Long‐Time Asymptotics for the Modified Camassa–Holm Positive Flow With the Schwartz Initial Data

ABSTRACT The Cauchy problem of the modified Camassa–Holm (CH) positive flow with the Schwartz initial data is studied by using the Riemann–Hilbert (RH) approach and nonlinear steepest descent method. Due to the existence of energy‐dependent potential, the spectral analysis of Lax pairs is extremely difficult. We have to go out of our way and introduce some spectral function transformations, from which a basic RH problem is constructed with the aid of the inverse scattering transformation. Then we convert the basic RH problem into a regular RH problem and take into account the reorientation, local parametrices, and main contributions. Finally, we obtain the long‐time asymptotic behavior of solution of the modified CH positive flow.