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Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schrödinger system

Xing Cheng, Zuyu Ma, Jiqiang Zheng

Abstract

In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schrödinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schrödinger system, we present a complete classification of minimal-mass blowup solutions. Specifically, we demonstrate that all such solutions must be either solitons or their pseudo-conformal transformations. To prove this result, we develop a modulation analysis that accounts for multi-component interactions to overcome the multiply phase transformations caused by the multi-component. A long time Strichartz estimate for vector-valued solutions is established to solve the difficulty posed by the Galilean transformations and spatial translation, where a new vector-valued bilinear estimate is proven to address the challenges caused by the coupled nonlinear interaction. For the infinitely coupled focusing nonlinear Schrödinger system when the mass is equal or slightly above the scattering threshold in \cite{CGHY}, we show that scattering is the only dynamical behavior of the solutions to the infinitely coupled system.

Classification of the minimal-mass blowup solutions to the two dimensional focusing cubic nonlinear Schrödinger system

Abstract

In this article, we study the two dimensional focusing finitely and infinitely coupled cubic nonlinear Schrödinger system when the mass is equal to the scattering threshold. For the focusing finitely coupled cubic nonlinear Schrödinger system, we present a complete classification of minimal-mass blowup solutions. Specifically, we demonstrate that all such solutions must be either solitons or their pseudo-conformal transformations. To prove this result, we develop a modulation analysis that accounts for multi-component interactions to overcome the multiply phase transformations caused by the multi-component. A long time Strichartz estimate for vector-valued solutions is established to solve the difficulty posed by the Galilean transformations and spatial translation, where a new vector-valued bilinear estimate is proven to address the challenges caused by the coupled nonlinear interaction. For the infinitely coupled focusing nonlinear Schrödinger system when the mass is equal or slightly above the scattering threshold in \cite{CGHY}, we show that scattering is the only dynamical behavior of the solutions to the infinitely coupled system.

Paper Structure

This paper contains 34 sections, 40 theorems, 586 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem.
  3. Finitely coupled NLSS
  4. Infinitely coupled NLSS
  5. On the mass critical nonlinear Schrödinger equation (NLS)
  6. On the mass critical coupled nonlinear Schrödinger system
  7. Finitely coupled NLSS
  8. Infinitely coupled NLSS
  9. Main results.
  10. Finitely coupled NLSS
  11. Infinitely coupled NLSS
  12. Summary of the proof.
  13. Modulation analysis
  14. A priori estimate of the reminder $\|\boldsymbol{\epsilon}(t)\|_{L_{x}^2l^2}$
  15. Virial inequality
  16. ...and 19 more sections

Key Result

Lemma 1.1

For any $\mathbf{u}\in H^{1}_{x} l^2(\mathbb R^2\times\mathbb Z_N)$, we have the equality holds if and only if $\mathbf{u}= \left(e^{i\gamma_1}\lambda \sqrt{\frac{1}{2N-1}}Q_0 (\lambda x+\tilde{x}),\cdots,e^{i\gamma_N}\lambda\sqrt{\frac{1}{2N-1}} Q_0 (\lambda x+\tilde{x}) \right)$ for some $(\gamma_1,\cdots,\gamma_N)\in [0,2\pi]^N,\lambda>0$ and $\tilde{x}\in \mathbb{R}^2$.

Theorems & Definitions (80)

  • Definition 1.1
  • Conjecture 1.1: Classification of minimal-mass blowup solutions
  • Lemma 1.1: Sharp Gagliardo-Nirenberg inequality
  • Lemma 1.2: Sharp Gagliardo-Nirenberg inequality for the infinite vector function
  • Theorem 1.1: Classification of minimal-mass blowup solutions to NLSS for $N< \infty$
  • Remark 1.1
  • Theorem 1.2: Global well-posedness and scattering for the cubic focusing infinitely coupled NLSS
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • ...and 70 more