Dans ce document, we study the following nonlinear Schr\”{o}dinger system of
Hamiltonian type \begin{équation*} \left\{\commencer{tableau}{je} -\Delta
u+V(x)u=\partial_v H(x,toi,v)+\oméga v, \ x \in \mathbb{R.}^N, \\ -\Delta
v+V(x)v=\partial_u H(x,toi,v)+\oméga toi,\ x \in \mathbb{R.}^N, \\
\displaystyle\int_{\mathbb{R.}^N}|z|^2dx=un^2, \fin{tableau}\droite. \fin{équation*}
où la fonction potentielle $V(x)$ est périodique,
$z:=(toi,v):\mathbb{R.}^N\rightarrow \mathbb{R.}\times\mathbb{R.}$, $\omega\in
\mathbb{R.}$ arises as a Lagrange multiplier, $a>0$ is a prescribed constant.
The main result in this paper establishes the existence and multiplicity of
$L^2$-normalized solutions for the above nonlinear Schr\”{o}dinger system with
a class of non-autonomous nonlinearity $H(x,toi,v)$. The proofs combine
Lyapunov-Schmidt reduction, perturbation argument and the multiplicity theorem
of Ljusternik-Schnirelmann. En outre, we obtain bifurcation results of this
problem.

Cet article explore les excursions dans le temps et leurs implications.

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