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# 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

#### 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