Discontinuity, Nonlinearity, and Complexity
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the ExtendedReduced Ostrovsky Equation: PhasePlane, MultiInfinite Series and Variational Formulations
Discontinuity, Nonlinearity, and Complexity 5(1) (2016) 85100  DOI:10.5890/DNC.2016.03.009
U. Tanriver$^{1}$, G. Gambino$^{2}$, S. Roy Choudhury$^{3}$
$^{1}$ Department of Mathematics, Texas A&M UniversityTexarkana, USA
$^{2}$ Department of Mathematics, University of Palermo, Italy
$^{3}$ Department of Mathematics, University of Central Florida, Orlando, USA
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Abstract
In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the socalled extendedreduced Ostrovsky Equation (exROE). A recent extension of phaseplane 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 multiinfinite 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 travelingwave equations, as well as ensure simultaneous convergence and continuity of the multiinfinite 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.
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