In this paper, we present the globally conservative solutions to the Cauchy problem for the modified Camassa–Holm (MOCH) equation. First, we transform the equation into an equivalent semi-linear system under new variables. Second, according to the standard ordinary differential equation theory with the aid of the conservation law, we give the global solutions of the semi-linear system. Finally, returning to the original variables, we obtain the globally conservative solutions to the MOCH equation.
Zhaonan Luo, Zhijun Qiao, and Zhaoyang Yin , "Globally conservative solutions for the modified Camassa–Holm (MOCH) equation", J. Math. Phys. 62, 091506 (2021) https://doi.org/10.1063/5.0048245
J. Math. Phys.