ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Extended-Reduced Ostrovsky Equation: Phase-Plane, Multi-Infinite Series and Variational Formulations

Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 85--100 | DOI:10.5890/DNC.2016.03.009

U. Tanriver$^{1}$, G. Gambino$^{2}$, S. Roy Choudhury$^{3}$

$^{1}$ Department of Mathematics, Texas A&M University-Texarkana, USA

$^{2}$ Department of Mathematics, University of Palermo, Italy

$^{3}$ Department of Mathematics, University of Central Florida, Orlando, USA

Abstract

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extendedreduced Ostrovsky Equation (exROE). A recent extension of phase-plane analysis is first employed to show the existence of breaking kink wave solutions and smooth periodic wave (compacton) solutions. Next, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of the travelingwave equations for the exROE equation. These correspond to pulse solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding traveling-wave equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. And finally, variational methods are employed to generate families of both regular and embedded solitary wave solutions for the exROE PDE. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and it is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the assumed ansatz for the trial functions). Thus, a direct error analysis is performed, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that not much is known about solutions of the family of generalized exROE equations considered here, the results obtained are both new and timely.

References

1.  [1] Parkes, E.J. (2008), Periodic and solitary travelling-wave solutions of an extended reduced Ostrovsky equation, SIGMA Symmetry Integrability Geom. Methods Appl., 4, Paper 053, 17.
2.  [2] Stepanyants, Y. A. (2008), Solutions classification to the extended reduced Ostrovsky equation, SIGMA Symmetry Integrability Geom. Methods Appl., 4, Paper 073, 19.
3.  [3] Stepanyants, Y.A. (2006), On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons, Chaos Solitons Fractals, 28(1), 193-204.
4.  [4] Ostrovskii, L.A. (1978), Nonlinear internal waves in the rotating ocean, Okeanologiia, 18, 181-191.
5.  [5] Choudhury, S.R. and Gambino, G. (2013), Convergent analytic solutions for homoclinic orbits in reversible and nonreversible systems, Nonlinear Dynam., 73(3), 1769-1782.
6.  [6] Rehman, T., Gambino, G., and Choudhury, S. R. (2014), Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations, Commun. Nonlinear Sci. Numer. Simulat., 19(6), 1746-1769.
7.  [7] Li, J. (2010), The dynamics of two classes of singular nonlinear travelling wave equations and loop solutions, In C. David and Z. Feng, editors, Solitary waves in fluid media, Bentham Science, Sharjah.
8.  [8] Li, J. and Dai, H. (2007), On the study of singular nonlinear traveling wave equations: dynamical approach, Science Press, Beijing.
9.  [9] Wang, X. (2009), Si'lnikov chaos and Hopf bifurcation analysis of Rucklidge system, Chaos Solitons Fractals, 42(4), 2208 - 2217.
10.  [10] Jacobi, C.G.J. (1844), Theoria novi multiplicatoris systemati aequationum differentalium vulgarium applicandi, J. für Math., 27,199.
11.  [11] Lie, S. (1874), Veralgemeinerung und neue Verwerthung der Jacobischen Multiplicator- Theorie. Fordhandlinger i Videnokabs - Selshabet i Christiania, 255-274.
12.  [12] Whittaker, E. T. (1918), Lezione sulla teoria dei gruppi continui finiti di transformazioni, Enrico Spoerri Ed., Pisa.
13.  [13] Madhava Rao, B.S. (1940), On the reduction of dynamical equations to the Lagrangian form, Proc. Benares Math. Soc., n. Ser., 2, 53-59.
14.  [14] Whittaker, E. T. (1988), A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge.
15.  [15] Nucci,M. C. (2005), Jacobi last multiplier and Lie symmetries: a novel application of an old relationship, J. Nonlinear Math. Phys., 12(2), 284-304.
16.  [16] Nucci, M. C. and Leach, P. G. L. (2008), Jacobi's last multiplier and Lagrangians for multidimensional systems, J. Math. Phys., 49(7), 073517-8.
17.  [17] Nucci, M. C. and Tamizhmani, K. M. (2010), Lagrangians for dissipative nonlinear oscillators: the method of Jacobi last multiplier, J. Nonlinear Math. Phys., 17(2), 167-178.
18.  [18] Choudhury, A. G. Guha, P. and Khanra, B. (2009), On the Jacobi last multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé-Gambier classification, J. Math. Anal. Appl., 360(2), 651-664.
19.  [19] Calogero, F. (2001), Isochronous dynamical systems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369(1939), 1118-1136.
20.  [20] Calogero, F. (2012), Isochronous Systems, Oxford University Press, Oxford.
21.  [21] Chalykh, O.A. and Veselov, A.P. (2005), A remark on rational isochronous potentials, J. Nonlinear Math. Phys., 12(suppl. 1), 179-183.
22.  [22] Nucci,M. C. and Leach, P. G. L. (2002), Jacobi's last multiplier and the complete symmetry group of the Euler-Poinsot system, J. Nonlinear Math. Phys., 9(suppl. 2), 110-121. Special issue in honour of P. G. L. Leach on the occasion of his 60th birthday.
23.  [23] Nucci, M. C. and Leach, P. G. L. (2004), Jacobi's last multiplier and symmetries for the Kepler problem plus a lineal story, J. Phys. A, 37(31), 7743-7753.
24.  [24] Nucci, M. C. (2007), Jacobi's last multiplier, Lie symmetries, and hidden linearity: profusion of "goldfish", Teoret. Mat. Fiz., 151(3), 495-509.
25.  [25] Tanriver, U., Choudhury, S. R. and G. Gambino (2015) Lagrangian dynamics and possible isochronous behavior in several classes of nonlinear second order oscillators via the use of Jacobi last multiplier, Int. J. Nonlinear Mech., 74, 100-107.
26.  [26] Conte, R. (2007), Partial integrability of the anharmonic oscillator, J. Nonlinear Math. Phys., 14(3), 454-465.
27.  [27] Hill, J. M., Lloyd, N. G. and Pearson, J. M. (2007) Algorithmic derivation of isochronicity conditions, Nonlinear Anal., 67(1), 52-69.
28.  [28] Cherkas, L. A. (1976), Conditions for a Lienard equation to have a center, Differential Equation, 12, 201-206.
29.  [29] Choudhury, A. G. and Guha, P. (2010), On isochronous cases of the Cherkas system and Jacobi's last multiplier, J. Phys. A, 43(12), 125202.
30.  [30] Guha, P. and Choudhury, A. G. (2011), The role of the Jacobi last multiplier and isochronous systems, Pramana, 77(5), 917-927.
31.  [31] Guha, P. and Choudhury, A. G. (2013), The Jacobi last multiplier and isochronicity of Liénard type systems, Rev. Math. Phys., 25(6), 1330009-31.
32.  [32] Kaup, D.J. and Malomed, B. A. (2003), Embedded solitons in lagrangian and semi-lagrangian systems, Physica D, 184(14), 153 - 161. Complexity and Nonlinearity in Physical Systems - A Special Issue to Honor Alan Newell.
33.  [33] Kaup, D.J. and Vogel, T.K. (2007), Quantitative measurement of variational approximations, Phys. Lett. A, 362(4), 289-297.
34.  [34] Whitam, G.B. (1974), Linear and Nonlinear Waves, Wiley, New York.