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In this paper, we are concerned with the existence and dynamics of solutionsto the equation with mixed fractional Laplacians $$ (-\Delta)^{s_1} u+(-\Delta)^{s_2} u + \lambda u=|u|^{p-2} u $$ under the constraint $$\int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<s_2<s_1<1$, $2+ \frac{4s_1}{N} \leq p< \infty $ if $N \leq 2s_1$, $2+ \frac {4s_1}{N} \leqp<\frac{2N}{N-2s_1}$ if $N >2s_1$, $\lambda \in \R$ appearing as Lagrangemultiplier is unknown. The fractional Laplacian $(-\Delta)^s$ is characterizedas $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s} \mathcal{F}(u)(\xi)$ for $\xi\in \R^N$, where $\mathcal{F}$ denotes the Fourier transform. First weestablish the existence of ground state solutions and the multiplicity of boundstate solutions. Then we study dynamics of solutions to the Cauchy problem forthe associated time-dependent equation. Moreover, we establish orbitalinstability of ground state solutions.

Existence and dynamics of normalized solutions to nonlinear Schr\"{o}dinger equations with mixed fractional Laplacians