper deals with the Camassa--Holm equation with variable coefficients (vcCH equation) that is a direct generalization of the well known Camassa--Holm equation. We focus on the mathematical description of particular solutions of the vcCH equation with a small dispersion that exhibit properties analogous to those of classical soliton and peakon solutions, and consider the construction of soliton- and peakon-like solutions in the form of asymptotic expansions, including both one-phase and two-phase cases. The solution is expressed as the sum of a regular background common to all soliton- and peakon-like solutions and a singular component that captures their distinctive features, with the precise definition of the main singular term playing a central role. In the one-phase case, this term is determined, and the solvability of higher-order singular corrections is established in suitable functional spaces, enabling the construction of asymptotic solutions to arbitrary accuracy in a small parameter. The study also addresses the construction of two-phase soliton- and peakon-like solutions. Theorems on the asymptotic accuracy of the constructed asymptotic solutions have been proved. Each of the considered cases is illustrated by nontrivial examples for which, in accordance with the obtained general results, approximate solutions are derived in explicit form and their graphs are presented.