The modified Camassa-Holm equation via Riemann-Hilbert

In 1993, Roberto Camassa and Darryl Holm introduced an equation modeling wave propagation on shallow water surfaces, termed the Camassa-Holm (CH) equation. This equation is distinguished by its ability to describe phenomena not captured by classical wave theories, notably the emergence of solutions with peak-like structures, known as peakons. The CH equation is heralded for its integrability and the rich mathematical structures, which has spurred interest in its study and the exploration of its generalizations and modifications, including the modified Camassa-Holm (mCH) equation and the two-component modified Camassa-Holm system (2-mCH). The main objects of the project are the mCH equation and the 2-mCH. The overall aims of the project research are: 1) the investigation of the global existence of the solution of the Cauchy problem for the mCH equation on the line when the solution is assumed to approach two different constants at the different infinities of the space variable; 2) the study of the long-time asymptotics for the mCH equation when the solution is assumed to approach two different constants at the different infinities of the space variable; 3) the development of the RH approach to the 2-mCH equation on the line in the case when the solution is assumed to approach non-zero constants at both infinities of the space variable. To address our research objectives, we will employ the Inverse Scattering Transform (IST) method in form of the Riemann-Hilbert (RH) problem. The specific feature of the study is that we consider these equations in the case of non-vanishing boundary conditions at infinity, a scenario that poses unique analytical challenges. The primary researchers involved are Iryna Karpenko (principal investigator) and Roland Donninger (mentor). The project benefits from collaboration with experts such as D. Shepelsky, G. Teschl, I. Egorova, V. Kotlarov, and J. Eckhardt, each of whom brings valuable expertise in inverse scattering transforms, the Riemann-Hilbert approach to integrable nonlinear equations, and asymptotics for soliton equations.