# On the long time asymptotic behavior of the modified Korteweg de Vries equation with step-like initial data

## TL;DR

This paper analyzes the long-term behavior of solutions to the modified Korteweg de Vries equation with step-like initial data, revealing the formation of dispersive shock waves and their interaction with solitons and breathers.

## Contribution

It provides a detailed description of the dispersive shock wave structure and its evolution, including the modulation by Whitham equations and the elastic interaction with solitons and breathers.

## Key findings

- Dispersive shock wave described by modulated periodic solutions.
- Oscillatory region expands within a specific cone in the (x,t) plane.
- Solution decomposes into solitons, breathers, and radiation with long-time behavior.

## Abstract

We study the long time asymptotic behaviour of the solution $q(x,t) $, to the modified Korteweg de Vries equation (MKDV) $q_t+6q^2q_x+q_{xxx}=0$ with step-like initial datum q(x,t=0)->c_- for x->-infinity and q(x,t=0)->c_+ for x-> +infinity. For the exact shock initial data q(x,t=0)=c_- for x<0 and q(x,t=0)=c_+ for x>0 the solution develops an oscillatory regions called dispersive shock wave that connects the two constant regions c_+ and c_-. We show that the dispersive shock wave is described by a modulated periodic travelling wave solution of the MKDV equation where the modulation parameters evolve according to the Whitham modulation equation. The oscillatory region is expanding within a cone in the $(x,t) plane defined as -6c_{-}^2+12c_{+}^2<x/t<4c_{-}^2+2c_{+}^2, with t\gg 1. For step like initial data we show that the solution decomposes for long times in three main regions: - a region where solitons and breathers travel with positive velocities on a constant background c_+; - an expanding oscillatory region that can contain generically breathers; - a region of breathers travelling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, it coincides up to a phase shift with the dispersive shock wave solution obtained for the exact step initial data. The phase shift depends on the solitons, breathers and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers and radiation.