Long-time asymptotics of solution for the fifth-order modified KdV equation in the presence of discrete spectrum

We investigate the Cauchy problem of an integrable focusing fifth-order modified Korteweg–de Vries (KdV) equation, which contains the fifth-order dispersion and relevant higher order nonlinear terms. The long-time asymptotics of solution is established in the case of initial conditions that lie in some low regularity weighted Sobolev spaces and allow for the presence of discrete spectrum. Our method is based on a $\overline{\partial }$ generalization of the nonlinear steepest descent method of Deift and Zhou. We show that the solution decomposes in the long time into three main regions: (i) an expanding oscillatory region where solitons and breathers travel with positive velocities, the leading order term has the form of a multisoliton/breather and soliton/breather–radiation interactions; (ii) a Painlevé region, which does not have traveling solitons and breathers, the asymptotics can be characterized with the solution of a fourth-order Painlevé II equation; (iii) a region of breathers traveling with negative velocities. Employing a global approximation via PDE techniques, the asymptotic behavior of solution is extended to lower regularity spaces with weights.