Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case. (April 2015) - Hayashi, Nakao; Naumkin, Pavel I.
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.,