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

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