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The Laplace-Adomian Decomposition Method (LADM) and Homotopy Perturbation Method (HPM) are both utilized in this research in order to obtain an approximate analytical solution to the nonlinear Schrödinger equation with harmonic oscillator. Accordingly, nonlinear Schrödinger equation in both one and two dimensions is provided to illustrate the effects of harmonic oscillator on the behavior of the wave function. The available literature does not provide an exact solution to the problem presented in this paper. Nevertheless, approximate analytical solutions are provided in this paper using LADM and HPM methods, in addition to comparing and analyzing both solutions.

The Schrödinger equation is often encountered in many branches of science and engineering, including quantum mechanics, nonlinear optics, plasma physics, hydrodynamics, and superconductivity. It is a mathematical partial differential equation used to describe the motion and behavior change of the physical system over time. In classical mechanics, it plays the role of Newton’s law and conservation of energy. In quantum mechanics, we describe systems using wave function. The Schrödinger equation has two “forms”; one is the time-dependent wave equation that describes how the wave function of a particle will evolve in time. The other is the time independent wave equation in which the time dependence has been “removed”; it describes what the allowed energies are of the particle [

In recent years, a considerable amount of research focused on finding analytical solution to the Schrödinger equations using various methods, among which are Adomian Decomposition Method [

This paper is organized in several sections. The HPM method is briefly explained in “Homotopy Perturbation Method”. Then the LADM model is described in the “Laplace-Adomian Decomposition Method”. Then in the “One-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator”, the solution to the One-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator equation in its nonlinear version is provided with a numerical example. Similarly, the solution of the Two-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator is presented in the “Two-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator”. Finally, in the “Conclusion”, we summarize our findings and present our final remarks. Since the exact solution to this problem is not available, we compare our numerical results with the results obtained using Mathematica function NDsolve.

The Adomian Decomposition Method (ADM) is a method to solve differential equations by expressing the analytic solution in terms of a series. The method separates the linear and nonlinear parts of a differential equation. The nonlinear part can be expressed in terms of what is called Adomian Polynomials [

The Laplace transform is an integral transform that is powerful and useful technique to solve differential equations, which transforms the original differential equation into an algebraic equation.

Below are the definitions of Laplace transform and inverse Laplace transform.

Given a function

and the inverse Laplace transform is defined as follows.

Given a continuous function

The Laplace-Adomian Decomposition Method (LADM) was first introduced by Suheil A. Khuri [

where

Laplace-Adomian Decomposition Method consists of applying Laplace transform to both sides of (

From Laplace transform of first derivative and substituting the initial condition, we get

Next step is replacing the wave function by an infinite series of terms to be determined later as per the Adomian Decomposition Method (ADM):

and the nonlinear terms are replaced by the series:

where

Substituting (

From (

The Homotopy Perturbation Method (HPM) is a special case of the homotopy analysis method (HAM) [

To demonstrate the idea of Homotopy Perturbation Method, we consider the general form nonlinear differential equation with initial conditions of the form [

where

To apply the Homotopy concept to (

Now, when the value of

Then the solution

The series in (

In the following two sections we apply the above two methods to solve the One- and Two-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator.

The nonlinear Schrödinger equation with harmonic oscillator described by

where

In this section we solve the One-Dimensional Nonlinear Schrödinger Equation with Harmonic Oscillator (

From the properties of Laplace transform of the first derivative and substituting the initial conditions (

Now substituting (

By applying inverse Laplace transform to (

we have

where

The Adomian Polynomials can be calculated using (

Now, using (

Therefore, the solution

In this section we apply the Homotopy Perturbation Method to obtain a solution to (

Substituting (

We finally obtain the general solution of (

The first few terms of the solution are calculated as follows:

and hence, the solution to (

The graphs of the one-dimensional wave function are shown in Figures

Real solution of the one-dimensional wave function obtained by LADM and HPM.

Real solution of the one-dimensional wave function obtained by NDsolve.

Imaginary solution of the one-dimensional wave function obtained by LADM and HPM.

Imaginary solution of the one-dimensional wave function obtained by NDsolve.

In this section we look at a particle movement in two dimensions, the nonlinear Schrödinger equation with harmonic oscillator when a particle moves in two dimensions with the initial condition can be written as [

Similar to the one-dimensional, after applying Laplace transform to (

Replacing the wave function

we get

Applying inverse Laplace transform, we have

and thus

Solving the above system of equations, we get

Therefore, the approximate solution can be written as

In this section we apply the HPM to solve the two-dimensional equation; first we construct the following suitable homotopy:

The general

Therefore, we can now evaluate the solution to the above system of differential equations:

Hence, the approximate solution to (

The graphs of the solution of the two-dimensional wave function are shown in Figures

Real solution of the two-dimensional wave function obtained by LADM and HPM.

Real solution of the two-dimensional wave function obtained by NDsolve.

Imaginary solution of the two-dimensional wave function obtained by LADM and HPM.

Imaginary solution of the two-dimensional wave function obtained by NDsolve.

In this paper, homotopy perturbation and Laplace-Adomian decomposition methods have proven successful when used to find the approximate solution to the nonlinear Schrödinger equation with harmonic oscillator in one and in two dimensions. Our theoretical analyses have shown that both methods have given equivalent analytical approximate solutions successfully and efficiently. Comparison between HPM and LADM shows that although the results of these two methods when applied to solve the Schrödinger equation are in good agreement, HPM can overcome the difficulties arising in calculation of Adomian’s polynomials. The solutions have been obtained and plotted for the real and imaginary wave function with the effect of adding the harmonic oscillator to the nonlinear Schrödinger equation in one and in two dimensions. HPM and LADM methods numerical results are in agreement with the solution obtained using Mathematica function NDsolve.

No data is involved in this research.

The authors declare no conflicts of interest.