Regularized degenerate multi-solitons

TL;DR

This paper introduces complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation, derived via Darboux-Crum transformations and Hirota's method, revealing new degenerate energy solutions with equal finite real energy.

Contribution

It presents a novel method to generate degenerate multi-soliton solutions for integrable systems using Darboux-Crum transformations and Hirota's method, applicable to various nonlinear equations.

Findings

Solutions contain asymptotic one-soliton structures

Solutions possess equal finite real energy

Method is applicable to other integrable systems

Abstract

We report complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. We demonstrate how these solutions originate from degenerate energy solutions of the Schroedinger equation. Technically this is achieved by the application of Darboux-Crum transformations involving Jordan states with suitable regularizing shifts. Alternatively they may be constructed from a limiting process within the context Hirota's direct method or on a nonlinear superposition obtained from multiple Baecklund transformations. The proposed procedure is completely generic and also applicable to other types of nonlinear integrable systems.