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A family of completely integrable multiHamiltonian systems explicitly related to some celebrated equationsע⣺ՓJOURNAL OF MATHEMATICAL PHYSICS
VOLUME 42, NUMBER 9 SEPTEMBER 2001:43274344l 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 KaupCNewelltype 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 KortewegdeVries ~MKdV! equation and the SharmaCTassoCOlver equation. It is shown that the hierarchy is integrable in Liouvilles sense and possesses multiHamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finitedimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the KaupCNewelltype hierarchy is also presented. In addition, an Nfold 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 firstorder ordinary differential equations. It is found that the KN, CLL, and GI equations can be described by a Kundutype derivative nonlinear Schrodinger equation involving a parameter. And then, we can construct the Hamiltonian formulations, Lax pairs and Nfold Darboux transformations for the Kundu, KN, CLL, and GI equations in explicit and unified ways. 1g[PDFʽȫҪʹܛAbode Acrobat(ܛ^Ҋ,վṩd) 2dՓȫՈc(113KB) վ䛵ıߵՓģ 1The zero curvature representation for hierarchies of nonlinear evolution equations 2Extended tanhfunction method and its applications to nonlinear equations 3Two new applications of the homogeneous balance method 4Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method

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