Exact envelope-soliton solutions of a nonlinear wave equation

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.