Multiple normalized solutions to Schrödinger systems with linear and nonlinear couplings

We establish the existence and multiplicity of normalized solutions to the coupled nonlinear Schrödinger system with linear and nonlinear couplings

$$\{\begin{array}{rlr}& -u^{″}+\lambda u=\mu u^3+\beta v^{2}u+\kappa v& \text{in}\mathbb{R},\\ & -v^{″}+\lambda v=\mu v^3+\beta u^{2}v+\kappa u& \text{in}\mathbb{R},\end{array}$$

satisfying the total mass constraint

$${\int }_{\mathbb{R}}(u^2+v^2)\mathrm{d}x=m,$$

where the nonlinear coupling parameter $\beta =\mu >0$. The system comes from the research on standing waves of coupled Gross–Pitaevskii equations for describing Bose–Einstein condensates. First, we show that, up to translations and sign symmetries, there exists exactly one class of normalized solutions when the linear coupling parameter $\kappa =0$. The least energy level is also explicitly determined. Second, we prove that at least three nontrivial normalized solutions exist under suitable conditions on the linear coupling parameter $\kappa \ne 0$. We present a new approach based on transforming the Schrödinger system with both linear and nonlinear couplings into a Schrödinger system with purely nonlinear coupling. This seems to be the first result concerning the multiplicity of normalized solutions to the Schrödinger system with both linear and nonlinear couplings.