Exact envelope-soliton solutions of a nonlinear wave equation

Abstract

Exact N‐envelope‐soliton solutions have been obtained for the following nonlinear wave equation, i∂ψ/∂t + i3α|ψ|² ∂ψ/∂x + β∂²ψ/∂x² + iγ∂³ψ/∂x³ + δ|ψ|²ψ = 0, where α, β, γ and δ are real positive constants with the relation αβ = γδ. In one limit of α = γ = 0, the equation reduces to the nonlinear Schrödinger equation which describes a plane self‐focusing and one‐dimensional self‐modulation of waves in nonlinear dispersive media. In another limit, β = δ = 0, the equation for real Ψ, reduces to the modified Korteweg‐de Vries equation. Hence, the solutions reveal the close relation between classical solitons and envelope‐solitons.