Novel Iterative Methods to Show Classical Solutions of the Rotational Two‐Component Camassa–Holm System

ABSTRACT The rotation two‐component Camassa–Holm (R2CH) system is an extension of the well‐known Camassa–Holm (CH) equation that incorporates both a second component (often a density or depth variable) and Coriolis‐type rotational effects (e.g., modeling Earth's rotation in shallow water dynamics). The paper is devoted to the study of the existence of classical solutions for the rotation two‐component Camassa‐Holm system, which is a generalization of the Camassa‐Holm equation. It is known that the nature of integrable equations allows for an extended search for their various exact solutions. Here, we propose and develop a new iterative method together with certain appropriate topological properties to establish one classical solution for the problem (1) and at least two non‐negative classical solutions.