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

N. Hayashi, E. I. Kaikina, P. I. Naumkin

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

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 ut - αuxx + 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 ∈ Hn,0 ∩ H0,1 have the mean value û0(0) = 1/√2π ∫ u0(x) dx ≠ 0 and the norm ∥u0Hn,0 + ∥u0H0,1 = ε is sufficiently small, where σ= 1 if Im α > 0 and σ = 2 if Im α = 0, and Hm,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 ut - αuxx + 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.

Original languageEnglish
Pages (from-to)1029-1043
Number of pages15
JournalRoyal Society of Edinburgh - Proceedings A
Volume130
Issue number5
Publication statusPublished - 2000 Dec 1
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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