Approximation of the DNLS equation by the cubic nonlinear Schrödinger equation

Abstract

We rigorously approach the Schrödinger equation of derivative type q_t + iq_xx + λ |q|²q_x + μq²q̄_x = 0, λ ∈ R, μ ∈ C, by the cubic nonlinear Schrödinger equation A_T + i A_XX + i k₀ (λ − μ) |A|²A = 0. We also study the case of the KdV-like equation q_t + i q_xx + aq_xxx + i |q|²q + λ̃ (|q|²q)_x + μ̃ |q|²q_x = 0, λ̃ μ̃ ∈ R, arising in optical physics.