TY - JOUR
T1 - Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation
AU - Hayashi, N.
AU - Kaikina, E. I.
AU - Naumkin, P. I.
PY - 2000/12/1
Y1 - 2000/12/1
N2 - 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 ∥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 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.
AB - 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 ∥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 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.
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M3 - Article
AN - SCOPUS:23044522241
VL - 130
SP - 1029
EP - 1043
JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
SN - 0308-2105
IS - 5
ER -