Proceedings of the Seventh International Conference on Management Science and Engineering Management pp 1009-1017 | Cite as

# The Exact Solutions to Analytical Model of Tsunami Generation by Sub-Marine Landslides

## Abstract

Nonlinear differential equations and its systems are used to describe various processes in physics, biology, economics etc. There are a lot of methods to look for exact solutions of nonlinear differential equations: the inverse scattering transform, Hirota method, the Backlund transform, the truncated Painleve expansion. It is well known that different types of exact solutions of an auxiliary equation produce new types of exact travelling wave solutions to nonlinear equations. In this paper, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with a sixth-degree nonlinear term are presented to obtain novel exact solutions of the analytical model of Tsunami generation by sub-marine landslides.

## Keywords

The auxiliary equation technique The analytical model of Tsunami generation by sub-marine landslides Wave solutions## Preview

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## References

- 1.Whitham GB (1965) A general approach to linear and nonlinear waves using a Lagrangian. Journal of Fluid Mechanics 22(2):273–283Google Scholar
- 2.Knobel R (2000) An introduction to the mathematical theory of waves. American Mathematical Society, Providence, RI 3Google 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(1):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(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(44):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 57(7):1107–1114Google Scholar
- 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(47):7011–7015Google Scholar
- 10.Ozis T, Koroglu C (2008) A novel approach for solving the fisher equation using exp-function method. Physics Letters A 372(21):3836–3840Google Scholar
- 11.Yalciner AC, Pelinovsky EN, Okal E et al (2003) Submarine landslides and tsunamis. Kluwer Academic Publisher, NetherlandsGoogle Scholar
- 12.Ma WX, Huang T, Zhang Y (2010) A multiple Exp-function method for nonlinear differential equations and its applications. Physica Scripta 82(6):065003Google 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(8):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 f‥ur Naturforschung-A 64(1):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(2):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(4):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 GJ, Han JH, Zhang WL et al (2007) New periodic wave solutions to generalized klein gordon and benjamin equations. Communications in Theoretical Physics 48:815–818Google Scholar
- 21.Sirendaoreji S (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 YH (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 unified 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 ZY (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 DJ, Zhang HQ (2006) New exact travelling waves solutions to the combined KdVMKdV 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.Chang HC, Demekhin EA (1999) Mechanism for drop formation on a coated vertical fibre. Journal of Fluid Mechanics 380:233–255Google 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 18:2177–2187Google Scholar
- 35.Fuhrman DR, Madsen PA (2009) Tsunami generation, propagation, and run-up with a highorder Boussinesq model. Coastal Engineering 56:747–758Google Scholar