Study of a Mixed Concave-Convex Sublinear-Superlinear Schrodinger System for Existence, Uniqueness, Classification and Computer Simulation

In this paper, we are concerned with the study of a system of NLS equations in the presence of a superlinear convex and a sublinear concave nonlinearities correlating the couple of solutions such as $iu_t+\Delta u+|u|^{p-1}u+|v|^{q-1}u=0$ and $iv_t+\Delta v+|v|^{p-1}v+|u|^{q-1}v=0$. Existence, uniqueness and classification of the solutions are investigated provided with some numerical simulations via illustrating graphs. The simulations proved that possible chaotic behavior may be investigated. Moreover, the study allowed to emphasize the influence of the power law exponents and their strong impact on the initial conditions as well as the behavior of the solution.