Large‐time asymptotics to the focusing nonlocal modified Kortweg‐de Vries equation with step‐like boundary conditions

Abstract

AbstractWe investigate the large‐time asymptotics of solution for the Cauchy problem of the nonlinear focusing nonlocal modified Kortweg‐de Vries (MKdV) equation with step‐like initial data, that is, as , as , where A is an arbitrary positive real number. We first develop the direct scattering theory to establish the basic Riemann‐Hilbert (RH) problem associated with step‐like initial data. Thanks to the symmetries , of nonlocal MKdV equation, we investigate the asymptotics as and , respectively. Our main technique is to use the steepest descent analysis to deform the original matrix‐valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally, we obtain the different large‐time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space‐time sectors , , , and on the whole ‐plane.