Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$

We consider the system of coupled elliptic equations $\begin{cases} -Δu - λ_1 u = μ_1 u^3+ βu v^2 \\ -Δv- λ_2 v = μ_2 v^3 +βu^2 v \end{cases} \text{in $\mathbb{R}^3$},$ and study the existence of positive solutions satisfying the additional condition $\int_{\mathbb{R}^3} u^2 = a_1^2 \quad \text{and} \quad \int_{\mathbb{R}^3} v^2 = a_2^2.$ Assuming that $a_1,a_2,μ_1,μ_2$ are positive fixed quantities, we prove existence results for different ranges of the coupling parameter $β>0$. The extension to systems with an arbitrary number of components is discussed, as well as the orbital stability of the corresponding standing waves for the related Schrödinger systems.

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