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Concentrating solutions of nonlinear Schrödinger systems with mixed interactions

Qing Guo, Angela Pistoia, Shixin Wen

TL;DR

The paper develops a Lyapunov–Schmidt reduction to construct concentrating solutions for a mixed-interaction nonlinear Schrödinger system arising in multispecies Bose–Einstein condensates and nonlinear optics. It builds an ansatz with a dominant single-bump component and a symmetric array of $k$ bumps for the other components, then reduces the problem to finite- and infinite-dimensional equations, controlling error terms via correction functions $\Phi_\epsilon,\Psi_\epsilon$. The leading-order balance between inter-bubble interactions and the shadow potential $\omega$ determines the feasibility and location of concentration, with distinct outcomes for $\alpha=0$ (two-component case) and $\alpha\neq0$ (multi-component case). The results establish the existence of solutions where the second (and further) components concentrate at $k$ points forming a regular polygon around the origin, under precise sign conditions on $\Delta\omega(0)$ and $\alpha$, and provide a framework extendable to more complex mixed-interaction systems.

Abstract

In this paper we study the existence of solutions to nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. In particular, we build solutions whose first component has one bump and the other components have several peaks forming a regular polygon around the single bump of the first component.

Concentrating solutions of nonlinear Schrödinger systems with mixed interactions

TL;DR

The paper develops a Lyapunov–Schmidt reduction to construct concentrating solutions for a mixed-interaction nonlinear Schrödinger system arising in multispecies Bose–Einstein condensates and nonlinear optics. It builds an ansatz with a dominant single-bump component and a symmetric array of bumps for the other components, then reduces the problem to finite- and infinite-dimensional equations, controlling error terms via correction functions . The leading-order balance between inter-bubble interactions and the shadow potential determines the feasibility and location of concentration, with distinct outcomes for (two-component case) and (multi-component case). The results establish the existence of solutions where the second (and further) components concentrate at points forming a regular polygon around the origin, under precise sign conditions on and , and provide a framework extendable to more complex mixed-interaction systems.

Abstract

In this paper we study the existence of solutions to nonlinear Schrödinger systems with mixed couplings of attractive and repulsive forces, which arise from the models in Bose-Einstein condensates and nonlinear optics. In particular, we build solutions whose first component has one bump and the other components have several peaks forming a regular polygon around the single bump of the first component.

Paper Structure

This paper contains 10 sections, 10 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mt1']} and Theorem \ref{['mt']}
  3. Preliminaries
  4. The correction of the ansatz
  5. The reduction process
  6. Solving equation (\ref{['equiv-sys1']})
  7. Solving equation (\ref{['equiv-sys2']})
  8. Proof of Theorems \ref{['mt1']} and \ref{['mt']}: completed
  9. Technical Lemma
  10. A regularity result

Key Result

Theorem 1.1

Assume $\Delta\omega(0)<0$. For any even integer $k$ there exists $\epsilon_k>0$ such that for any $\epsilon\in(0,\epsilon_k)$, p2 has a solution $(u_\epsilon,v_\epsilon)\in H_{s}^{2}\left(\mathbb{R}^N\right)\times H_{s}^{2}\left(\mathbb{R}^N\right)$ as where and $\Theta,\Xi$ are small functions in suitable norms.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • ...and 9 more