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Infinitely many new solutions for a nonlinear coupled Schrödinger system

Qingfang Wang, Mingxue Zhai

TL;DR

The paper studies a semiclassical two-component nonlinear Schrödinger system in $\mathbb{R}^3$ with positive potentials $P(x)$ and $Q(x)$ and interspecies coupling $\beta$. By combining Lyapunov–Schmidt reduction with a finite-dimensional reduction, the authors glue prior synchronized and segregated multi-peak solutions to produce new dichotomous solutions that concentrate both at the origin and at infinity, under symmetry assumptions on the potentials. They establish non-degeneracy of the base multi-peak solutions and construct two families of dichotomous solutions (synchronized and segregated) for large spike counts $k$, with precise energy expansions and interior maximizers in the reduction parameters. Moreover, they prove the non-degeneracy of the synchronized dichotomous peaks, which is essential for iterative gluing and stability analyses. These results extend the landscape of spike-interaction patterns in coupled Schrödinger systems and highlight the role of interspecies coupling in shaping complex concentration phenomena.

Abstract

We revisit the following nonlinear Schrödinger system \begin{align*}\begin{cases} -ε^{2}Δu +P(x) u= μ_1 u^3 +βuv^2, &~\text{in}\;\mathbb {R}^3,\\ -ε^{2}Δv+Q(x) v= μ_2 v^3 +βu^2v, &~\text{in}\;\mathbb{ R}^3, \end{cases} \end{align*} where $ε$ is a positive parameter, $P(x),\,Q(x)$ are the potential functions, $μ_1>0$, $μ_2>0$ and $β\in\mathbb R$ is a coupling constant. Employing the finite dimensional reduction method, we prove that there are new kind of synchronized and segregated solutions, which concentrate both in a bounded domain and near infinity, and present a special structure. Moreover, by applying the local Pohozaev identities and some point-wise estimates of the errors, we prove that the new kind of synchronized solutions are non-degenerate, which is of great interest independently. One of the main difficulties of Schrödinger system come from the interspecies interaction between the components, which never appear in the study of single equation. Secondly, prior to the construction of new solutions, we shall verify the non-degeneracy of the solutions established in [Peng-Pi, Discrete Contin. Dyn. Syst., 2016] for the Schrödinger systems.

Infinitely many new solutions for a nonlinear coupled Schrödinger system

TL;DR

The paper studies a semiclassical two-component nonlinear Schrödinger system in with positive potentials and and interspecies coupling . By combining Lyapunov–Schmidt reduction with a finite-dimensional reduction, the authors glue prior synchronized and segregated multi-peak solutions to produce new dichotomous solutions that concentrate both at the origin and at infinity, under symmetry assumptions on the potentials. They establish non-degeneracy of the base multi-peak solutions and construct two families of dichotomous solutions (synchronized and segregated) for large spike counts , with precise energy expansions and interior maximizers in the reduction parameters. Moreover, they prove the non-degeneracy of the synchronized dichotomous peaks, which is essential for iterative gluing and stability analyses. These results extend the landscape of spike-interaction patterns in coupled Schrödinger systems and highlight the role of interspecies coupling in shaping complex concentration phenomena.

Abstract

We revisit the following nonlinear Schrödinger system \begin{align*}\begin{cases} -ε^{2}Δu +P(x) u= μ_1 u^3 +βuv^2, &~\text{in}\;\mathbb {R}^3,\\ -ε^{2}Δv+Q(x) v= μ_2 v^3 +βu^2v, &~\text{in}\;\mathbb{ R}^3, \end{cases} \end{align*} where is a positive parameter, are the potential functions, , and is a coupling constant. Employing the finite dimensional reduction method, we prove that there are new kind of synchronized and segregated solutions, which concentrate both in a bounded domain and near infinity, and present a special structure. Moreover, by applying the local Pohozaev identities and some point-wise estimates of the errors, we prove that the new kind of synchronized solutions are non-degenerate, which is of great interest independently. One of the main difficulties of Schrödinger system come from the interspecies interaction between the components, which never appear in the study of single equation. Secondly, prior to the construction of new solutions, we shall verify the non-degeneracy of the solutions established in [Peng-Pi, Discrete Contin. Dyn. Syst., 2016] for the Schrödinger systems.
Paper Structure (8 sections, 24 theorems, 256 equations)

This paper contains 8 sections, 24 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Non-degeneracy of the solutions in Theorem A
  3. Existence of dichotomous peak-solutions
  4. Dichotomy for Synchronized solutions
  5. Dichotomy for Segregated solutions
  6. The non-degeneracy of the dichotomous synchronized peak-solutions
  7. some useful estimates
  8. Estimates of the energy functional

Key Result

Theorem 1.1

Assume that ($A_1)-(A_2),\,(H_1)-(H_2$) and $(A_\beta)$ hold. Let $(u_{\epsilon},v_{\epsilon})$ be a pair of solution obtained in Theorem A. Then system eq0 has infinitely many pairs of synchronized dichotomous concentrated solutions. More specifically, for any integer $k>k_{0}$ large enough, system where and $\|(\varphi_{k},\psi_{k})\| \rightarrow 0$ as $k\rightarrow\infty$.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • ...and 37 more