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
VL - 54
SP - 1065
EP - 1085
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
SN - 0008-414X
IS - 5
ER -