Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

Justin Holmer; Chang Liu

doi:10.3934/cpaa.2020264

Article Contents

2021, Volume 20, Issue 1: 215-242. Doi: 10.3934/cpaa.2020264

This issue Previous Article Spectrum of the Laplacian on regular polyhedra Next Article High-order Wong-Zakai approximations for non-autonomous stochastic

$ p $

-Laplacian equations on

$ \mathbb{R}^N $

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

Justin Holmer^, and
Chang Liu

Brown University, Department of Mathematics, Box 1917,151 Thayer St. Providence, RI 02912, USA

^* Corresponding author
^* Corresponding author

Received: April 2020

Revised: August 2020

Early access: November 2020

Published: January 2021

The first author was supported in part by NSF grants DMS-1200455, DMS-1500106. The second author was supported in part by NSF grant DMS-1200455 (PI Justin Holmer).

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Abstract

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,

$ \begin{equation} i\partial_t\psi + \partial_x^2\psi + \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $

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 cylinder functions (Weber functions) and solving the jump condition at $ x = 0 $ imposed by the $ \delta $ term in (0.1). 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 steepest descent method in the integral formulae for the parabolic cylinder functions.

Keywords:

Mathematics Subject Classification: Primary: 35Q55, 35B44; Secondary: 35C15, 35C20, 33C05.

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Figure 1. (left) plot of numerical solution of $ \sigma(h) $ versus $ h^{-1} $, (right) plot of numerical solution of $ \log \sigma(h) $ versus $ h^{-1} $ in solid black, together with $ h^{-1}\to \infty $ asymptotic $ \log \sigma(h) = \log 2 -\pi h^{-1} + \log h^{-1} $, from Theorem 1.4, in dashed red, showing good agreement for $ h^{-1}>1 $

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Figure 2. Graph of $ S(\omega) = \omega\sqrt{1-\omega^2}+\arcsin \omega $

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Figure 3. The case of $ 0<\omega<1 $. There are two relevant stationary points $ \zeta_\pm = 2\omega( \omega \pm i \sqrt{1-\omega^2}) $. The contour $ i\mathbb{R}_+ $ is deformed to the contour indicated in gray. It starts by going down on the imaginary axis until it picks up a descent curve corresponding stationary point $ \zeta_- $. At $ \zeta_- $, it follows the curve of ascent which is the curve of descent corresponding to $ \zeta_+ $. At $ \zeta_+ $, it follows the descent curve upward until it reaches the imaginary axis. The contour heat plot shows the value of $ \operatorname{Re} \phi(z) $. Along the chosen contour, it reaches its maximum at $ \zeta_+ $

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Figure 4. The case $ \omega>1 $. There is one relevant stationary point $ \zeta = 2\omega (\omega - \sqrt{\omega^2-1}) $ on the real axis. The contour $ i\mathbb{R}_+ $ is deformed to the contour illustrated in purple. Starting at $ 0 $, it proceeds down the negative imaginary axis until it links up with the descent curve corresponding to the stationary point $ \zeta $. Following this curve upward to the positive imaginary axis, it then transitions to the imaginary axis itself to continue to $ +i \infty $. Along this curve, the value of $ \operatorname{Re} \phi(z) $ reaches its maximum value of $ 0 $ at $ z = \zeta $

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Figure 5. The case $ \omega = 1 $. There is one relevant stationary point $ \zeta = 2 $. It is degenerate: $ \phi'(\zeta) = 0 $ and $ \phi''(\zeta) = 0 $ but $ \phi'''(\zeta)\neq 0 $. Thus there are three descent curves and three ascent curves emanating from the stationary point, alternating and separated from each other by $ 60 $ degrees. The desired contour follows two descent arcs into the stationary point, as indicated. The value of $ \operatorname{Re} \phi(z) $ reaches its maximum value of $ 0 $ at $ z = \zeta $

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