Asymptotic behavior for large time of solutions to the nonlinear nonlocal Schrödinger equation on half-line

Source abstract

We study the following initial-boundary value problem for the nonlinear nonlocal Schrödinger equation (NNS) ut+N(u)+bu+Ku=0 (t,x)∈R+×R+u(0,x)=u¯(x), x>0, ∂xj−1u(0,t), t>0, for j=1,…,n, with the compatibility condition ∂xj−1u¯(0)=u~j(0),j=1,2…,n, where n=[ α2 ],[ s ] denotes the largest integer less than s,b≥0, the nonlinear term N(u)=ia(t)|u|ρu,ρ>1, the coefficient a(t)∈C1 and K is the pseudodifferential operator on the half line R+ of order α>1. We prove that if xδu¯∈L1, with 0<δ<12 and the norm ‖u¯‖X of the initial data and the norms ‖u~j‖Y, j=1,…n of the boundary data are sufficiently small then there exists a unique solution u∈C([0,∞);L2)∩C(R+;W2[α]−1∩C[α]−1) of the initial-value problem (NNS). Here X=W∞[α]∩W1[α]+1 and Y=W∞1∩W12 and Wpk is the Sobolev space with the norm ‖ϕ‖Wpk=‖(1−∂x2)k/2ϕ(x)‖ Lp. We also find the large time asymptotics of the solutions.