On non-existence of continuous families of stationary nonlinear modes for a class of complex potentials

Abstract

There are two cases when the nonlinear Schrödinger equation (NLSE) with an external complex potential is well-known to support continuous families of localized stationary modes: the $\mathcal{P}\mathcal{T}$-symmetric potentials and the Wadati potentials. Recently Y. Kominis and coauthors [Chaos, Solitons and Fractals, 118, 222-233 (2019)] have suggested that the continuous families can be also found in complex potentials of the form $W(x)=W_1(x)+iC{W}_{1,x}(x)$, where $C$ is an arbitrary real and $W_1(x)$ is a real-valued and bounded differentiable function. Here we study in detail nonlinear stationary modes that emerge in complex potentials of this type (for brevity, we call them W-dW potentials). First, we assume that the potential is small and employ asymptotic methods to construct a family of nonlinear modes. Our asymptotic procedure stops at the terms of the ${\epsilon }^2$ order, where small $\epsilon$ characterizes amplitude of the potential. We therefore conjecture that no continuous families of authentic nonlinear modes exist in this case, but "pseudo-modes" that satisfy the equation up to ${\epsilon }^2$-error can indeed be found in W-dW potentials. Second, we consider the particular case of a W-dW potential well of finite depth and support our hypothesis with qualitative and numerical arguments. Third, we simulate the nonlinear dynamics of found pseudo-modes and observe that, if the amplitude of W-dW potential is small, then the pseudo-modes are robust and display persistent oscillations around a certain position predicted by the asymptotic expansion. Finally, we study the authentic stationary modes which do not form a continuous family, but exist as isolated points. Numerical simulations reveal dynamical instability of these solutions.

title changed; several corrections made; 20 pages, 9 figures; accepted for Studies in Applied Mathematics

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