Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 1

Asymptotic behavior of solutions for a class of two-coupled nonlinear fractional Schrödinger equations

Pages: 11 – 32

DOI: https://dx.doi.org/10.4310/DPDE.2021.v18.n1.a2

Author

Brahim Alouini (Analysis, Probability and Fractals research laboratory, University of Monastir, Faculty of Sciences, Monastir, Tunisia; and Department of Mathematics, I.P.E.I., Monastir, Tunisia)

Abstract

In the current issue, we consider two coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities that reads\[\begin{cases}u_t - i(-\Delta)^{\frac{\alpha}{2}} u + i ({\lvert u \rvert}^2 + {\lvert v \rvert}^2) u + \gamma u = f \\v_t - i(-\Delta)^{\frac{\alpha}{2}} v + i ({\lvert u \rvert}^2 + {\lvert v \rvert}^2) v + \delta v_x + \gamma v = g\end{cases}\]We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor in the phase space with finite fractal dimension.

Keywords

fractional Schrödinger equation, dynamical systems, asymptotic behavior, global attractor, fractal dimension

2010 Mathematics Subject Classification

76B15, 35B40, 35Q55

Received 4 August 2020

Published 19 February 2021