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

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.

50 pages, 39 figures (compressed in order to comply with arXiv policy; higher resolutions maybe obtained from the authors upon request)

References in corpus (7)

• Making Sense of Non-Hermitian Hamiltonians
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• Complexified Dynamical Systems
• PT-symmetric deformations of Calogero models
• From real fields to complex Calogero particles
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• Euler Incognito

Cited by in corpus (17)

• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Observation of Asymmetric Transport in Structures with Active Nonlinearities
• Stationary states of a PT-symmetric two-mode Bose-Einstein condensate
• PT-symmetric deformations of integrable models
• Bohmian quantum trajectories from coherent states
• Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• PT-symmetric interpretation of unstable effective potentials
• Complex solitons with real energies
• Time-delay and reality conditions for complex solitons
• Competing PT potentials and re-entrant PT symmetric phase for a particle in a box
• Complex BPS solitons with real energies from duality
• N-site-lattice analogues of $V(x)=i{x}^3$
• PT-symmetrically deformed shock waves
• PT-symmetry in quasi-integrable models
• Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model
• Bloch oscillations in a Bose-Hubbard chain with single-particle losses