Existence of global solutions to the Cauchy problem for the modified Camassa-Holm equation with a linear dispersion term

Yiling Yang, Engui Fan, Yue Liu

Published 2023-06-10Version 1

In this paper, we address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation with a linear dispersion term \begin{align*} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+\kappa u_{x}=0, \quad m=u-u_{x x}, &m(x, 0)=m_{0}(x),\nonumber \end{align*} where $\kappa$ is a positive constant characterizing the effect of the linear dispersion. The key to prove this result is to establish a bijective maps between potentials and reflection coefficients with the inverse scattering transform theory, in which the Volttera integral operator and Cauchy integral operator play an important role. Based on the spectral analysis of the Lax pair associated with the mCH equation the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale $(y,t)$. By using the reconstruction formula and estimates on the solution of the time-dependent RH problem, we further affirm the existence of a unique global solution to the Cauchy problem for the mCH equation.