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

Author： Hayashi N. Kaikina E. I. Naumkin P. I.

ISSN： 1473-7124

Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.130, Iss.5, 2000-10, pp. : 1029-1043

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Abstract

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipationu_t + Lu + i|u|²u = 0, x ∈ R, t > 0,u(0,x) = u0(x), x ∈ R,(A)where L is a linear pseudodifferential operator with dissipative symbolReL(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, Imα ≥ 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))^-1/2σunder the conditions that u₀ ∈ H^n,0 ∩ H^0,1 have the mean valueû₀(0) = (1/√(2π)) ∫ u₀(x)dx ≠ 0and the norm ||u₀||_Hn,0 + ||u₀||_H0,1 = ε is sufficiently small, where σ = 1 if Imα > 0 and σ = 2 if Imα = 0, andH^m,s = {&phis; ∈ S′; ||&phis;||m,s = ||(1 + x²)^s/2(1 - ∂²_x)^m/2&phis;|| < ∞}, 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 equationu_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.