Homotopy Analysis Method and Time-fractional NLSE

U. Ghosh ; T. Das ; S. Sarkar

Russian journal of nonlinear dynamics, Tome 18 (2022) no. 2, pp. 309-328

Résumé

A brief outline of the derivation of the time-fractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study time-fractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential $[V(x)=V_1^{}\cos x+V_2^{}\cos 2x]$, the second one is the Morse potential $[V(x)=V_1^{}e^{-2\beta x}+V_2^{}e^{-\beta x}]$, and a hyperbolic potential $[V(x)=V_0^{}\tanh(x)sech(x)]$ is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order.

Détail

Keywords: time fractional nonlinear Schrödinger equation (NLSE), homotopy analysis method (HAM), Caputo derivative, Riemann – Liouville fractional integral operator, trapping potential.