Variational Approach to studying solitary waves in the nonlinear Schrodinger equation with Complex Potentials

Abstract

We discuss the behavior of solitary wave solutions of the nonlinear Schr{ö}dinger equation (NLSE) as they interact with complex potentials, using a four parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope of $dp(t)/dv(t)$ becomes negative when plotted parametrically as a function of time, where $p(t)$ is the momentum of the solitary wave and $v(t)$ the velocity.

19 pages, 10 figures

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Cited by in corpus (6)

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• Symmetry breaking bifurcations and excitations of solitons in linearly coupled NLS equations with PT-symmetric potentials
• A Solvable Model of a Nonlinear extension of Quantum Mechanics
• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• Behavior of solitary waves of coupled nonlinear Schrödinger equations subjected to complex external periodic potentials with anti-$\mathcal{P}\mathcal{T}$ symmetry
• Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential