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<p>After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain forcing. 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.</p>

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A Note on the Existence of Self-Similar Profiles of the Hydrodynamic Formulation of the Focusing Nonlinear Schrödinger Equation

  • 期刊论文

  • 发表日期：

    2026年1月8日

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

摘要

After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain forcing. 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.