On the new critical exponent for the nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension iut + 1 2 uxx = |u| p , x ∈ R, t > 0, u (0, x) = u0 (x) , x ∈ R, where p > ps = 3+ √ 17 2. We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if ps < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4.

Some non-existence results for the semilinear Schrödinger equation without gauge invariance

“…On the other hand, ill-posedness to (NLS) has not been less understood, though in the case F (z) = c 1 z 2 + c 2z 2 for some c 1 , c 2 ∈ C\{0}, ill-posedness results were studied in [1,9,37]. For more information about (NLS) with F (z) = |z| p , see [15,[23][24][25]42,43]. At the end of this section, we introduce some function spaces and notations used throughout this paper.…”

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

“…As far as we know, was widely used for the study of the large time asymptotic behavior of solutions to the nonlinear Schrödinger equations with a gauge invariant nonlinearity such that i∂ t u + 1 2 Δu = |u| p−1 u, (t, x) ∈ R + × R n , u (0, x) = u 0 (x) , x ∈ R n , (1.3) where it works well. However it is known from [7] that the operator e…”