2015/10/13 by Justin Holmer, Chang Liu, Holmer, Justin +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1510.03491
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, (δNLS) i∂tψ+ ∂x2ψ+ δ|ψ|p-1ψ= 0, where δ=δ(x) is the delta function supported at the origin. We show that δNLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS (NLS) i∂t ψ+ Δψ+ |ψ|p-1ψ=0 . The critical Sobolev space Hσc for δNLS is σc=\frac12-(1)/(p-1), whereas for NLS it is σc=(d)/(2)-(2)/(p-1). In particular, the L2 critical case for δNLS is p=3. We prove several results pertaining to blow-up for δNLS 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 σc=0 and Duyckaerts, Holmer, & Roudenko (2008), Guevara (2014), and Fang, Xie, & Cazenave (2011) for 0