Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The uniqueness and concentration behavior of solutions for a nonlinear fractional Schrödinger system

Chungen Liu, Zhigao Zhang, Jiabin Zuo

TL;DR

The study analyzes a two-component nonlinear fractional Schrödinger system on $\mathbb{R}$ with attractive intra- and interspecies interactions under an $L^2$-mass constraint. By formulating a constrained energy minimization problem, it proves uniqueness of nonnegative minimizers for small interaction parameters via the implicit function theorem and derives detailed concentration behavior as the total interaction strength approaches a critical value, with blow-up occurring at the flattest minima of the trapping potentials. Sharp energy and $L^4$-norm estimates are established, showing a precise blow-up rate and linking the limiting profiles to the ground state of the associated limit equation. The rescaled minimizers converge to the classical ground state $Q$ up to translation and dilation, providing a rigorous description of mass concentration and symmetry-breaking phenomena in this nonlocal multi-component Bose-Einstein condensate model. These results contribute to the understanding of nonlocal, normalized solutions and their concentration dynamics under competing interactions and trapping landscapes.

Abstract

The paper is concerned with a nonlinear system of two coupled fractional Schrödinger equations with both attractive intraspecies and attractive interspecies interactions in $\mathbb{R}$. By analyzing an associated $L^2$-constrained minimization problem, the uniqueness of solutions to this system is proved via the implicit function theorem. Under a certain type of trapping potential, by establishing some delicate energy estimates, we present a detailed analysis on the concentration behavior of the solutions as the total strength of intraspecies and interspecies interactions tends to a critical value, where each component of the solutions blows up and concentrates at a flattest common minimum point of the associated trapping potentials. An optimal blow-up rate of solutions to the system is also given.

The uniqueness and concentration behavior of solutions for a nonlinear fractional Schrödinger system

TL;DR

The study analyzes a two-component nonlinear fractional Schrödinger system on with attractive intra- and interspecies interactions under an -mass constraint. By formulating a constrained energy minimization problem, it proves uniqueness of nonnegative minimizers for small interaction parameters via the implicit function theorem and derives detailed concentration behavior as the total interaction strength approaches a critical value, with blow-up occurring at the flattest minima of the trapping potentials. Sharp energy and -norm estimates are established, showing a precise blow-up rate and linking the limiting profiles to the ground state of the associated limit equation. The rescaled minimizers converge to the classical ground state up to translation and dilation, providing a rigorous description of mass concentration and symmetry-breaking phenomena in this nonlocal multi-component Bose-Einstein condensate model. These results contribute to the understanding of nonlocal, normalized solutions and their concentration dynamics under competing interactions and trapping landscapes.

Abstract

The paper is concerned with a nonlinear system of two coupled fractional Schrödinger equations with both attractive intraspecies and attractive interspecies interactions in . By analyzing an associated -constrained minimization problem, the uniqueness of solutions to this system is proved via the implicit function theorem. Under a certain type of trapping potential, by establishing some delicate energy estimates, we present a detailed analysis on the concentration behavior of the solutions as the total strength of intraspecies and interspecies interactions tends to a critical value, where each component of the solutions blows up and concentrates at a flattest common minimum point of the associated trapping potentials. An optimal blow-up rate of solutions to the system is also given.
Paper Structure (4 sections, 11 theorems, 182 equations)

This paper contains 4 sections, 11 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Uniqueness of non-negative minimizers
  3. Optimal estimates
  4. Proof of Theorem \ref{['blow']}.

Key Result

Theorem 1.1

If $V_i(x)$ satisfies $(\mathcal{D})$ for $i=1,2$, then (problem 1.7) has a unique non-negative minimizer if $|(a_1, a_2, \beta)|$ is suitably small.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • proof
  • Proposition 3.1
  • ...and 11 more