Solitary wave solutions of nonlinear PDEs using Kudryashov's R function method

Abstract

Applications of a new function introduced by Kudryashov [Optik. 2020;206:163550] to obtain solitary wave solutions of nonlinear PDEs through their travelling wave reductions are considered. The Kudryashov function, R, satisfying a first-order second degree ODE has several features which significantly assist symbolic calculations, especially for highly dispersive nonlinear equations. A remarkable feature of the Kudryashov function R, is that its even order derivatives are polynomials in R only while its odd order derivatives turn out to be polynomials in R and R_z. The procedure has been illustrated by means of the Schrödinger–Hirota equation, a quartic NLS equation and the fifth-order Kawahara equation as examples. A comparison with the Rayleigh–Ritz variational approach has also been considered for the purposes of illustration. The results obtained here are novel and span the family of solutions for such kind of equations.

Keywords:

• Kudryashov function
• Schrödinger–Hirota equation
• quartic NLS equation
• Kawahara equation
• travelling wave solutions
• solitary waves

PACS:

• 02.30.Hq
• 42.65.Tg
• 05.45.Yv
• 11.30.Er

Acknowledgments

We wish to thank Professors Nikolay Kudryashov and Anjan Biswas for drawing our attention to the new approach and for encouragement.

Author's contributions

All authors' contributed equally to this work.

Data availability statement

The simulation materials which support the findings of this study are available from the corresponding author (SG) upon reasonable request via mail.

Disclosure statement

No potential conflict of interest was reported by the author(s).