Proceedings of the Seventh International Conference on Management Science and Engineering Management pp 1351-1358 | Cite as

# The Travelling Wave Solutions of the Active-Dissipative Dispersive Media Equation by (*G*′/*G*)-Expansion Method

## Abstract

Over the past decades a number of approximate methods for finding travelling wave solutions to nonlinear evolution equations have been proposed. Among these methods, one of the current methods is so called (*G*′/*G*)-expansion method. In this paper, we will examine the (*G*′/*G*)-expansion method for determining the solutions of the active-dissipative dispersive media equation. The active-dissipative dispersive media equation is given by *μt* +*μμx* + *αμxx* + *βμxxx* + *γμxxxx* = 0, where for positive constants *α* and *γ* in equation are small-amplitude. This equation describe long waves on a viscous fluid flowing down along an inclined plane, unstable drift waves in plasma and stress waves in fragmented porous media. When *β* = 0, equation is reduced to the Kuramoto–Sivashinsky equation, which is the simplest equations that appears in modelling the nonlinear behaviour of disturbances for a sufficiently large class of active dissipative media. It represents the evolution of concentration in chemical reactions, hydrodynamic instabilities in laminar flame fronts and at the interface of two viscous fluids.

## Keywords

(*G*′/

*G*)-Expansion method The active-dissipative dispersive media equation Wave solutions

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Whitham GB (1965) A general approach to linear and nonlinear waves using a Lagrangian. Journal of Fluid Mechanics 22:273–283Google Scholar
- 2.Knobel R (2000) An Introduction to the Mathematical Theory of Waves. American Mathematical Society, Providence, RIGoogle Scholar
- 3.Shen SS (1994) A course on nonlinear waves. Kluwer, DordrechtGoogle Scholar
- 4.Yong C, Biao L, Hong-Quing Z (2003) Generalized Riccati equation expansion method and its application to Bogoyaylenskii’s generalized breaking soliton equation. Chinese Physics 12:940–946Google Scholar
- 5.Zhou Y, Wang M, Wang Y (2003) Periodic wave solutions to a coupled KdV equations with variable coefficient. Physics Letters A 308:31–36Google Scholar
- 6.Cai G,Wang Q, Huang J (2006) A modified F-expansion method for solving breaking soliton equation. International Journal of Nonlinear Science 2:122–128Google Scholar
- 7.Zeng X, Yong X (2008) A new mapping method and its applications to nonlinear partial differential equations. Physics Letters A 372:6602–6607Google Scholar
- 8.Yong X, Zeng X, Zhang Z et al (2009) Symbolic computation of Jacobi elliptic function solutions to nonlinear differential-difference equations. Computers & Mathematics with Applications doi: 10.1016/j.camwa2009.01.008
- 9.Ozis T, Aslan I (2008) Exact and explicit solutions to the (3 + 1)-dimensional Jimbo-Miwa equation via the Exp-function method. Physics Letters A 372:7011–7015Google Scholar
- 10.Ozis T, Koroglu C (2008) A novel approach for solving the Fisher equation using Expfunction method. Physics Letters A 372:3836–3840Google Scholar
- 11.Topper J, Kawahara T (1978) Approximate equations for long nonlinear waves on a viscous fluid. Journal of the Physical Society of Japan 44:663–666Google Scholar
- 12.Ma W, Huang T, Zhang Y (2010) A multiple Exp-function method for nonlinear differential equations and its applications. Physica Scripta 82(2010) 065003:8Google Scholar
- 13.Ozis T, Aslan I (2010) Application of the G’/G-expansion method to Kawahara type equations using symbolic computation. Applied Mathematics and Computation 216:2360–2365Google Scholar
- 14.Ozis T, Aslan I (2009) Symblic computation and construction of New exact traveling wawe solutions to Fitzhugh-Nagumo and Klein Gordon equation. Zeitschrift für Naturforschung 64a:15–20Google Scholar
- 15.Aslan I, Ozis T (2009) An analytical study on nonlinear evolution equations by using the (G’/G)- expansion method. Appled Mathematics and Computation 209:425–429Google Scholar
- 16.Ozis T, Aslan I (2009) Symbolic computations and exact and explicit solutions of some nonlinear evolution equations in mathematical physics. Communications in Theoretical Physics 51:577–580Google Scholar
- 17.Zhang H (2009) A note on some sub-equation methods and new types of exact travelling wave solutions for two nonlinear partial differential equations. Acta Applicandae Mathematicae 106:241–249Google Scholar
- 18.Lia B, Chena Y, Lia YQ (2008) A generalized sub-equation expansion method and some analytical solutions to the inhomogeneous higher-order nonlinear schrodinger equation. Z. Naturforsch. 63a:763–777Google Scholar
- 19.Yomba E (2006) The modified extended Fan sub-equation method and its application to the (2 + 1)-dimensional Broer-Kaup-Kupershmidt equation. Chaos Solitons Fractals 27:187–196Google Scholar
- 20.Wu G, Han J, ZhangWet al (2007) New periodic wave solutions to generalized Klein Gordon and Benjamin Equations. Communications in Theoretical Physics 48:815–818Google Scholar
- 21.Sirendaoreji (2007) Auxiliary equation method and new solutions of Klein-Gordon equations. Chaos, Solitons and Fractals 31:943–950Google Scholar
- 22.Jang B (2009) New exact travelling wave solutions of nonlinear Klein-Gordon equations. Chaos, Solitons and Fractals 41:646–654Google Scholar
- 23.Lv X, Lai S,Wu Y (2009) An auxiliary equation technique and exact solutions for a nonlinear Klein-Gordon equation. Chaos, Solitons and Fractals 41:82–90Google Scholar
- 24.Yomba E (2008) A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations. Physics Letters A 372:1048–1060Google Scholar
- 25.Nickel J (2007) Elliptic solutions to a generalized BBM equation. Physics Letters A 364:221–226Google Scholar
- 26.Whittaker ET, Watson GN (1927) A course of modern analysis. Cambridge University Press, CambridgeGoogle Scholar
- 27.Fan EG (2002) Multiple travelling wave solutions of nonlinear evolution equations using a united algebraic method. Journal of Physics A: Mathematical and General 35:6853–6872Google Scholar
- 28.Yomba E (2005) The extended Fan’s sub-equation method and its application to KdV-MKdV, BKK and variant Boussinesq equations. Physics Letters A 336:463–476Google Scholar
- 29.Yan Z (2004) An improved algebra method and its applications in nonlinear wave equations. Chaos Solitons & Fractals 21:1013–1021Google Scholar
- 30.Abdou MA (2008) A generalized auxiliary equation method and its applications. Nonlinear Dynamics 52:95–102Google Scholar
- 31.Huang D, Zhang H (2006) New exact travelling waves solutions to the combined KdV- MKdV and generalized Zakharov equations. Reports on Mathematical Physics 57:257–269Google Scholar
- 32.Yomba E (2008) A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations. Physics Letters A 372:1048–1060Google Scholar
- 33.Kudryashov NA, Zargaryan ED (1996) Solitary waves in active-dissipative dispersive media. Journal of Physics A: Mathematical and General 29:8067–8077Google Scholar
- 34.Pinar Z, Ozis T (2013) An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term. Communications in Nonlinear Science and Numerical Simulation doi: 10.1016/j.cnsns.2012.12.025
- 35.Tatsumi T (1984) Turbulence and chaotic phenomena of fluids. In: Iutam symposia proceedings, Kyoto, Japan, North-Holland, AmsterdamGoogle Scholar
- 36.Kudryashov NA, Zargaryan ED (1996) Solitary waves in active-dissipative dispersive media. Journal of Physics A: Mathematical and General 29:8067–8077Google Scholar
- 37.Kuramoto Y, Tsuzuki T (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progress of Theoretical Physics 55(2):356–369Google Scholar
- 38.Sivashinsky GI (1983) Instabilities, pattern formation, and turbulence in flames. Annual Review of Fluid Mechanics 15:179–199Google Scholar