Chen-Lee-Liu Research Articles

Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations

Abstract Based a spectral problem with an arbitrary parameter and Lenard operator pairs, we derive a generalized 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 MKdV equation and the Sharma–Tasso–Olver equation. Furthermore, the separation of variables for x - and t m -constrained flows of the the generalized Kaup–Newell hierarchy is shown. Especially the Kundu, KN, CLL and GI equations are uniformly decomposed into systems of solvable ordinary differential equations. A hyperelliptic Riemann surface and Abel–Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of these equations are explicitly constructed in terms of the Riemann theta functions by standard Jacobi inversion technique.

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

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 Schrödinger 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.

Integrable systems of derivative nonlinear Schrödinger type and their multi-Hamiltonian structure

A spectral problem and the associated hierarchy of Schrodinger type equations are proposed. It is shown that the hierarchy is integrable in Liouville's sense and possesses multi-Hamiltonian structure. It is found that several kinds of important equation such as the Kaup-Newell (KN) equation, the Chen-Lee-Liu (CLL) equation, the Gerdjikov-Ivanov (GI) equation, the modified Korteweg-de Vries equation and the Sharma-Tasso-Olever equation are members in the hierarchy as its special reductions. Moreover, KN, CLL and GI equations are described by using a unified generalized derivative Schrodinger equation involving a parameter, and their Hamiltonian structure and Lax pairs are also given by unified and explicit formulae.