Nonlinear Green's function method for the Tzitzéica and related equations

Abstract

In this short note we apply the nonlinear Green's function method for the solution of the Tzitzéica type equation hierarchies arising in nonlinear science. Using the travelling wave ansatz, we first transform the nonlinear partial differential equations to nonlinear ordinary differential equations. Then, we establish a general representation formula for nonlinear Green's function of these equations. Eventually, using Frasca's short time expansion, we obtain the exact solution to these equations. Numerical analysis shows that the obtained Green's function solution is sufficiently close to the numerical solution obtained by the well-known method of lines. Finally, we involve the inverse transform and study the full nature of the Tzitzéica equation.

KEYWORDS:

• Travelling wave
• nonlinear wave
• nonlinear optics
• Frasca's method
• short time expansion
• nonlinear Green's function

Acknowledgments

We express our sincere gratitude to Marco Frasca, the founder of the nonlinear Green's function method, for many fruitful discussions. We highly appreciate the useful suggestions made by an anonymous referee improving the quality of the presented material.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Asatur Zh. Khurshudyan http://orcid.org/0000-0003-2037-8919