Convergent analytic solutions for homoclinic orbits in reversible and non-reversible systems

Communications in Nonlinear Science and Numerical Simulation

Tanriver

Guha

et al. 2015

Self Cite

In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane analysis is first employed to show the existence of breaking kink wave solutions in certain parameter regimes. Secondly, smooth traveling waves are derived using a recent technique to derive convergent multi-infinite series solutions for the homoclinic (heteroclinic) orbits of the traveling-wave equations for the SPE equation, as well as for its generalized version with arbitrary coefficients. These correspond to pulse (kink or shock) solutions respectively of the original PDEs.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 SPE 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 SPE equations considered here, the results obtained are both new and timely.

Section: Singular Solutions Of the Spementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Regular and singular pulse and front solutions and possible isochronous behavior in the short-pulse equation: Phase-plane, multi-infinite series and variational approaches

Communications in Nonlinear Science and Numerical Simulation

Tanriver

Guha

et al. 2015

Self Cite

“…As it is well known, homoclinic and heteroclinic orbits of the traveling wave ODE (of any PDE) correspond to pulse and front (shock or kink) solutions of the governing PDE. In particular, we apply a recently developed technique [5,24] to analytically compute convergent multi-infinite series solutions for the possible homoclinic and heteroclinic orbits of the traveling-wave ODEs of Eqs. (1.4)- (1.6).…”

Section: Introductionmentioning

confidence: 99%

“…Since the later terms in the series fall off exponentially, we show high accuracy may be obtained for the pulse and front shapes using only small number of terms. The actual convergence of such series is analogous to the earlier treatments [5,24], and it is omitted here. The plan of the paper is the following: in Section 2, the recently developed theory for singular traveling-wave ODEs is reviewed [17,18].…”

Section: Introductionmentioning

confidence: 99%

Smooth and non-smooth traveling wave solutions of some generalized Camassa–Holm equations

Rehman

Communications in Nonlinear Science and Numerical Simulation

Choudhury

2014

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In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa-Holm (GCH) equations. A recent, novel application of phaseplane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M -wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH 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. We also show the traveling wave nature of these pulse and front solutions to the GCH NLPDEs.

“…In Section 2, the traveling wave ODE of the exROE equation is considered. A recently developed technique (see [7], [27]) is employed to construct convergent series solutions for its homoclinic and heteroclinic orbits, corresponding to solitary wave and front (pulse) solutions of the original exROE NLPDE. A Lagrangian for the exROE equation is developed in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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

Tanriver

Choudhury

2016

DNC

Self Cite

In this paper we employ three recent analytical approaches to investigate several classes of traveling wave solutions of the so-called extended-reduced 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 traveling-wave 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.