Peakons and periodic cusp wave solutions in a generalized Camassa–Holm equation

The Bäcklund transformation (BT) for the Camassa-Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent variables. Superposition principles are given for the action of double BTs on the variables of the CH and the potential CH equations. Applications of the BT and its superposition principles are presented, specifically the construction of travelling wave solutions, a new method to construct multi-soliton, multi-cuspon and solitoncuspon solutions, and a derivation of generating functions for the local symmetries and conservation laws of the CH hierarchy.

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“…where c 1 , c 2 , c 3 , c 4 are constants. (This generalized equation is referred to in [26,64] as the "CH-r equation".) The BT is…”

Section: The Bäcklund Transformation For the Camassa-holm Equationmentioning

confidence: 99%

Bäcklund Transformations for the Camassa–Holm Equation

Rasin

Schiff

2016

J Nonlinear Sci

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“…Finally, for the parameters as in Fig. 3(e) periodic cusp wave solutions (both of peak and valley type) arise (see also [32]).…”

Section: The Spe Equation With Arbitrary Coefficientsmentioning

confidence: 81%

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

Gambino

Tanriver

Guha

et al. 2015

Communications in Nonlinear Science and Numerical Simulation

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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.

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“…They derived the two kinds of peakons in three different ways. In [17], Zhang et al proved that for α = 0, (1.2) has one peakon of the form…”

Section: Introductionmentioning

confidence: 98%

Single peak solitary wave solutions for the generalized Camassa–Holm equation

Wang

2013

Applicable Analysis

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The solitary wave solutions to class of nonlinear fourth order variant of a generalized Camassa-Holm equation is investigated by employing the interesting perturbation method. The solitary wave solutions to Camassa-Holm equation is given in the form of Jacobi functions and expressed by the hyperbolic functions, the trigonometric functions with different modulus k. The numerical results for different types of solutions of the nonlinear Camassa-Holm equation are simply discussed.

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Peakons and periodic cusp wave solutions in a generalized Camassa–Holm equation

Cited by 15 publications

References 11 publications

Bäcklund Transformations for the Camassa–Holm Equation

Bäcklund Transformations for the Camassa–Holm Equation

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

Single peak solitary wave solutions for the generalized Camassa–Holm equation

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