[Paper Review] Nonlinear differential equations with exact solutions

Nikolay A. Kudryashov|ArXiv.org|Nov 27, 2003

Nonlinear Waves and Solitons3 references3 citations

TL;DR

This paper presents a novel method to construct nonlinear ordinary differential equations (ODEs) with exact solutions by leveraging the general solution of the Riccati equation as a building block. The approach systematically generates second-, third-, and fourth-order polynomial ODEs—many relevant to physics and nonlinear wave theory—whose exact solutions are expressed via hyperbolic functions, with explicit examples including a fifth-order equation derived from a chaos model, demonstrating the method's utility in finding closed-form solutions for non-integrable equations.

ABSTRACT

New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption that nonlinear differential equations have exact solution which is general solution of the simplest integrable equation. The Riccati equation is shown to be a building block to find a lot of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutions are given. Most of these equations are used at the description of processes in physics and in theory of nonlinear waves.

Motivation & Objective

• To address the challenge of finding exact solutions for nonlinear differential equations that are not integrable but are widely used in physics and nonlinear wave theory.
• To shift focus from integrability to the existence of special exact solutions, particularly those expressible via the general solution of the Riccati equation.
• To develop a systematic method for generating polynomial nonlinear ODEs of second, third, and fourth order with known exact solutions.
• To provide a catalog of such equations with explicit solutions, including a fifth-order case derived from a chaos model.
• To demonstrate the method's applicability by deriving exact solutions for a sixth-order nonlinear evolution equation used in turbulence modeling.

Proposed method

• Assume that the exact solution of a nonlinear ODE is the general solution of a simpler, integrable equation—specifically, the Riccati equation.
• Use the Riccati equation’s general solution, involving hyperbolic tangent functions, as a trial function to substitute into higher-order ODEs.
• Substitute the ansatz into the target ODE and equate coefficients of like powers of the solution and its derivatives to derive constraints on the ODE’s parameters.
• Derive conditions under which the ansatz satisfies the ODE, leading to a system of algebraic equations for the coefficients.
• Solve the resulting system to identify specific ODEs (e.g., second, third, fourth, and fifth-order) that admit exact solutions.
• Apply transformations to reduce complex equations to standard forms, enabling the derivation of exact solutions for equations like the sixth-order model in turbulence.

Experimental results

Research questions

• RQ1 Can we systematically generate nonlinear ODEs with exact solutions without requiring full integrability?
• RQ2 How can the general solution of the Riccati equation be used as a foundation to construct exact solutions for higher-order nonlinear ODEs?
• RQ3 What classes of second-, third-, and fourth-order nonlinear ODEs admit exact solutions expressible via hyperbolic functions?
• RQ4 Can this method be extended to fifth-order and higher-order ODEs, such as those arising in chaos and turbulence models?
• RQ5 What are the specific parameter constraints that allow a nonlinear ODE to admit exact solutions derived from the Riccati equation?

Key findings

• The method successfully generates a class of second-, third-, and fourth-order nonlinear ODEs with exact solutions, all expressible via the general solution of the Riccati equation.
• For the fifth-order ODE derived from a chaos model, six distinct parameter sets (α values) were found, including real and complex conjugate pairs, enabling exact solutions.
• Explicit exact solutions were derived for three distinct parameter cases of the fifth-order equation, with solutions in the form of polynomials in tanh functions: y(z) = C₀ + A₁Y + A₂Y³ + A₃Y⁵, where Y = ±(1/2√110) tanh(±z/(2√110) + φ₀) for α₁.
• The fifth-order equation was reduced to standard form y_zzzzz - y_zzz + σ y_z + ½ y² - C₀ y + C₁ = 0, with σ = (-92400α + 10204656α² + 213811840α³ + 2045)/(121(9240α - 79)) for α₁.
• For the sixth-order nonlinear evolution equation used in turbulence modeling, exact solutions were obtained by transforming the equation and applying the method, yielding solutions with specific coefficients and wave speed parameters.
• The derived exact solutions for the fifth-order equation are valid for specific values of C₀ and C₁, with C₁ = -321489/322102000 + ½ C₀² for α₁, and similar expressions for α₂ and α₃.