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A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation

Gonzalo Cao-Labora, Javier Gómez-Serrano, Jia Shi, Gigliola Staffilani

TL;DR

This work investigates self-similar blow-up profiles for the focusing nonlinear Schrödinger equation by mapping it to a hydrodynamic system through the Madelung transform. The authors derive an autonomous ODE system for radially symmetric self-similar variables and construct solutions via a convergent power-series expansion around $\xi=-\infty$, valid for mass-supercritical exponents where $s_c>0$ (i.e., $p>1+4/d$). The main theorem proves existence of a smooth self-similar profile under the condition $\alpha d>r-1$ with $\alpha=(p-1)/4$, including parity and decay properties, and shows the solution decays to $(0,0)$ as the similarity variable grows. The results provide candidate blow-up profiles for the focusing NLS in the mass-supercritical regime and extend the defocusing-inspired methodology to the focusing setting, potentially informing stability and instability analyses.

Abstract

After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain dissipation. In this short note, we construct self-similar solutions of such system in the focusing case for any mass supercritical exponent. To the best of our knowledge these solutions are new, and may formally arise as potential blow-up profiles of the focusing NLS equation.

A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation

TL;DR

This work investigates self-similar blow-up profiles for the focusing nonlinear Schrödinger equation by mapping it to a hydrodynamic system through the Madelung transform. The authors derive an autonomous ODE system for radially symmetric self-similar variables and construct solutions via a convergent power-series expansion around , valid for mass-supercritical exponents where (i.e., ). The main theorem proves existence of a smooth self-similar profile under the condition with , including parity and decay properties, and shows the solution decays to as the similarity variable grows. The results provide candidate blow-up profiles for the focusing NLS in the mass-supercritical regime and extend the defocusing-inspired methodology to the focusing setting, potentially informing stability and instability analyses.

Abstract

After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain dissipation. In this short note, we construct self-similar solutions of such system in the focusing case for any mass supercritical exponent. To the best of our knowledge these solutions are new, and may formally arise as potential blow-up profiles of the focusing NLS equation.

Paper Structure

This paper contains 10 sections, 1 theorem, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical Background
  3. Setup of the problem
  4. Self-similar equations
  5. Main result
  6. Acknowledgements
  7. Proof of the Main Theorem
  8. Expansion of the solution as a series
  9. Convergence of the series
  10. Existence of self-similar solutions

Key Result

Theorem 1.1

Assume that $\alpha d > r-1$, $\alpha=\frac{p-1}{4}$, with $p$ an odd natural number. Then there exists a smooth solution $( U_R(\zeta ), S(\zeta ))$ to eq:SSeq_zeta with decay at infinity. Moreover, $U_R$ admits a $C^\infty$ odd extension and $S$ admits a $C^\infty$ even extension with respect to

Figures (1)

  • Figure 1: Green: Taylor series expansion starting from $P_0$. The trajectory will reach $(0,0)$. Red: barrier $\overline U = -1$. The solution needs to start from $(\overline U, \overline S) = (\overline U_0, \infty)$ with $\overline U_0 > -1$. There will exist values of $r > 2$ such that this happens if and only if $(p, d)$ lie in the mass supercritical regime.

Theorems & Definitions (5)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5