On bi-Hamiltonian structure of two-component Novikov equation

Li, Nianhua; Liu, Q. P.

doi:10.1016/j.physleta.2012.11.023

Cited by 60 publications

(36 citation statements)

References 12 publications

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“…Finally, we show that the Geng-Xue equation (18) and (19) in the transformed variables is associated with the first negative flow in the modified Boussinesq hierarchy. To this end, we start with the following Lax pair:…”

Section: Provides Us a Miura Transformationmentioning

confidence: 80%

“…(18) and (19), thus they do not constitute the Lax representation of (18) and (19). To find a proper Lax representation, we introduce the vector = (φ, φ y , φ yy ) T and find…”

Section: A Reciprocal Transformationmentioning

confidence: 99%

“…18 Furthermore, its spectral and inverse spectral problem is solved by Lundmark and Szmigielski. 15 Although the CH type equations and the KdV type equations are quite different, they are closely related.…”

Section: Introductionmentioning

confidence: 99%

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A reciprocal transformation for the Geng-Xue equation

Li¹,

Niu²

2014

Journal of Mathematical Physics

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Articles you may be interested inGeneralized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transformIn this paper, we construct a reciprocal transformation for the Geng-Xue equation and show that, with help of this transformation, we relate the first negative flow of the modified Boussinesq hierarchy to the Geng-Xue equation. Furthermore, we analyze the construction of conserved quantities and present new ones. C 2014 AIP Publishing LLC. [http://dx.

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Section: Provides Us a Miura Transformationmentioning

confidence: 80%

“…(18) and (19), thus they do not constitute the Lax representation of (18) and (19). To find a proper Lax representation, we introduce the vector = (φ, φ y , φ yy ) T and find…”

Section: A Reciprocal Transformationmentioning

confidence: 99%

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A reciprocal transformation for the Geng-Xue equation

Li¹,

Niu²

2014

Journal of Mathematical Physics

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“…Further, Hone and Wang [2] presented the matrix Lax representation, the biHamiltonian structure, an infinite number of conservation laws and established the relationship between the Novikov equation and the Sawada-Kotera equation by a reciprocal transformation. As Li and Liu [3] stated, Eq. (1.1) is the Camassa-Holm type equation with cubic nonlinearity and they also introduced the two-component Novikov equation with a bi-Hamiltonian structure.…”

Section: Introductionmentioning

confidence: 99%

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

Pan¹,

Yi²

2021

JNMP

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In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UU τyy − U y U τy + U 2 U τ + 3U y = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.

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“…This is a two-component CH type equation which is derived by Geng and Xue, and the authors also calculated this equation also admits multi-peakons and infinite many conservation laws, but the bi-Hamiltonian structure was constructed by Li and Liu [4]. In fact, (2) is an extension of Novikov (Nov) equation [5] if we take u = v,…”

Section: Introductionmentioning

confidence: 99%

Local well-posedness and global existence for a multi-component Novikov equation

2020

Applicable Analysis

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Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence for some two-component case in H s . Finally we verify that the system possesses peakons and periodic peakons.Correspondence should be addressed to Yuxi Hu; hu-yuxi@163.comNovikov equation was derived by Novikov, used the perturbative symmetry approach in classification of nonlocal PDES with three order nonlinearity. It should be note that Novikov equation is not symmetrical, which means (u, x) (−u, −x). Another interesting reduction for (2) is DP equation if we take v = 1,The DP equation was derived by Degasperis and Procesi [6] by applying the method of asymptotic integrability to a three order dispersive PDE. A big feature for DP equation is shock peakon [7,8] u(t, x) = − 1 t + k sgn(x)e −|x| .

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On bi-Hamiltonian structure of two-component Novikov equation

Cited by 60 publications

References 12 publications

A reciprocal transformation for the Geng-Xue equation

A reciprocal transformation for the Geng-Xue equation

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

Local well-posedness and global existence for a multi-component Novikov equation

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