On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries

Abstract

We explore the $N_{\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+\kappa u_x=0,\quad m=u-u_{xx}$ with $\lim_{x\rightarrow \pm \infty }u(x,t)=0$. We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with $\ell=n=1$, the corresponding $N_{\infty}$-soliton is the one-soliton solution which the discrete spectral point is the center of the region; ii) when the region is a quadrature domain with $\ell=n$, the corresponding $N_{\infty}$-soliton is an $n$-soliton solution; iii) when the discrete spectra lie in the line region, we provide its corresponding Riemann-Hilbert problem,; and iv) when the discrete spectra lie in an elliptic region, it is equivalent to the case of the line region.