On the solvability of a space-time fractional nonlinear Schrödinger system

Abstract

This paper is devoted to the theoretical analysis of a coupled nonlinear system of fractional Schrödinger equations in $R^n\times R^{+},$ $n\ge 1,$ considering time fractional derivative in the Caputo sense, and a fractional spatial dispersion defined in terms of the Fourier transform. We prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, considering power-type nonlinearities and initial data in the framework of weak-$L^p$ spaces, which contain singular functions with infinite energy. As consequence of the embedding of weak-$L^p$ spaces in $L_{loc}^2,$ for $p>2,$ the obtained solutions have finite local $L^2$-mass. In addition, we discuss the scenario in which it is possible to obtain the existence of self-similar solutions, which is a symmetric property that reproduces the structure of physical phenomena in different spatio-temporal scales. Our results are applicable, in the fractional setting, to the nonlinear Schrödinger and Biharmonic equations, as well as in a large class of dispersive systems appearing in nonlinear optics.