Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation (Formula Presented) where ℒ is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C₁|ξ|²/(1 + ξ²) and |L′(ξ)| ≤ C₂(|ξ| + |ξ|ⁿ) for all ξ ∈R. Here, C₁,C₂ > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ² + O(|ξ|^2+γ) for all |ξ| < 1, where γ > 0, Reα > 0, Im α ≥ 0. When L(ξ) = αξ², equation (A) is the nonlinear Schrödinger equation with dissipation u_t - αu_xx + i|u|²u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate ∥u(t)∥_∞ ≤ C(1 + t)^-1/2(1 + log(1 + t))^-1/2σ under the conditions that u₀ ∈ H^n,0 ∩ H^0,1 have the mean value û₀(0) = 1/√2π ∫ u₀(x) dx ≠ 0 and the norm ∥u₀∥_Hn,0 + ∥u₀∥_H0,1 = ε is sufficiently small, where σ= 1 if Im α > 0 and σ = 2 if Im α = 0, and H^m,s = {φ ∈ S′; ∥φ∥_m,s = ∥(1 + x²)^s/2(1 - ∂_x²)^m/2φ∥ < ∞}, m,s ∈ R. Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation u_t - αu_xx + i|u|^p-1u = 0, with p > 3 have the same time decay estimate ∥u∥_L∞ = O(t^-1/2) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.