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A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations

ע⣺ՓJOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 9 SEPTEMBER 2001:4327-4344l
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(SCI䛱)

Engui Fan

Institute of Mathematics, Fudan University, Shanghai 200433, Peoples Republic of China

~Received 2 October 2000; accepted for publication 4 June 2001

By introducing a spectral problem with an arbitrary parameter, we derive a KaupCNewell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the KaupCNewell ~KN! equation, the ChenCLeeCLiu ~CLL! equation, the GerdjikovCIvanov ~GI! equation,the Burgers equation, the modified Korteweg-deVries ~MKdV! equation and the SharmaCTassoCOlver equation. It is shown that the hierarchy is integrable in Liouvilles sense and possesses multi-Hamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the  KaupCNewell-type hierarchy is also presented. In addition, an N-fold Darboux transformation of the Kundu equation is constructed with the help of its Lax pairs and a reduction technique. According to the Darboux transformation, the solutions of the Kundu equation is reduced to solving a linear algebraic system and two first-order ordinary differential equations. It is found that the KN, CLL, and GI equations can be described by a Kundu-type derivative nonlinear Schrodinger equation involving a parameter. And then, we can construct the Hamiltonian formulations, Lax pairs and N-fold Darboux transformations for the Kundu, KN, CLL, and GI equations in explicit and unified ways.

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