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Collision of two solitons for $1d$ Nonlinear Schrodinger Equation with the same mass

Abdon Moutinho

Abstract

We study the global dynamics of the collision of two solitons having the same mass for one-dimensional Nonlinear Schrödinger models with multi-power nonlinearity. For any natural number k, it is verified that if the incoming speed v between the two solitary waves is small enough, then, after the collision, the two solitons will move away with an outcoming speed v_{f}=v+O(v^{k}) and the remainder of the solution will also have energy and weighted norms of order O(v^{k}). This is applied to the one-dimensional models with polynomial odd nonlinearity having a stable soliton such as the cubic NLS and the cubic-quintic NLS.

Collision of two solitons for $1d$ Nonlinear Schrodinger Equation with the same mass

Abstract

We study the global dynamics of the collision of two solitons having the same mass for one-dimensional Nonlinear Schrödinger models with multi-power nonlinearity. For any natural number k, it is verified that if the incoming speed v between the two solitary waves is small enough, then, after the collision, the two solitons will move away with an outcoming speed v_{f}=v+O(v^{k}) and the remainder of the solution will also have energy and weighted norms of order O(v^{k}). This is applied to the one-dimensional models with polynomial odd nonlinearity having a stable soliton such as the cubic NLS and the cubic-quintic NLS.
Paper Structure (10 sections, 33 theorems, 384 equations)

This paper contains 10 sections, 33 theorems, 384 equations.

Table of Contents

  1. Introduction
  2. Resume of the proof
  3. Background
  4. Approximate solutions
  5. Energy Estimate
  6. Dynamics of the Modulation Parameters
  7. Energy Estimate Functional
  8. Orbital Stability of two Solitary Waves
  9. Proof of Theorem \ref{['main']}
  10. Proof of Theorem \ref{['uniq']}

Key Result

Theorem 1.1

Let $\omega>0,$ if satisfies for some $y_{0}>0$ then the ordinary differential equation has a unique positive solution $\phi_{\omega}\in H^{1}(\mathbb{R}).$

Theorems & Definitions (79)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 69 more