A new Liouville transformation for the Geng-Xue system

Li, Hongmin; Chai, Wang Yin

doi:10.1016/j.cnsns.2017.01.027

Cited by 9 publications

(12 citation statements)

References 23 publications

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“…Turning to the AGX integrable hierarchy, based on Theorem 2.5, one can readily construct two Hamiltonian operators for the transformed system (3.6) by applying the Liouville transformation (3.7) from the given Hamitonian pair K and J introduced in (3.8) for the GX system. Indeed, resulting Hamiltonian operators admitted by (3.6) were given in [38] as follows:…”

Section: The Liouville Correspondence Between the Gx And Agx Integrab...mentioning

confidence: 99%

“…which was introduced by Geng and Xue [22], and so is referred to be the Geng-Xue (GX) system; see [43] and references therein. As a prototypical multi-component integrable system with cubic nonlinearity, the GX system (1.9) admits special peakon solutions and has recently attracted much attention [38,39,40,43,44]. In [38], it was shown that there exists a certain Liouville transformation converting the Lax-pair of the GX system (1.9) into the Lax-pair of the following integrable system…”

Section: Introductionmentioning

confidence: 99%

“…where q = v m 2/3 n −1/3 and p = u m −1/3 n 2/3 . The system (1.10) is bi-Hamiltonian, whose structure is derived in [38]. We shall elucidate the entire associated Geng-Xue (AGX) integrable hierarchy in which (1.10) is the first negative flow.…”

Section: Introductionmentioning

confidence: 99%

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Liouville Correspondences between Integrable Hierarchies

Kang

Liu

Olver³

et al. 2017

SIGMA

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Abstract. In this paper, we study explicit correspondences between the integrable Novikov and Sawada-Kotera hierarchies, and between the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada-Kotera equations, and the isospectral problems of the Degasperis-Procesi and Kaup-Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada-Kotera and KaupKupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis-Procesi equations.

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Section: The Liouville Correspondence Between the Gx And Agx Integrab...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Liouville Correspondences between Integrable Hierarchies

Kang

Liu

Olver³

et al. 2017

SIGMA

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“…It is mentioned that the homogeneous and local properties of the Hamiltonian functionals were discussed [20]. Also, the Geng-Xue equation is related to a negative flow in a modified Boussinesq hierarchy by a reciprocal transformation [23] and the behaviour of the bi-Hamiltonian structures under the transformation was studied [21]. Moreover, the Geng-Xue equation was shown to admit multi-peakon solutions [28,29,34] and its Cauchy problem was considered [35,13].…”

Section: Introductionmentioning

confidence: 99%

Smooth multisoliton solutions of the Geng-Xue equation

Li¹,

Liu²

2022

Preprint

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We present a reciprocal transformation which links the Geng-Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. We discuss two reductions of the reciprocal transformation for the Degasperis-Procesi and Novikov equations, respectively. With the aid of the Darboux transformation and the reciprocal transformation, we obtain a compact parametric representation for the smooth soliton solutions such as multi-kink solutions of the Geng-Xue equation.

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“…Li et al proved it is bi-Hamiltonian [29] and reciprocal linked to a negative flow in the modified Boussinesq hierarchy [30]. Very recently, we constructed a Liouville transformation to connect it with another negative modified Boussinesq equation, and Lax pairs as well as bi-Hamiltonian structures of them are connected [27]. Recently, we make the vector prolongation of the Lax pair (1.5) as follows [28] Φ…”

Section: Introductionmentioning

confidence: 99%