TY - JOUR

T1 - Subcritical quadratic nonlinear schrödinger equation

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

N1 - Funding Information:
We are very grateful to an unknown referee for many useful suggestions and comments. The work of N. Hayashi was partially supported by KAKENHI (No. 19340030) and the work of P. I. Naumkin was partially supported by CONACYT and PAPIIT.

PY - 2011/12

Y1 - 2011/12

N2 - We study the initial value problem for the quadratic nonlinear Schrödinger equation iut+1/2uxx=-4iπtγ-1/2u 2, x R, t > u(1,x) = u 1 (x), x ε R, where γ > 0. Suppose that the Fourier transform û 1 of the initial data u 1 satisfies estimates û1 L∞ ≤ ε, d/d (ei/2ε 2û1(ε))L∞,1 ≤ ε> 0 is sufficiently small. Also suppose that Re(eu/2ε 2 û1(ε))≥Cε5/4 for |ξ| ≤ 1. Assume that γ > 0 is small: γ = O(5/4). Then we prove that there exists a unique solution u ∈ C([1, ∞);L 2) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equation (*).

AB - We study the initial value problem for the quadratic nonlinear Schrödinger equation iut+1/2uxx=-4iπtγ-1/2u 2, x R, t > u(1,x) = u 1 (x), x ε R, where γ > 0. Suppose that the Fourier transform û 1 of the initial data u 1 satisfies estimates û1 L∞ ≤ ε, d/d (ei/2ε 2û1(ε))L∞,1 ≤ ε> 0 is sufficiently small. Also suppose that Re(eu/2ε 2 û1(ε))≥Cε5/4 for |ξ| ≤ 1. Assume that γ > 0 is small: γ = O(5/4). Then we prove that there exists a unique solution u ∈ C([1, ∞);L 2) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equation (*).

KW - Asymptotics of solutions

KW - quadratic nonlinear Schrödinger equation

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U2 - 10.1142/S021919971100452X

DO - 10.1142/S021919971100452X

M3 - Article

AN - SCOPUS:84855240671

SN - 0219-1997

VL - 13

SP - 969

EP - 1007

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 6

ER -