TY - JOUR

T1 - Large time behavior for the cubic nonlinear Schrödinger equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

PY - 2002/10

Y1 - 2002/10

N2 - We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension (1) {iut + 1/2uxx + u-3 = 0, t ∈ R, x ∈ R, u(0, x) = u0(x), x ∈ R.} Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data u0 ∈ H1,0 ∩ H0,1 are small and such that sup|ξ| ≤ 1 | arg ℱu0(ξ) - πn/2 | < π/8 for some n ∈ Z, and inf|ξ| ≤ 1 |ℱu0(ξ)| > 0, then the solution has an additional logarithmic time-decay in the short range region |x| ≤ √t. In the far region |x| > √t the asymptotics have a quasilinear character.

AB - We consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimension (1) {iut + 1/2uxx + u-3 = 0, t ∈ R, x ∈ R, u(0, x) = u0(x), x ∈ R.} Cubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data u0 ∈ H1,0 ∩ H0,1 are small and such that sup|ξ| ≤ 1 | arg ℱu0(ξ) - πn/2 | < π/8 for some n ∈ Z, and inf|ξ| ≤ 1 |ℱu0(ξ)| > 0, then the solution has an additional logarithmic time-decay in the short range region |x| ≤ √t. In the far region |x| > √t the asymptotics have a quasilinear character.

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U2 - 10.4153/CJM-2002-039-3

DO - 10.4153/CJM-2002-039-3

M3 - Article

AN - SCOPUS:0036800092

SN - 0008-414X

VL - 54

SP - 1065

EP - 1085

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

IS - 5

ER -