Evolution Solutions for the Ostrovsky Equation: A Perspective from an Auxiliary Elliptic Equation

Abstract:

The Ostrovsky equation is used to describe the oceanic waves affected by Earth rotation. The auxiliary elliptic equation was employed as a direct method to construct the evolution solutions for the Ostrovsky equation in this manuscript. Detailed analysis of evolution solutions of the Ostrovsky equation is presented. The periodic solutions, which read Jacobi elliptic function solutions, hyperbolic solutions and trigonal solutions, are obtained. More parameters are included in the evolution solutions, the parameter space is enlarged. This method adds a new route to explore evolution solutions of nonlinear differential equation in a perspective from auxiliary elliptic equation.

• Periodical: Advanced Materials Research (Volumes 538-541)
• Pages: 621-624
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.538-541.621
• Online since: June 2012
• Authors: Chun Huan Xiang
• Keywords: Auxiliary Elliptic Equation, Evolution Solutions, Numerical Simulation, Ostrovsky Equation, Periodic Solution

References

[1] J.P. Boyd, G.Y. Chen, Five regimes of the quasi-cnoidal steadily translating waves of the rotation-modified Korteweg-de Vries (Ostrovsky) equation. Wave Motion, 35 (2002), 141-155.

DOI: 10.1016/s0165-2125(01)00097-x

[2] V.M. Galkin, Y.A. Stepanyants, On the existence of stationary solitary waves in a rotating °uid. Prikl Matamat i Mekhanika, 55 (1991), 1051- 1055 (in Russian); (English translation: J Appl Maths Mechs, 55 (1991), 939-943).

DOI: 10.1016/0021-8928(91)90148-n

[3] R.H.J. Grimshaw, Adjustment processes and radiating solitary waves in a regularised Ostrovsky equation, Eur J Mech B/Fluids, 18 (1999), 535-543

DOI: 10.1016/s0997-7546(99)80048-x

[4] S.P. Nikitenkova, Y.A. Stepanyants, L.M. Chikhladze, Solutions of the modified Ostrovskii equation with cubic non-linearity, Prikl Matamat I Mekhanika 64 (2000), 276-284 (in Russian); (English translation: J Appl Maths Mechs 64 (2000), 267-274).

DOI: 10.1016/s0021-8928(00)00048-4

[5] C. Ramirez, D. Renouard, Y.A. Stepanyants, Propagation of cylindrical waves in a rotating fluid, Fluid Dyn Res, 30 (2002), 169-196.

DOI: 10.1016/s0169-5983(02)00040-0

[6] Singh. Jitendra, Praveen Kumar Gupta and K.N. Rai, Solution of fractional bioheat equations by finite difference method and HPM, Mathematical and Computer Modelling, 54 (2011), 2316-2325.

DOI: 10.1016/j.mcm.2011.05.040

[7] S.V. Muzylev, Nonlinear equatorial waves in the ocean, Digest of Reports 2nd All-Union Congress of oceanographers, Sebastopol, USSR, 2 (1982), 26-27 (in Russian).

[8] Yu-Chao Tang, Ji-Gen Peng and Li-Wei Liu, Strong convergence of shrinking projection methods for a family of Pseudocontractive mappings in Hilbert spaces(2011), doi:10.1016 /j.mcm.2011. 07.038.

DOI: 10.1016/j.mcm.2011.07.038

[9] Y.A. Stepanyants, On stationary solutions of the reduced Ostrovsky equation: Periodic waves, compactons and compound solitons, Chaos, Solitons and Fractals, 28 (2006), 193-204.

DOI: 10.1016/j.chaos.2005.05.020

[10] V.O. Vakhnenko, High-frequency soliton-like waves in a relaxing medium, J Math Phys 40 (1999), 2011-2020.

DOI: 10.1063/1.532847

[11] S.K. Liu, S.D. Liu, Nonlinear Equat ions in Physics, Beijing: Peking University Press, 2000.

[12] G.T. Liu, A New Auxiliary Elliptic Equation for Solving Nonlinear Evolution Equations, J. Inner. M. Norm. Univer., 34 (2005), 383-389.

[13] W. Malfliet, The tanh method: A tool for solving certain classes of nonlinear evolution and wave equations, J. Comput. Appl. Math.,164 (2004), 529-541.

DOI: 10.1016/s0377-0427(03)00645-9