Dark solitons of the Qiao's hierarchy

Ivanov, Rossen I.; Lyons, Tony

doi:10.1063/1.4764859

Cited by 19 publications

(13 citation statements)

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“…An exact method of solution presented here is also applicable to a variant of the mCH In conclusion, we comment on a paper by Ivanov and Lyons. 8 Applying the IST to an initial value problem of the mCH equation under the boundary condition m → m 0 (= u 0 )…”

Section: Discussionmentioning

confidence: 99%

“…5,7 Quite recently, smooth dark soliton solutions of the integrable hierarchy including the mCH equation were obtained for the variable m using the inverse scattering transform (IST) method and the properties of the one-and two-soliton solutions were explored. 8 The purpose of the present paper is to construct the bright N-soliton solutions of the mCH equation under the boundary condition u → u 0 as |x| → ∞, where N is an arbitrary positive integer and u 0 is a positive constant. We employ an exact method of solution which worked effectively for constructing the N-soliton solutions of the CH and Degaperis-Procesi equations.…”

Section: Introductionmentioning

confidence: 99%

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Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

Matsuno

2013

Journal of Mathematical Physics

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We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel Bäcklund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz at al. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Korteweg-de Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one-and two-soliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Last, we use the Bäcklund transformation to derive an infinite number of conservation laws of the mCH equation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

Matsuno

2013

Journal of Mathematical Physics

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“…Recently, the so called "white" solitons and "dark" ones of Eq. (1.2) have been presented in [32] and [22], respectively. In [2], the authors apply the geometric and analytic approaches to give a geometric interpretation to the variable m(t, x) and construct an infinite-dimensional Lie algebra of symmetries to Eq.…”

Section: Introductionmentioning

confidence: 99%

On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

Liu¹,

Zhang²,

Yin³

2014

Communications on Pure &Amp; Applied Analysis

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In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.2000 Mathematics Subject Classification: 35G25, 35L05

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“…Actually the Qiao's equation together with the CH equation belong to the bi-Hamiltonian hierarchy of equations described by Fokas and Fuchssteiner [25]. These equations also are negative flows (after change of the x-variable) of some known soliton hierarchies [37,39]. This feature has motivated us to explore other examples of integrable non-evolutionary equations by looking at the negative flows of existing hierarchies of soliton equations.…”

Section: Introductionmentioning

confidence: 99%

Integrable negative flows of the Heisenberg ferromagnet equation hierarchy

Ivanov

2020

Eur. Phys. J. Plus

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We study the negative flows of the hierarchy of the integrable Heisenberg Ferromagnet model and their soliton solutions. The first negative flow is related to the so-called short pulse equation. We provide a framework which generates Lax pairs for the other members of the hierarchy. The Inverse scattering, based on the dressing method is illustrated with the derivation of the one-soliton solution.

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Dark solitons of the Qiao's hierarchy

Cited by 19 publications

References 14 publications

Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

Integrable negative flows of the Heisenberg ferromagnet equation hierarchy

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