Blow-up for the 1D nonlinear Schr\"odinger equation with point nonlinearity II: Supercritical blow-up profiles

TL;DR

This paper constructs self-similar blow-up solutions for the 1D nonlinear Schrödinger equation with point nonlinearity in the supercritical regime, using special functions and algebraic conditions to analyze existence and uniqueness.

Contribution

It introduces a method to find explicit self-similar blow-up profiles for supercritical point nonlinear Schrödinger equations, employing Weber functions and gamma function analysis.

Findings

Constructed explicit blow-up solutions in the energy space

Derived algebraic jump conditions involving gamma functions

Analyzed solutions in the slightly supercritical case using asymptotic methods

Abstract

We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0$$ where $\delta=\delta(x)$ is the delta function supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap \dot H_x^1$. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at $x=0$ imposed by the $\delta$ term. This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case $0<p-3 \ll 1$ using the log Binet formula for the gamma function, and contour deformation and stationary phase/Laplace method in the integral formulae for the parabolic Weber functions.