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

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

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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.

元の言語English
ページ(範囲)1029-1043
ページ数15
ジャーナルRoyal Society of Edinburgh - Proceedings A
130
発行部数5
出版物ステータスPublished - 2000 12 1
外部発表Yes

ASJC Scopus subject areas

  • Mathematics(all)

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