Minimal mass blow-up solutions for the $L^2$-critical NLS with the Delta potential for radial data in one dimension

Abstract

We consider the $L^2$-critical nonlinear Schrödinger equation (NLS) with the delta potential $$i{\partial }_{t}u+{\partial }_x^{2}u+\mu \delta u+|u{|}^{4}u=0,t\in \mathbb{R},x\in \mathbb{R},$$ where $\mu \in \mathbb{R}$, and $\delta$ is the Dirac delta distribution at $x=0$. Local well-posedness theory together with sharp Gagliardo-Nirenberg inequality and the conservation laws of mass and energy implies that the solution with mass less than $\Vert Q{\Vert }_2$ is global existence in $H^1(\mathbb{R})$, where $Q$ is the ground state of the $L^2$-critical NLS without the delta potential (i.e. $\mu =0$). We are interested in the dynamics of the solution with threshold mass $\Vert u_0{\Vert }_2=\Vert Q{\Vert }_2$ in $H^1(\mathbb{R})$. First, for the case $\mu =0$, such blow-up solution exists due to the pseudo-conformal symmetry of the equation, and is unique up to the symmetries of the equation in $H^1(\mathbb{R})$ from \cite{Me93:NLS:mini sol} (see also \cite{HmKe05:NLS:mini blp}), and recently in $L^2(\mathbb{R})$ from \cite{Dod:NLS:L2thrh1}. Second, for the case $\mu <0$, simple variational argument with the conservation laws of mass and energy implies that radial solutions with threshold mass exist globally in $H^1(\mathbb{R})$. Last, for the case $\mu >0$, we show the existence of radial threshold solutions with blow-up speed determined by the sign (i.e. $\mu >0$) of the delta potential perturbation since the refined blow-up profile to the rescaled equation is stable in a precise sense. The key ingredients here including the Energy-Morawetz argument and compactness method as well as the modulation analysis are close to the original one in \cite{RaS11:NLS:mini sol} (see also \cite{KrLR13:HalfW:nondis, LeMR:CNLS:blp, Mart05:Kdv:N sol, MaP17:BO:mini sol, MeRS14:NLS:blp}).

39 pages, 3 figures. Welcome any comment and suggestion. arXiv admin note: text overlap with arXiv:1406.6002 by other authors

References in corpus (2)

• Fast soliton scattering by delta impurities
• Blow up for the critical gKdV equation III: exotic regimes