A reciprocal transformation for the Geng-Xue equation

Abstract

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.

Reciprocal link for a three-component Camassa-Holm type equation

“…which is the spectral problem of an extended modified KdV hierarchy [20], and this extended MKdV hierarchy is related to the extended KdV hierarchy(the Yajima-Oikawa hierarchy [21,22]) via a Miura transformation [20]. Furthermore, as we already mentioned [13], the transformation between (Q 1 , Q 2 , Q 3 ) and (u, v, w) may be also connected via a Miura transformation. Indeed, the Miura transformation…”

“…Based on the Miura link (13), the compatible Hamiltonian operators for another extend MKdV hierarchy of Q = (Q 1 , Q 2 , Q 3 ) T may be obtained, which arẽ…”

“…Besides, the CH equation has a reciprocal link to the first negative flow of the KdV hierarchy [7], and the conservation laws of each may be connected [8]. Subsequently, many other CH type equations (the equations possess peakon solutions) are proposed and studied, such as the DP equation, the Novikov's equation and the Geng-xue equation [9,10,11,12,13,14,15]. Although many of them is integrable but it has some nonstandard features such as the weak Painlevé property, and is still much to understand via reciprocal transformation [16].…”

Two-component generalizations of the Novikov equation

“…They also gave an infinite sequence of conserved quantities of the system (1.4) (also called Geng-Xue equation). 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].…”

“…Next we will utilize the procedure in Ref. [30] to obtain the required Hamiltonians. Substituting these equalities (3.1) to the spectral problem (1.5) yields…”