C. Li - Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, R7A 6A9, Canada. K. Nonlaopon - Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand. A. Hrytsenko - Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, R7A 6A9, Canada. J. Beaudin - Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, R7A 6A9, Canada.
In this work, we are devoted to the following fractional nonlinear Schrödinger equation with the initial conditions in the Caputo sense for \(1 < \alpha \leq 2\): \begin{equation}\label{J1} \begin{cases} \displaystyle i \frac{ _c \partial^{\alpha}}{\partial \theta^{\alpha}} W(\theta, \sigma) + \beta_1 \frac{ \partial^{2}}{\partial \sigma^{2}} W(\theta, \sigma) \\ \hspace{0.5in} + \gamma (\theta, \sigma) W(\theta, \sigma) + \beta_2 |W(\theta, \sigma)|^2 W (\theta, \sigma) + \beta_3 W^2 (\theta, \sigma) = 0,\\ W(0, \sigma) = \phi_1(\sigma), \quad W_\theta'(0, \sigma) = \phi_2(\sigma), \end{cases} \end{equation} where \(\theta > 0, \sigma \in \mathbb{R}\), \(\gamma(\theta, \sigma)\) is a continuous function and \(\beta_1, \beta_2, \beta_3\) are constants. Our analysis for deriving analytic and approximate solutions to the Schrödinger equation relies on the Adomian decomposition method and fractional calculus. Several illustrative examples are presented to demonstrate the solution constructions. Finally, the variant and symmetric system of the fractional nonlinear Schrödinger equations are studied.
C. Li, K. Nonlaopon, A. Hrytsenko, J. Beaudin, On the analytic and approximate solutions for the fractional nonlinear Schrödinger equations, Journal of Nonlinear Sciences and Applications, 16 (2023), no. 1, 51--59
Li C., Nonlaopon K., Hrytsenko A., Beaudin J., On the analytic and approximate solutions for the fractional nonlinear Schrödinger equations. J. Nonlinear Sci. Appl. (2023); 16(1):51--59
Li, C., Nonlaopon, K., Hrytsenko, A., Beaudin, J.. "On the analytic and approximate solutions for the fractional nonlinear Schrödinger equations." Journal of Nonlinear Sciences and Applications, 16, no. 1 (2023): 51--59