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

Tamara Grava, Alexander Minakov

Research output: Contribution to journalArticle (Academic Journal)peer-review

7 Downloads (Pure)

Abstract

We study the long time asymptotic behaviour of the solution q(x,t) of the modified Korteweg de Vries equation (MKdV) with step-like initial datum asymptotic to c_+ at +infinity and to c_- at -infinity. We show that the solution for long times decomposes in the (x,t) plane in three main regions:
1. a region where solitons and breathers travel with positive velocities on a constant background c_+;
2. an expanding oscillatory region {\color{black} (that generically contains breathers)};
3. a region of breathers travelling with negative velocities on the constant background c_-.

When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the 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.
Original languageEnglish
Pages (from-to)5892–5993
Number of pages102
JournalSIAM Journal on Mathematical Analysis
Volume52
Issue number6
DOIs
Publication statusPublished - 24 Nov 2020

Keywords

  • integrable system
  • Riemann-Hilbert problem
  • dispersive shock waves
  • long time asymptotic analysis

Fingerprint Dive into the research topics of 'On the long time asymptotic behaviour of the modified Korteweg de Vries equation with step-like initial data'. Together they form a unique fingerprint.

Cite this