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

TL;DR

This paper extends the analysis of integrable defects in KdV and sine-Gordon models to finite-gap solutions using Darboux and Backlund transformations, revealing new features while recovering known soliton results in the appropriate limits.

Contribution

It introduces a novel method to construct finite-gap solutions with integrable defects, utilizing Darboux transformations and Backlund relations, expanding the understanding of soliton-defect interactions.

Findings

Finite-gap solutions can be constructed with integrable defects using Darboux transformations.

The method applies to sine-Gordon and KdV models, including genus 1 solutions in terms of Jacobi elliptic functions.

Known soliton results are recovered in the multi-soliton limit of finite-gap solutions.

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 ansatze 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 Backlund 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.