Stationary modes and integrals of motion in nonlinear lattices with PT-symmetric linear part

Abstract

We consider finite-dimensional nonlinear systems with linear part described by a parity-time (PT-) symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of PT-symmetric nonlinearities allowing for existence of the families of nonlinear modes. We pay particular attention to the situations when the underlying linear PT-symmetric operator is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for the system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinear operator.

21 pages, 2 figures; submitted to J. Phys. A: Math. Theor

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• Necessity of PT symmetry for soliton families in one-dimensional complex potentials
• PT-symmetry in optics beyond the paraxial approximation
• Nonlinear modes in a generalized $\mathcal{P}\mathcal{T}$-symmetric discrete nonlinear Schrödinger equation
• Stable two-dimensional soliton complexes in Bose-Einstein condensates with helicoidal spin-orbit coupling
• Nonlinear Topological Edge States in a non-Hermitian Array of Optical Waveguides Embedded in an Atomic Gas
• $\mathcal{P}\mathcal{T}$ symmetry breaking in waveguide arrays with competing loss/gain pairs
• $\mathcal{P}\mathcal{T}$-symmetric coupler with ${\chi }^{(2)}$ nonlinearity
• Exact control of parity-time symmetry in periodically modulated nonlinear optical couplers
• Global existence of solutions to coupled $\mathcal{P}\mathcal{T}$-symmetric nonlinear Schrödinger equations
• Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons