Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory

Abstract

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, $$(\delta \text{NLS})i{\partial }_t\psi +{\partial }_x^2\psi +\delta |\psi {|}^{p-1}\psi =0,$$ where $\delta =\delta (x)$ is the delta function supported at the origin. We show that $\delta$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$(\text{NLS})i{\partial }_t\psi +\Delta \psi +|\psi {|}^{p-1}\psi =0.$$ The critical Sobolev space ${\dot{H}}^{{\sigma }_c}$ for $\delta$NLS is ${\sigma }_c=\frac{1}{2}-\frac{1}{p-1}$, whereas for NLS it is ${\sigma }_c=\frac{d}{2}-\frac{2}{p-1}$. In particular, the $L^2$ critical case for $\delta$NLS is $p=3$. We prove several results pertaining to blow-up for $\delta$NLS that correspond to key classical results for NLS. Specifically, we (1) obtain a sharp Gagliardo-Nirenberg inequality analogous to Weinstein (1983), (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein (1983), Glassey (1977) in the case ${\sigma }_c=0$ and Duyckaerts, Holmer, & Roudenko (2008), Guevara (2014), and Fang, Xie, & Cazenave (2011) for $0<{\sigma }_c<1$, (3) prove a sharp mass concentration result in the $L^2$ critical case analogous to Tsutsumi (1990), Merle & Tsutsumi (1990) and (4) show that minimal mass blow-up solutions in the $L^2$ critical case are pseudoconformal transformations of the ground state, analogous to Merle (1993).

22 pages, 1 figure

References in corpus (1)

• Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schrödinger Equations

Cited by in corpus (8)

• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• Blow-up for the pointwise NLS in dimension two: absence of critical power
• Time dependent delta-prime interactions in dimension one
• The 1-D Dirac equation with concentrated nonlinearity
• Mathematical Study of Scattering Resonances
• Scattering for the $L^2$ supercritical point NLS
• Blow-up and scattering for the 1D NLS with point nonlinearity above the mass-energy threshold