Stability of 2-soliton solutions in the modified Camassa–Holm equation

The modified Camassa–Holm equation with cubic nonlinearity is a completely integrable model, utilized to describe the unidirectional propagation of shallow-water waves. In this study, we show that the 2-soliton, when employed as a solution to the initial-value problem for the modified Camassa–Holm equation, is nonlinearly stable to perturbations within the Sobolev space $$H^2$$ related to the momentum density. This stability is demonstrated through the conservation of crucial quantities in the Sobolev spaces $$H^2$$ , $$H^{1}$$ , and $$L^{1}$$ in terms of the momentum density.