PT-symmetry breaking in complex nonlinear wave equations and their deformations

TL;DR

This paper explores PT-symmetry breaking in complex nonlinear wave equations, constructing solutions, analyzing energy reality conditions, and revealing how symmetry breaking and restoration affect the models' physical properties.

Contribution

It introduces new classes of invariant extensions of complex nonlinear wave models and examines their PT-symmetry breaking and energy properties.

Findings

PT-symmetry can be spontaneously broken by domain choices or parameter manipulation.

Energy reality can be restored through further symmetry breaking at the Hamiltonian level.

Fixed points in solutions can undergo Hopf bifurcations during symmetry breaking.

Abstract

We investigate complex versions of the Korteweg-deVries equations and an Ito type nonlinear system with two coupled nonlinear fields. We systematically construct rational, trigonometric/hyperbolic, elliptic and soliton solutions for these models and focus in particular on physically feasible systems, that is those with real energies. The reality of the energy is usually attributed to different realisations of an antilinear symmetry, as for instance PT-symmetry. It is shown that the symmetry can be spontaneously broken in two alternative ways either by specific choices of the domain or by manipulating the parameters in the solutions of the model, thus leading to complex energies. Surprisingly the reality of the energies can be regained in some cases by a further breaking of the symmetry on the level of the Hamiltonian. In many examples some of the fixed points in the complex solution for the field undergo a Hopf bifurcation in the PT-symmetry breaking process. By employing several different variants of the symmetries we propose many classes of new invariant extensions of these models and study their properties. The reduction of some of these models yields complex quantum mechanical models previously studied.