Time-delay and reality conditions for complex solitons

TL;DR

This paper analyzes complex multi-soliton solutions of the Korteweg de-Vries equation, focusing on their scattering properties, internal structures, and the reality of conserved charges, using analytical methods and symmetry considerations.

Contribution

It provides explicit formulas for soliton displacements and delays, clarifies distinctions between solution techniques, and demonstrates the reality of conserved charges through PT-symmetry and integrability.

Findings

Explicit formulas for soliton time-delays and displacements.

Distinction between solution methods clarified.

Conservation laws are real due to PT-symmetry and integrability.

Abstract

We compute lateral displacements and time-delays for a scattering processes of complex multi-soliton solutions of the Korteweg de-Vries equation.The resulting expressions are employed to explain the precise distinction between solutions obtained from different techniques, Hirota's direct method and a superposition principle based on Baecklund transformations. Moreover they explain the internal structures of degenerate compound multi-solitons previously constructed. Their individual one-soliton constituents are time-delayed when scattered amongst each other. We present generic formulae for these time-dependent displacements. By recalling Gardner's transformation method for conserved charges, we argue that the structure of the asymptotic behaviour resulting from the integrability of the model together with its PT-symmetry ensure the reality of all of these charges, including in particular the mass, the momentum and the energy.