A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations
注意：本論文已在《JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 42, NUMBER 9 SEPTEMBER 2001:4327-4344》發表
Institute of Mathematics, Fudan University, Shanghai 200433, People’s 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 Kaup–Newell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell ~KN! equation, the Chen–Lee–Liu ~CLL! equation, the Gerdjikov–Ivanov ~GI! equation,the Burgers equation, the modified Korteweg-deVries ~MKdV! equation and the Sharma–Tasso–Olver equation. It is shown that the hierarchy is integrable in Liouville’s 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 Kaup–Newell-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 Schro¨dinger 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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