Normalized ground state solutions of Schrödinger-KdV system in \(\mathbb {R}^3\)

In this paper, we study the coupled Schrödinger-KdV system \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u +\uplambda _1 u=u^3+\upbeta uv~~& \text {in}~~\mathbb {R}^{3},\\ -\Delta v +\uplambda _2 v=\frac{1}{2}v^2+\frac{1}{2}\upbeta u^2~~& \text {in}~~\mathbb {R}^{3} \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+{\mathrm{\lambda }}_{1}u=u^3+\mathrm{\beta }uv\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }v+{\mathrm{\lambda }}_{2}v=\frac{1}{2}{v}^2+\frac{1}{2}\mathrm{\beta }{u}^2\hfill & \text{in}{\mathbb{R}}^3\hfill \end{array}\right)\end{array}$$ subject to the mass constraints \(\begin{aligned} \int \limits _{\mathbb {R}^{3}}|u|^2 dx=a,\quad \int \limits _{\mathbb {R}^{3}}|v|^2 dx=b, \end{aligned}\) $$\begin{array}{c}\hfill \underset{{\mathbb{R}}^3}{\int }{|u|}^{2}dx=a,\underset{{\mathbb{R}}^3}{\int }{|v|}^{2}dx=b,\end{array}$$ where \(a, b>0\) $a,b>0$ are given constants, \(\upbeta >0\) $\mathrm{\beta }>0$ , and the frequencies \(\uplambda _1,\uplambda _2\) ${\mathrm{\lambda }}_1,{\mathrm{\lambda }}_2$ arise as Lagrange multipliers. The system exhibits \(L^2\) $L^2$ -supercritical growth. Using a novel constraint minimization approach, we demonstrate the existence of a local minimum solution to the system. Furthermore, we establish the existence of normalized ground state solutions.