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Self-similar blowup for the cubic Schrödinger equation

Roland Donninger, Birgit Schörkhuber

TL;DR

The authors prove the existence of a nontrivial, finite-energy self-similar blowup solution for the 3D focusing cubic NLS by constructing an accurate, radially symmetric self-similar profile $Q$ and a corresponding self-similar solution $\\psi_*(t,x)$. They develop a rigorous, computer-assisted fixed-point argument around a numerically computed approximation, combining a Chebyshev pseudo-spectral method with fraction-precise arithmetic to obtain explicit, verifiable bounds. The core innovation lies in formulating a robust abstract framework with admissible left/right fundamental matrices, a regularized inverse, and a carefully designed function space pair $(X,Y)$ that allow contraction despite the problem’s unbounded domain and symmetry-induced noninvertibility. This approach yields both qualitative and quantitative evidence that the constructed profile matches the numerically observed universal blowup shape, contributing a rigorous foundation for the universality of the breakdown pattern in the 3D cubic NLS. The methodology blends deep analytic structure with exact, reproducible numerics, offering a blueprint for rigorous computer-assisted proofs in nonlinear dispersive equations.

Abstract

We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schrödinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows the existence of a solution in the vicinity of a numerically constructed approximation. The latter is obtained by a standard pseudo-spectral method. The computer-assisted part of the rigorous proof uses nothing but fraction arithmetic in order to obtain quantitative bounds for the fixed point argument.

Self-similar blowup for the cubic Schrödinger equation

TL;DR

The authors prove the existence of a nontrivial, finite-energy self-similar blowup solution for the 3D focusing cubic NLS by constructing an accurate, radially symmetric self-similar profile and a corresponding self-similar solution . They develop a rigorous, computer-assisted fixed-point argument around a numerically computed approximation, combining a Chebyshev pseudo-spectral method with fraction-precise arithmetic to obtain explicit, verifiable bounds. The core innovation lies in formulating a robust abstract framework with admissible left/right fundamental matrices, a regularized inverse, and a carefully designed function space pair that allow contraction despite the problem’s unbounded domain and symmetry-induced noninvertibility. This approach yields both qualitative and quantitative evidence that the constructed profile matches the numerically observed universal blowup shape, contributing a rigorous foundation for the universality of the breakdown pattern in the 3D cubic NLS. The methodology blends deep analytic structure with exact, reproducible numerics, offering a blueprint for rigorous computer-assisted proofs in nonlinear dispersive equations.

Abstract

We give a rigorous proof for the existence of a finite-energy, self-similar solution to the focusing cubic Schrödinger equation in three spatial dimensions. The proof is computer-assisted and relies on a fixed point argument that shows the existence of a solution in the vicinity of a numerically constructed approximation. The latter is obtained by a standard pseudo-spectral method. The computer-assisted part of the rigorous proof uses nothing but fraction arithmetic in order to obtain quantitative bounds for the fixed point argument.

Paper Structure

This paper contains 56 sections, 62 theorems, 442 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History of the problem
  3. Related results
  4. Computer-assisted proofs
  5. Idea of proof
  6. Setup of the problem
  7. Difficulties
  8. Strategy of the proof
  9. Function spaces
  10. Computer assistance
  11. Notation
  12. Structure of the technical part
  13. Theoretical framework
  14. Function spaces and basic properties
  15. Perturbative decomposition
  16. ...and 41 more sections

Key Result

Theorem 1.1

There exists a nontrivial, radial function $Q\in L^4(\mathbb{R}^3)\cap \dot H^1(\mathbb{R}^3)\cap C^\infty(\mathbb{R}^3)$ and an $\alpha>0$ such that $\psi_*: (-\infty,1)\times \mathbb{R}^3\to\mathbb{C}$, defined by satisfies for all $(t,x)\in (-\infty,1)\times \mathbb{R}^3$. In particular, $Q$ satisfies the profile equation for all $\xi\in \mathbb{R}^3$.

Figures (1)

  • Figure 1: A plot of $|Q_*(re_1)|$ for $r\in [0,10]$, where $Q_*: \mathbb{R}^3\to\mathbb{C}$ is given by $Q_*(x):=(1+|x|)^{-1-\frac{i}{a_*}}g_*(|x|,0),$see Definition \ref{['def:gstar']}. The function $Q_*$ is an approximation to the profile $Q$ of Theorems \ref{['thm:qual']} and \ref{['thm:quant']}.

Theorems & Definitions (203)

  • Theorem 1.1: Main theorem, qualitative version
  • Definition 1.2: Chebyshev polynomials
  • Definition 1.3
  • Theorem 1.4: Main theorem, quantitative version
  • Remark 1.5
  • Remark 1.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 193 more