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Non-synchronized solutions to nonlinear elliptic Schrödinger systems on a closed Riemannian manifold

Saikat Mazumdar, Jérôme Vétois

Abstract

On a smooth, closed Riemannian manifold, we study the question of proportionality of components, also called synchronization, of vector-valued solutions to nonlinear elliptic Schrödinger systems with constant coefficients. In particular, we obtain bifurcation results showing the existence of branches of non-synchronized solutions emanating from the constant solutions.

Non-synchronized solutions to nonlinear elliptic Schrödinger systems on a closed Riemannian manifold

Abstract

On a smooth, closed Riemannian manifold, we study the question of proportionality of components, also called synchronization, of vector-valued solutions to nonlinear elliptic Schrödinger systems with constant coefficients. In particular, we obtain bifurcation results showing the existence of branches of non-synchronized solutions emanating from the constant solutions.

Paper Structure

This paper contains 3 sections, 5 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Bifurcation results
  3. Synchronization and non-existence results

Key Result

Theorem 1.1

Let $\lambda,a,b,c\in\mathbb{R}$ and $q\in$2,∞$$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2