Forced nonlinear Schrödinger equation with arbitrary nonlinearity

Abstract

We consider the nonlinear Schrödinger equation (NLSE) in 1+1 dimension with scalar-scalar self-interaction g 2 κ + 1 ( ψ ☆ ψ ) κ + 1 in the presence of the external forcing terms of the form r e − i ( k x + θ ) − δ ψ . We find new exact solutions for this problem and show that the solitary wave momentum is conserved in a moving frame where v k = 2 k . These new exact solutions reduce to the constant phase solutions of the unforced problem when r → 0 . In particular we study the behavior of solitary wave solutions in the presence of these external forces in a variational approximation which allows the position, momentum, width, and phase of these waves to vary in time. We show that the stationary solutions of the variational equations include a solution close to the exact one and we study small oscillations around all the stationary solutions. We postulate that the dynamical condition for instability is that d p ( t ) / d ˙ q ( t ) < 0 , where p ( t ) is the normalized canonical momentum p ( t ) = 1 M ( t ) ∂ L ∂ ˙ q , and ˙ q ( t ) is the solitary wave velocity. Here M ( t ) = ∫ d x ψ ☆ ( x , t ) ψ ( x , t ) . Stability is also studied using a “phase portrait” of the soliton, where its dynamics is represented by two-dimensional projections of its trajectory in the four-dimensional space of collective coordinates. The criterion for stability of a soliton is that its trajectory is a closed single curve with a positive sense of rotation around a fixed point. We investigate the accuracy of our variational approximation and these criteria using numerical simulations of the NLSE. We find that our criteria work quite well when the magnitude of the forcing term is small compared to the amplitude of the unforced solitary wave. In this regime the variational approximation captures quite well the behavior of the solitary wave.