Schrödinger systems arising in nonlinear optics and quantum mechanics. I (Q2896916)

The author studies the existence (local and global) and uniqueness of solutions of a general nonlinear system of \(m\)-coupled Schrödinger equations in the presence of diamagnetic field, local and nonlocal nonlinearities: NEWLINE\[NEWLINE\left\{ \begin{aligned} & i\partial_t\Phi_j = L_A\Phi_j +V(x)\Phi_j -g_j(|x|,|\Phi_1|^2,...,|\Phi_m|^2)\Phi_j-\sum\limits_{j=1}^m W_{ij}*h(|\Phi_i|) \frac{h'(|\Phi_j|)}{|\Phi_j|}\Phi_j, \\ & \Phi_j(0,x)=\Phi_j^0(x), \\ & 1\leq j \leq m, \end{aligned} \right. NEWLINE\]NEWLINE where \(L_A=\left(\frac{1}{i}\nabla-A(x)\right)^2\), for all \(1\leq j \leq m, \Phi_j^0:{\mathbb R}^N\to {\mathbb C}, \Phi_j:{\mathbb R}^+\times{\mathbb R}^N\to {\mathbb C}\), \(h:{\mathbb R}^+\to{\mathbb R}^+\) continuous and non-decreasing, \(V:{\mathbb R}^N\to {\mathbb R}\) and \(A:{\mathbb R}^N\to {\mathbb R}^N\) represent electric and magnetic potentials. The case \(m=1\) has been studied in detail in the literature, but there is lack of results concerning the general case \(m>1\).NEWLINENEWLINEThis kind of systems model many important phenomena in nonlinear optics; multimodal optical fibers, optical pulse propagation, ferromagnetic film and optical pulse propagation in the birefringent fibers. They also govern the interaction of electron and nucleii through Coulombic potential and under the action of external magnetic field in quantum mechanics.