Existence of global solutions for the modified Camassa-Holm equation with a nonzero background

TL;DR

This paper proves the existence of unique global solutions for the modified Camassa-Holm equation with a nonzero background, using inverse scattering transform and Riemann-Hilbert problem techniques in weighted Sobolev spaces.

Contribution

It establishes the global existence and regularity of solutions for the mCH equation with nonzero background initial data via advanced inverse scattering methods.

Findings

Existence of unique global solutions in weighted Sobolev spaces.

Application of Riemann-Hilbert problem to the mCH equation.

Boundedness and regularity of solutions proved through refined estimates.

Abstract

Consideration in the present paper is the existence of global solutions for the modified Camassa-Holm (mCH) equation with a nonzero background initial value. The mCH equation is completely integrable and can be considered as a model for the unidirectional propagation of shallow-water waves. By applying the inverse scattering transform with an application of the Cauchy projection operator, the existence of a unique global solution to the mCH equation in the line with a nonzero background initial value is established in the weighted Sobolev space $ H^{2, 1} (\mathbb{R})\cap H^{1, 2} (\mathbb{R})$ based on the representation of a Riemann-Hilbert (RH) problem associated with the Cauchy problem to the mCH equation. A crucial technique used is to derive the boundedness of the solution in the Sobolev space $ W^{1,\infty}(\mathbb{R}),$ then reconstruct a new RH problem for the Cauchy projection operator of reflection coefficients. The regularity of the global solution is achieved by the refined estimate arguments on those solutions of the corresponding RH problem.