Exact Soliton Solution for Superfluid Film Dynamics

Traveling wave solutions are examined for the nonlinear wave equation i∂ψ/ ∂t=-∂²ψ/∂x² + [-(1+d/1+d|ψ|²)²]ψ, which describes the dynamics of condensate wave function ψ(x, t) in superfluid film of mean thickness d. An exact one-soliton solution is obtained analytically for arbitrary amplitude, and this suggests that the "quasi-solitons" found in the previous numerical work are stable at least in the asymptotic situation where quasi-solitons are essentially non-overlapping. It is shown explicitly that our solution reduces, in small amplitude regime, to the Korteweg-de Vries one-soliton solution.