Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case. (April 2015)

• Record Type: Journal Article
• Authors: Hayashi, Nakao; Naumkin, Pavel I.
• Is Part Of: Nonlinear analysis. Volume 116(2015)
• Page Start: 112
• Page End: 131
• Publication Date: 2015-04
• Subjects: Fourth-order nonlinear Schrödinger equation -- Large time asymptotics -- Self similar solution; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2014.12.024
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation { i ∂ t u + 1 4 ∂ x 4 u = i λ ∂ x ( | u | 3 u ), t > 0, x ∈ R, u ( 0, x ) = u 0 ( x ), x ∈ R with critical nonlinearity, where λ ∈ R. We assume that the initial data are such that u 0 ∈ H 1, 1, with sufficiently small norm | u 0 | H 1, 1 . We prove that there exists a unique global solution e − i t 4 ∂ x 4 u ∈ C ( [ 0, ∞ ) ; H 1, 1 ) of the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation such that | u ( t ) | L ∞ ≤ C ( 1 + t ) − 1 4 . Moreover we show that if the total mass 1 2 π ∫ R u 0 ( x ) d x ≠ 0, then the large time asymptotics is determined by the self-similar solution.