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Small-amplitude self-similar solutions for one-dimensional nonlinear dispersive equations

Simão Correia, Gonçalo Pereira, Thyago S. R. Santos

TL;DR

The work addresses the existence of small-amplitude self-similar solutions for 1D scale-invariant dispersive equations, including the cubic NLS, mBO, DNLS, and 4KdV. It develops a robust Fourier-analytic framework that absorbs divergent high-frequency interactions through explicit ansätze (e.g., $S_A$ and $S_{A,a,B}$) and employs fixed-point theory in Fourier-Lebesgue-type spaces, supported by detailed multilinear estimates. The main contributions are the construction of self-similar profiles for all four models with precise frequency-space asymptotics, the establishment of data-to-scattering maps linking low- and high-frequency data, and the demonstration that these maps yield unique small-amplitude self-similar solutions. The results illuminate the role of self-similar structures in dispersive dynamics, providing a path toward finite-energy self-similar solutions and informing long-time behavior and potential blow-up scenarios in 1D systems.

Abstract

Given a nonlinear dispersive equation which admits a scaling invariance, there may exist self-similar solutions. In this work, we present a systematic approach for the construction of small-amplitude self-similar solutions, together with precise asymptotic descriptions at both small and large frequency scales. These ideas are then applied to three classic dispersive models: the modified Benjamin-Ono, the quartic Korteweg-de Vries and the cubic nonlinear Schrödinger equations.

Small-amplitude self-similar solutions for one-dimensional nonlinear dispersive equations

TL;DR

The work addresses the existence of small-amplitude self-similar solutions for 1D scale-invariant dispersive equations, including the cubic NLS, mBO, DNLS, and 4KdV. It develops a robust Fourier-analytic framework that absorbs divergent high-frequency interactions through explicit ansätze (e.g., and ) and employs fixed-point theory in Fourier-Lebesgue-type spaces, supported by detailed multilinear estimates. The main contributions are the construction of self-similar profiles for all four models with precise frequency-space asymptotics, the establishment of data-to-scattering maps linking low- and high-frequency data, and the demonstration that these maps yield unique small-amplitude self-similar solutions. The results illuminate the role of self-similar structures in dispersive dynamics, providing a path toward finite-energy self-similar solutions and informing long-time behavior and potential blow-up scenarios in 1D systems.

Abstract

Given a nonlinear dispersive equation which admits a scaling invariance, there may exist self-similar solutions. In this work, we present a systematic approach for the construction of small-amplitude self-similar solutions, together with precise asymptotic descriptions at both small and large frequency scales. These ideas are then applied to three classic dispersive models: the modified Benjamin-Ono, the quartic Korteweg-de Vries and the cubic nonlinear Schrödinger equations.

Paper Structure

This paper contains 27 sections, 26 theorems, 427 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Previous works
  4. Main results
  5. Overview of the method
  6. Review of the \ref{['mKdV']} case.
  7. The \ref{['4Kdv']} case
  8. The \ref{['dnls']} case.
  9. The \ref{['3NLS']} case
  10. Future perspectives
  11. Small self-similar solutions for 1d dispersive equations.
  12. Finite-energy self-similar solutions
  13. Self-similar solutions for 4KdV
  14. Self-similar solutions for mBO
  15. Self-similar solutions for NLS
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $\kappa \in \left(\frac{5}{8}, \frac{2}{3}\right)$ and $c \in \mathbb{C}$ with $|c| \ll 1$. Then, there exist a constant $A = A(c) \in \mathbb{C}$ and a tempered distribution $V_c \in \mathcal{S}^\prime$ such that the function is a (real-valued) self-similar solution to 4Kdv. Furthermore, the function $V_c$ is characterized by the relation where and the remainder term $z(\xi)= \overline{z(-

Theorems & Definitions (51)

  • Theorem 1.2: Self-similar solutions for \ref{['4Kdv']}
  • Theorem 1.3: Self-similar solutions for \ref{['dnls']}
  • Theorem 1.4: Self-similar solutions for \ref{['3NLS']}
  • Remark 1.1
  • Example 1.2: Generalized Korteweg–de Vries equation
  • Example 1.3: Nonlinear Schrödinger equation
  • Example 1.4: Generalized Benjamin–Ono equation
  • Definition 1.5: Self-similar solution for \ref{['eq dispersiva']}
  • Theorem 2.1: Multilinear Estimates
  • Theorem 2.1
  • ...and 41 more