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A new class of critical solutions for 1D cubic NLS

Anatole Guérin

TL;DR

This work constructs a new class of solutions to the 1D cubic NLS with initial data given by a sum of Dirac masses at the critical Fourier-Lebesgue regularity $\mathcal{F}(L^\infty)$ and in $\dot H^s$ for $s<-\tfrac12$. The authors develop a scattering framework around a multi-Dirac-mass profile by applying a Wick renormalization and a pseudo-conformal transform, then perform a detailed fixed-point analysis of a Duhamel-type integral around a leading term $v_1$, with an oscillatory-integral-driven decomposition into terms $I(v)$ and $J_\ast$. They prove the existence and decay of the perturbation $v-v_1$ in a weighted Sobolev space, and, after reversing the transform, establish precise convergence rates for the original solution $u$ toward the Dirac-mass profile $u_{\{\alpha_j\}}$, including refined $J^k$-weighted asymptotics. The approach avoids reliance on Strichartz estimates, instead leveraging sharp oscillatory integral estimates to control slowest-decaying nonlinear interactions, yielding a robust stability/line-scattering result in the critical setting.

Abstract

The aim of this article is to prove the existence of a new class of solutions of 1D cubic NLS with an initial data related to a sum of Dirac masses, of critical regularity $F(L^\infty)$, and belonging to $\dot H^s$ for any $s <-1/2$. This problem is motivated by the lack of result for critical regularity initial condition. Our result is based on a scattering approach, after performing a pseudo-conformal transformation, and on fine estimations of oscillatory integrals.

A new class of critical solutions for 1D cubic NLS

TL;DR

This work constructs a new class of solutions to the 1D cubic NLS with initial data given by a sum of Dirac masses at the critical Fourier-Lebesgue regularity and in for . The authors develop a scattering framework around a multi-Dirac-mass profile by applying a Wick renormalization and a pseudo-conformal transform, then perform a detailed fixed-point analysis of a Duhamel-type integral around a leading term , with an oscillatory-integral-driven decomposition into terms and . They prove the existence and decay of the perturbation in a weighted Sobolev space, and, after reversing the transform, establish precise convergence rates for the original solution toward the Dirac-mass profile , including refined -weighted asymptotics. The approach avoids reliance on Strichartz estimates, instead leveraging sharp oscillatory integral estimates to control slowest-decaying nonlinear interactions, yielding a robust stability/line-scattering result in the critical setting.

Abstract

The aim of this article is to prove the existence of a new class of solutions of 1D cubic NLS with an initial data related to a sum of Dirac masses, of critical regularity , and belonging to for any . This problem is motivated by the lack of result for critical regularity initial condition. Our result is based on a scattering approach, after performing a pseudo-conformal transformation, and on fine estimations of oscillatory integrals.
Paper Structure (7 sections, 8 theorems, 96 equations)

This paper contains 7 sections, 8 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['stab']}
  3. Fixed point functional
  4. Estimates on the fixed point argument terms
  5. The stability
  6. The contraction
  7. End of the proof: estimates \ref{['decayschr0']} and \ref{['decay0']}

Key Result

Theorem 1.1

Let $s\in\mathbb N^*$, $(\alpha_j)\in l^{2,q}$ with $q-s>\frac{1}{2}$, $u_+ \in H^s\cap \dot H^{-2} \cap W^{1,s}$ such that Let $u_{\{\alpha_j\}}$ the solution of eq defined in solA on $[0,T]$. Then, if $\|\alpha_j\|_{l^{2,q}}$ is small enough, there exists $T_1<T$ depending on $\|\alpha_j\|_{l^{2,q}}$ and $u_+$, and there exists $u$ a unique solution of eq on $(0,T_1]$, such that: with the foll

Theorems & Definitions (15)

  • Theorem 1.1
  • Lemma 2.1: Estimation of $I(v)$
  • proof
  • Lemma 2.2: Estimation of $\nabla^kI(v)$
  • proof
  • Lemma 2.3: Estimation of $\nabla^k J_a$
  • proof
  • Lemma 2.4: Estimation of $\nabla^k J_b$
  • proof
  • Lemma 2.5: Estimation of $\nabla^k J_c$
  • ...and 5 more