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<p>We focus on the singularly perturbed nonlinear Schrödinger system given by <Equation ID="Equa"> <EquationSource Format="TEX">\(\begin{cases}-\varepsilon^{2}\Delta u+P_{1}(x)u=\mu_{1}u^{3}+\beta u v^{2} &amp; \text{in}\ \mathbb{R}^{3}, \\-\varepsilon^{2}\Delta v+P_{2}(x)v=\mu_{2}v^{3}+\beta v u^{2} &amp; \text{in}\ \mathbb{R}^{3},\end{cases}\)</EquationSource> </Equation> where <i>ε</i> is a small positive parameter, <i>P</i><sub><i>i</i></sub>(<i>x</i>) (<i>i</i> = 1, 2) are potential functions, <i>μ</i><sub>1</sub> &gt; 0, <i>μ</i><sub>2</sub> &gt; 0, and β ∈ <i>ℝ</i> is the coupling constant.</p><p>Previous work on the Schrödinger-Newton equations and single nonlinear Schrödinger equations explored the construction of concentrated solutions in two stages. For sufficiently small <i>ε</i> &gt; 0, it was shown that the ground state <i>u</i><sub><i>ε</i></sub> remains non-degenerate. Furthermore, solutions with multiple spikes were constructed by gluing ground states, with the number of spikes growing as <i>k</i> → +∞. In this process, part of the energy escapes to infinity, while the remaining energy stays near the origin.</p><p>In contrast to this approach, we investigate a more intricate and genuine dichotomy phenomenon, where solutions concentrate both at the origin and at infinity simultaneously as <i>ε</i> → 0. Specifically, we construct (<i>k</i> + 1)-peak solutions in a perturbative framework, considering both limits <i>ε</i> → 0 and <i>k</i> → +∞ simultaneously. This dual concentration at both the origin and infinity introduces new behavior in the coupling between <i>u</i> and <i>v</i>, revealing either synchronized or segregated coupling dynamics as the solutions approach infinity. These results extend the analysis from single equations to systems, providing deeper insights into the complex interactions in coupled systems.</p>

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New type of dichotomy behavior in synchronized and segregated solutions for nonlinear Schrödinger systems

  • 期刊论文

  • 发表日期：

    2026年3月30日

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  • Xinan Duan,
  • Zijuan Gao,
  • Qing Guo

摘要

We focus on the singularly perturbed nonlinear Schrödinger system given by \(\begin{cases}-\varepsilon^{2}\Delta u+P_{1}(x)u=\mu_{1}u^{3}+\beta u v^{2} & \text{in}\ \mathbb{R}^{3}, \\-\varepsilon^{2}\Delta v+P_{2}(x)v=\mu_{2}v^{3}+\beta v u^{2} & \text{in}\ \mathbb{R}^{3},\end{cases}\) where ε is a small positive parameter, Pi(x) (i = 1, 2) are potential functions, μ1 > 0, μ2 > 0, and β ∈ ℝ is the coupling constant.

Previous work on the Schrödinger-Newton equations and single nonlinear Schrödinger equations explored the construction of concentrated solutions in two stages. For sufficiently small ε > 0, it was shown that the ground state uε remains non-degenerate. Furthermore, solutions with multiple spikes were constructed by gluing ground states, with the number of spikes growing as k → +∞. In this process, part of the energy escapes to infinity, while the remaining energy stays near the origin.

In contrast to this approach, we investigate a more intricate and genuine dichotomy phenomenon, where solutions concentrate both at the origin and at infinity simultaneously as ε → 0. Specifically, we construct (k + 1)-peak solutions in a perturbative framework, considering both limits ε → 0 and k → +∞ simultaneously. This dual concentration at both the origin and infinity introduces new behavior in the coupling between u and v, revealing either synchronized or segregated coupling dynamics as the solutions approach infinity. These results extend the analysis from single equations to systems, providing deeper insights into the complex interactions in coupled systems.