Multi-peakons and a theorem of Stieltjes

Beals, Richard; Sattinger, D. H.; Szmigielski, Jacek

doi:10.1088/0266-5611/15/1/001

Cited by 260 publications

(303 citation statements)

References 4 publications

(9 reference statements)

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“…As well as in the present system, in that equation the peakons coexist with regular solitons [20]. In the next subsection, we demonstrate that the peakons, which are found only as limit-form solutions in the no-SPM case σ 3 = 0, become generic solutions in the case σ 3 = 0.…”

Section: (B)supporting

confidence: 69%

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Singular and regular gap solitons between three dispersion curves

2002

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A general model is introduced to describe a wave-envelope system for the situation when the linear dispersion relation has three branches, which in the absence of any coupling terms between these branches, would intersect pair-wise in three nearly-coincident points. The system contains two waves with a strong linear coupling between them, to which a third wave is then coupled. This model has two gaps in its linear spectrum. As is typical for wave-envelope systems, the model also contains a set of cubic nonlinear terms. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, either a planar waveguide, or a bulk-layered medium resembling a photonic-crystal fiber, which carry a triple spatial Bragg grating. Another physical system described by the same general model is a set of three internal wave modes in a density-stratified fluid, whose phase speeds come into close coincidence for a certain wavenumber. A nonlinear analysis is performed for zero-velocity solitons, that is, they have zero velocity in the reference frame in which the third wave has zero group velocity. If one may disregard the self-phase modulation (SPM) term in the equation for the third wave, we find an analytical solution which shows that there simultaneously exist two different families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons, which have finite amplitude and energy, but a singularity in the first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two nearly vanishes, the soliton family remains drastically different from that in the uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there may again simultaneously exist two different families of generic stable soliton solutions, namely, regular ones and peakons. Direct simulations show that both types of solitons are stable in this case.

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Section: (B)supporting

confidence: 69%

“…shows that the cuspons look like peakons; that is, except for the above-mentioned narrow region of the width |x| ∼ κ 2 , where the cusp is located, they have the shape of a soliton with a discontinuity in the first derivative of S(x) and a jump in the phase φ(x), which are the defining features of peakons ( [17,20]). …”

Section: Cuspons the Case σ 3 =mentioning

confidence: 99%

Singular and regular gap solitons between three dispersion curves

2002

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“…The system for N = 2 peakons is (we assume that x 1 < x 2 for all y and t, i.e. the case for which the N -peakon solution for the CH equation is obtained in [22,23]…”

Section: Peakon Solutions Of a (1+2) -Dimensional Equation From The Cmentioning

confidence: 99%

“…The CH solitary waves are stable solitons if ω > 0 [6,12,13,21] or peakons if ω = 0 [1,22,23]. The KdV and CH equations can also be interpreted as geodesic flow equations for the respective L 2 and H 1 metrics on the Bott-Virasoro group [24,25,26,27,28,29].…”

Section: Introductionmentioning

confidence: 99%

Equations of the Camassa-Holm hierarchy

Ivanov¹

2009

Theor Math Phys

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The squared eigenfunctions of the spectral problem associated with the CamassaHolm (CH) equation represent a complete basis of functions, which helps to describe the inverse scattering transform for the CH hierarchy as a generalized Fourier transform (GFT). All the fundamental properties of the CH equation, such as the integrals of motion, the description of the equations of the whole hierarchy, and their Hamiltonian structures, can be naturally expressed using the completeness relation and the recursion operator, whose eigenfunctions are the squared solutions. Using the GFT, we explicitly describe some members of the CH hierarchy, including integrable deformations for the CH equation. We also show that solutions of some (1 + 2) -dimensional members of the CH hierarchy can be constructed using results for the inverse scattering transform for the CH equation. We give an example of the peakon solution of one such equation.

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“…The N -peakon solutions (1.4) are weak solutions with discontinuous first derivatives at the positions q j of the peaks; q j , p j are canonical coordinates and momenta in an integrable finite-dimensional Hamiltonian system; the interpretation of weak solutions is discussed further in [12]. The N -peakon interaction was obtained in [3], with the explicit peakon-antipeakon interaction being given in [4]. The stability of peakons in the Camassa-Holm equation was proved in [13,14].…”

Section: Introductionmentioning

confidence: 99%

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

Holm¹,

Hone²

2005

JNMP

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We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x). When b = 2 and g is the peakon kernel (i.e. g(x) = exp(−|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b = 3. Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable. However, for b = 2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses. On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g. The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b = 1.

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Multi-peakons and a theorem of Stieltjes

Abstract: A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

Cited by 260 publications

References 4 publications

Singular and regular gap solitons between three dispersion curves

Singular and regular gap solitons between three dispersion curves

Equations of the Camassa-Holm hierarchy

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

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