Complex solitons with real energies

TL;DR

This paper constructs explicit complex soliton solutions for several integrable equations, revealing PT-symmetry properties and showing that total energy remains positive despite non-Hermitian Hamiltonians.

Contribution

It introduces new complex soliton solutions for the KdV, mKdV, and sine-Gordon equations using Hirota's method and Baecklund transformations, highlighting PT-symmetry and energy positivity.

Findings

PT-symmetric one-soliton solutions with imaginary constants

Multi-soliton solutions from Hirota's method break PT-symmetry

Total energy remains positive despite non-Hermitian Hamiltonians

Abstract

Using Hirota's direct method and Baecklund transformations we construct explicit complex one and two-solutions to the complex Korteweg-de Vries equation, the complex modified Korteweg-de Vries equation and the complex sine-Gordon equation. The one-soliton solutions of trigonometric and elliptic type turn out to be PT-symmetric when a constant of integration is chosen to be purely imaginary with one special choice corresponding to solutions recently found by Khare and Saxena. We show that alternatively complex PT-symmetric solutions to the Korteweg-de Vries equation may also be constructed alternatively from real solutions to the modified Korteweg-de Vries by means of Miura transformations. The multi-soliton solutions obtained from Hirota's method break the PT-symmetric, whereas those obtained from Baecklund transformations are PT-invariant under certain conditions. Despite the fact that some of the Hamiltonian densities are non-Hermitian, the total energy is found to be positive in all cases, that is irrespective of whether they are PT-symmetric or not. The reason is that the symmetry can be restored by suitable shifts in space-time and the fact that any of our N-soliton solutions may be decomposed into N separate PT-symmetrizable one-soliton solutions.