[Paper Review] The Laplace-Adomian Decomposition Method Applied to the Kundu-Eckhaus Equation

O. González-Gaxiola|arXiv (Cornell University)|Apr 25, 2017

Nonlinear Waves and Solitons24 references23 citations

TL;DR

This paper proposes the Laplace-Adomian Decomposition Method (LADM) to solve the nonlinear Kundu-Eckhaus equation, combining Laplace transforms with Adomian polynomials to decompose nonlinear terms. The method yields highly accurate approximate solutions that closely match the only known exact solution in the literature, demonstrating high convergence with few terms and strong accuracy even at larger time values.

ABSTRACT

The Kundu-Eckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Besides, we compare the behaviour of the solutions obtained with the only exact solutions given in the literature through fractional calculus. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

Motivation & Objective

• To develop an efficient analytical approach for solving the nonlinear Kundu-Eckhaus equation, which models phenomena in nonlinear optics and quantum field theory.
• To combine the Adomian Decomposition Method (ADM) with the Laplace transform (Laplace-ADM) to enhance solution accuracy and convergence for nonlinear partial differential equations.
• To validate the proposed LADM approach by comparing its approximate solutions with the only known exact solution from Arzu (2018).
• To demonstrate the method's effectiveness and simplicity in solving complex nonlinear evolution equations in mathematical physics.

Proposed method

• The method decomposes the Kundu-Eckhaus equation into linear and nonlinear parts, applying the Laplace transform to the highest-order time derivative.
• Adomian polynomials are used to represent the nonlinear terms, enabling recursive computation of solution components.
• The inverse Laplace transform is applied to the resulting algebraic equations to reconstruct the solution as a series of terms.
• Initial conditions are incorporated directly into the first term of the series, ensuring consistency with the problem's boundary conditions.
• The solution is constructed iteratively using recurrence relations derived from the transformed equation.
• The method is applied numerically to a test case with specific initial conditions, and results are compared with the exact solution from Arzu.

Experimental results

Research questions

• RQ1 Can the LADM effectively solve the nonlinear Kundu-Eckhaus equation with high accuracy and minimal computational effort?
• RQ2 How does the LADM approximate solution compare in precision to the only known exact solution in the literature?
• RQ3 Does the LADM maintain high accuracy as time increases, despite the nonlinear and dispersive nature of the equation?
• RQ4 To what extent does the method converge rapidly with only a few terms in the series expansion?

Key findings

• The LADM approximate solution shows excellent agreement with the exact solution from Arzu, with maximum absolute errors in the real and imaginary parts consistently below 2.2×10⁻⁴ across all tested time points and spatial values.
• At t=1.0, the maximum error in the real part is 2.05×10⁻⁴, and at t=2.0, it is 2.39×10⁻⁴, indicating stable accuracy over time.
• For the imaginary part, the maximum error is 2.39×10⁻⁴ at t=2.0, and the smallest error observed is 5.14×10⁻⁶ at x=4.5, t=2.0, showing high precision in localized regions.
• The method converges rapidly, with high accuracy achieved using only a few terms in the series expansion, confirming its efficiency.
• Visual comparisons in Figures 1 and 2 show that the real and imaginary parts of the LADM solution closely track the exact solution across all time steps.
• The study concludes that LADM is a powerful, direct, and effective method for solving nonlinear partial differential equations, especially those arising in mathematical physics.