Non-uniform dependence on initial data for the periodic modified Camassa–Holm equation

Abstract

In this paper, we study the periodic Cauchy problem for the modified Camassa–Holm equation $$m_t+um_x+2u_xm=0,\quad m=(1-\partial_x^2)^2u$$ , and show that the solution map is not uniformly continuous in Sobolev spaces \({H^s(\mathbb T)}\) for s > 7/2. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.