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$\mathrm{𝒫𝒯}$-symmetry breaking in complex nonlinear wave equations and their deformations. (English) Zbl 1226.81090
Summary: We investigate complex versions of the Korteweg-de Vries equations and an Ito-type nonlinear system with two coupled nonlinear fields. We systematically construct rational, trigonometric/hyperbolic and elliptic solutions for these models including those which are physically feasible in an obvious sense, that is, those with real energies, but also those with complex energy spectra. The reality of the energy is usually attributed to different realizations of an antilinear symmetry, as for instance $\mathrm{𝒫𝒯}$-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 $\mathrm{𝒫𝒯}$-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 previously-studied complex quantum mechanical models.
##### MSC:
 81R40 Symmetry breaking (quantum theory) 81Q05 Closed and approximate solutions to quantum-mechanical equations 35Q55 NLS-like (nonlinear Schrödinger) equations 81R12 Relations of groups and algebras in quantum theory with integrable systems 35Q53 KdV-like (Korteweg-de Vries) equations 34C23 Bifurcation (ODE)