Type I integrable defects and finite-gap solutions for KdV and sine-Gordon models

Francis Edward Corrigan, Robert Charles Parini

Research output: Contribution to journalArticlepeer-review

Abstract

The main purpose of this paper is to extend results, which have been obtained previously to describe the classical scattering of solitons with integrable defects of type I, to include the much larger and intricate collection of finite-gap solutions defined in terms of generalised theta functions. In this context, it is generally not feasible to adopt a direct approach, via ansätze for the fields to either side of the defect tuned to satisfy the defect sewing conditions. Rather,
essential use is made of the fact that the defect sewing conditions themselves are intimately related to Bäcklund transformations in order to set up a strategy to enable the calculation of the field on one side by suitably transforming the field on the other side. The method is implemented using Darboux transformations and illustrated in detail for the sine-Gordon and KdV models. An exception, treatable by both methods, indirect and direct, is provided by the genus 1 solutions. These can be expressed in terms of Jacobi elliptic functions, which satisfy a number of useful identities of relevance to this problem. There are new features to the solutions obtained in the finite-gap context but, in all cases, if a (multi)soliton limit is taken within the finite-gap solutions previously known results are recovered.
Original languageEnglish
Article number284001
Number of pages41
JournalJournal of Physics A: Mathematical and Theoretical
Volume50
Issue number28
Early online date31 May 2017
DOIs
Publication statusPublished - 19 Jun 2017

Bibliographical note

© 2017 IOP Publishing Ltd. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.

Keywords

  • Finite gap solutions
  • KdV
  • Sine-Gordon
  • integrable defects

Cite this