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arXiv · 1811.00789

Bifurcations of nontrivial solutions of a cubic Helmholtz system

Rainer Mandel·Dominic Scheider
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Abstract

This paper presents local and global bifurcation results for radially symmetric solutions of the cubic Helmholtz system \begin{equation*} \begin{cases} -Δu - μu = \left( u^2 + b \: v^2 \right) u &\text{ on } \mathbb{R}^3, \\ -Δv - νv = \left( v^2 + b \: u^2 \right) v &\text{ on } \mathbb{R}^3. \end{cases} \end{equation*} It is shown that every point along any given branch of radial semitrivial solutions $(u_0, 0, b)$ or diagonal solutions $(u_b, u_b, b)$ (for $μ= ν$) is a bifurcation point. Our analysis is based on a detailed investigation of the oscillatory behavior of solutions at infinity that are shown to decay like $\frac{1}{|x|}$ as $|x|\to\infty$.

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Rainer Mandel, Dominic Scheider. 2018-11-02. Bifurcations of nontrivial solutions of a cubic Helmholtz system. https://arxiv.org/abs/1811.00789

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