International Journal of Emerging Multidiciplinaries: Mathematics Homotopy Analysis Method for Non-Linear Schrödinger Equations

This paper applies Homotopy Analysis Method (HAM) to obtain analytical solutions of nonlinear Schrödinger equations. Numerical results clearly reflect complete compatibility of the proposed algorithm and discussed problems. Several examples are presented to show the efficiency and simplicity of the method.


Introduction
Differential equations arise in almost all areas of the applied, physical and engineering sciences . Recently [16][17][18][19][20][21], lot of attention is being paid on fractional differential equations and it has been observed that number of physical problems is better modeled by such equations. Several numerical and analytical techniques including Perturbation, Modified Adomian's Decomposition (MADM), Variational Iteration (VIM), Homotopy Perturbation (HPM) have been developed to solve such equations, see [16][17][18][19][20][21] and the references therein. Inspired and motivated by the ongoing research in this area, we apply Homotopy Analysis Method (HAM)   Consider the Schrödinger equation with the following initial condition where ( ) is the trapping potential and is a real constant. Numerical results are very encouraging and reveal the efficiency of proposed scheme (HAM).

2.
Homotopy Analysis Method (HAM)  We consider the following equation where is a nonlinear operator, denotes dependent variables and ( ) is an unknown function. For simplicity, we ignore all boundary and initial conditions, which can be treated in the similar way. By means of HAM L  constructed zero-order deformation equation Where ℒ is a linear operator, ( ) is an initial guess, ℏ ≠ 0 is an auxiliary parameter and ∈ [0,1] is the embedding parameter. It is obvious that when p=0 and 1, it holds respectively. The solution ∅( ; ) varies from initial guess ( ) to solution ( ). Liao [18] expanded ∅( ; ) in Taylor series about the embedding parameter where The convergence of (5) depends on the auxiliary parameter ℏ. If this series is convergent at p=1, 86 International Journal of Emerging Multidiciplinaries ∅(τ; 1) = u (τ) + u (τ) , Define vector If we differentiate the zeroth-order deformation equation Eq. (2) -times with respect to and then divide them ! and finally set = 0, we obtain the following th-order deformation equation where If we multiply with ℒ each side of Eq. (8), we will obtain the following th order deformation equation

Numerical Applications
In this section, we apply Homotopy Analysis Method (HAM) on the required problems. Numerical results are highly encouraging.
Example 1: Consider the following one dimensional Schrödinger equation subject to the initial condition To solve equation (1) The zeroth order deformation is, where ∈ [ 0 , 1 ] is an ebeding parameter, ℎ ≠ 0 is a non-zero auxiliary parameter; ( , ) is initial guess.
subject to the initial condition The zeroth order deformation is, where ∈ [ 0 , 1 ] is an ebeding parameter, ℎ ≠ 0 is a non-zero auxiliary parameter; ( , ) is initial guess.
The zeroth-order deformation is,