[Paper Review] Variable separated ODE method--A powerful tool for testing traveling wave solutions of nonlinear equations

Sirendaoreji|arXiv (Cornell University)|Nov 11, 2018

Nonlinear Waves and Solitons6 references4 citations

TL;DR

This paper extends the variable separated ODE method by using the general elliptic equation as an auxiliary equation to systematically derive exact traveling wave solutions for nonlinear PDEs. By comparing coefficients and leveraging known solutions of the auxiliary equation, the method efficiently generates a wide range of exact solutions without solving the original nonlinear equation directly, significantly simplifying the solution process for diverse nonlinear wave equations.

ABSTRACT

The variable separated ODE method is extended by choosing the additional variable separated equation as the general elliptic equation. More exact traveling wave solutions of nonlinear equations are obtained by using the method of comparison of coefficients and the known solutions of the auxiliary equation.

Motivation & Objective

• To extend the variable separated ODE method beyond sine/cosine-type equations to general nonlinear PDEs.
• To address the challenge of finding exact traveling wave solutions when standard trigonometric or hyperbolic functions are not present in the equation.
• To develop a systematic method that avoids solving the original nonlinear PDE directly.
• To generate a broader class of exact solutions by using the general elliptic equation as an auxiliary equation.
• To simplify the solution process by eliminating the need for homogeneous balancing and expansion-based ansatzes.

Proposed method

• The method applies the wave transformation $ u(x,t) = u(\xi) $, $ \xi = x - \omega t $, reducing the PDE to an ODE.
• An auxiliary equation $ (u')^2 = c_0 + c_1 u + c_2 u^2 + c_3 u^3 + c_4 u^4 $ is used as the variable separated ODE, representing the general elliptic equation.
• Known exact solutions of the general elliptic equation are classified into 5 cases based on parameter conditions, providing a library of base functions.
• The method compares coefficients of polynomial terms between the transformed ODE and the assumed solution ansatz to determine unknown parameters.
• Solutions are constructed by substituting the known forms of the general elliptic equation solutions into the ansatz, yielding exact traveling wave solutions.
• The approach avoids the need for expansion methods or homogeneous balancing, streamlining the solution derivation.

Experimental results

Research questions

• RQ1 Can the variable separated ODE method be generalized to nonlinear equations that do not involve sine, cosine, or hyperbolic functions?
• RQ2 How can the method be extended to generate more exact solutions using a richer class of auxiliary equations?
• RQ3 Can the general elliptic equation serve as a universal auxiliary equation to unify and extend existing solution techniques?
• RQ4 What is the systematic procedure to derive exact solutions without solving the original nonlinear PDE?
• RQ5 How can coefficient comparison and known elliptic function solutions be combined to yield new traveling wave solutions?

Key findings

• The method successfully generates 23 distinct exact traveling wave solutions for the nonlinear Schrödinger equation with cubic nonlinearity, including hyperbolic, trigonometric, and Jacobi elliptic function forms.
• Solutions such as $ u_{16}(x,t) = -\frac{\alpha}{2\beta}\left[1 + \varepsilon \mathrm{cn}\left(\cdots\right)\right] $ and $ u_{22}(x,t) = -\frac{\alpha}{2\beta}\left[1 + \varepsilon \mathrm{dn}\left(\cdots\right)\right] $ are derived under specific parameter conditions.
• The method enables derivation of solutions involving $ \mathrm{sn}, \mathrm{cn}, \mathrm{dn} $, and rational forms, demonstrating broad applicability.
• For the case $ c_1 = \frac{c_3^3 m^2}{32 c_4^2 (m^2 - 1)} $, the solution $ u_{18}(x,t) $ is obtained with $ \omega = -\frac{\alpha^2 (5m^2 - 4)}{4\beta (m^2 - 1)} $, showing dependence on modulus $ m $.
• The method avoids solving the original nonlinear equation and does not require ansatz expansion, reducing complexity and increasing efficiency.
• The approach is general and can be applied to a wide class of nonlinear equations by selecting appropriate parameters in the general elliptic auxiliary equation.