Asymptotic behavior of rapidly oscillating solutions of the modified Camassa–Holm equation

We consider the modernized Camassa–Holm equation with periodic boundary conditions. The quadratic nonlinearities in this equation differ substantially from the nonlinearities in the classical Camassa–Holm equation but have all its main properties in a certain sense. We study the so-called nonregular solutions, i.e., those that are rapidly oscillating in the spatial variable. We investigate the problem of constructing solutions asymptotically periodic in time and more complicated solutions whose leading terms of the asymptotic expansion are multifrequency. We study the problem of the possibility of a compact form of these asymptotic expansions and the problem of reducing the construction of the leading terms of the asymptotic expansions to the analysis of the solutions of special nonlinear boundary-value problems. We show that this is possible only for the classical Camassa–Holm equation.