On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and \(b\)-equation
type: Item
instance of: scholarly article
title: On the class of nonlinear PDEs that can be treated by the modified method of simplest equation. Application to generalized Degasperis-Processi equation and \(b\)-equation
zbMATH Open document ID: 1229.35028
author: Nikolay K. Vitanov
author: Zlatinka I. Dimitrova
author: Kaloyan N. Vitanov
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2011-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1304.1164
review text: The method of simplest equation relies on exact solutions of some simple equations such as the equations of Bernoulli and Riccati, which are embedded into much more complicated equations such as the Degasperi-Processi equation. This method was proposed by \textit{N.A. Kudryashov} in [Phys. Lett., A 342, No.~1--2, 99--106 (2005; Zbl 1222.35054)]. The authors propose a modification of this method, where the procedure analogous to the search for a Painlevé property is developed to identify the balance equations and the underlying ``simplest'' equation. The authors apply this modification to the construction of exact traveling wave solutions to two complicated equations, the Degasperis-Processi equation and the \(b\)-equation. The solutions are constructed analytically and represented graphically. The authors also compare their method with the known exp-function method.
Mathematics Subject Classification ID: 35C07
Mathematics Subject Classification ID: 35C05
Mathematics Subject Classification ID: 35A25
zbMATH DE Number: 5920355
zbMATH Keywords: Painlevé property
zbMATH Keywords: exp-function method
zbMATH Keywords: exact traveling-wave solutions
reviewed by: Dmitry Pelinovsky
MaRDI profile type: Publication
OpenAlex ID: W1967473206
arXiv ID: 1304.1164
cites work: Q3995225
cites work: Q4158861
cites work: Q4429919
cites work: Q3686255
cites work: Upper bounds on the heat transport in a porous layer
cites work: Method for Solving the Korteweg-deVries Equation
cites work: Nonlinear-evolution equations of physical significance
cites work: The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
cites work: Exact Solution of the Korteweg—de Vries Equation for Multiple Collisions of Solitons
cites work: New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations
cites work: Extended tanh-function method and its applications to nonlinear equations
cites work: Modified method of simplest equation and its application to nonlinear PDEs
cites work: Simplest equation method to look for exact solutions of nonlinear differential equations
cites work: Explicit solutions of Fisher's equation for a special wave speed
cites work: Influence of adaptation on the nonlinear dynamics of a system of competing populations
cites work: Dynamical consequences of adaptation of the growth rates in a system of three competing populations
cites work: Adaptation and its impact on the dynamics of a system of three competing populations
cites work: Chaotic pairwise competition
cites work: On nonlinear dynamics of interacting populations: coupled kink waves in a system of two populations
cites work: On nonlinear population waves
cites work: Exact solitary waves of the Fisher equation
cites work: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity
cites work: Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation
cites work: Symmetry analysis and exact solutions of the \(2+1\) dimensional sine-Gordon system
cites work: The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations
cites work: New class of running-wave solutions of the (2+1)-dimensional sine-Gordon equation
cites work: On travelling waves and double-periodic structures in two-dimensional sine - Gordon systems
cites work: Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system
cites work: Breather and soliton wave families for the sine–Gordon equation
cites work: The (2 + 1)-dimensional sine - Gordon equation; integrability and localized solutions
cites work: Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations
cites work: Analytical and numerical investigation of two families of Lorenz-like dynamical systems
cites work: Q4237094
cites work: Extended simplest equation method for nonlinear differential equations
cites work: Be careful with the Exp-function method
cites work: Seven common errors in finding exact solutions of nonlinear differential equations
cites work: Painlevé Tests, Singularity Structure and Integrability
cites work: Modified method of simplest equation: powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs
cites work: Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics
cites work: A transformed rational function method and exact solutions to the \(3+1\) dimensional Jimbo-Miwa equation
cites work: Q4293617
cites work: Water waves and integrability
cites work: The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations
cites work: Q4508328
cites work: Exp-function method for nonlinear wave equations
cites work: Complexiton solutions to the Korteweg-de Vries equation
cites work: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
DOI: 10.1016/J.CNSNS.2010.11.013
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