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[en] The prolongation structure technique of Wahlquist and Estanbrook is improved and applied to a new equation proposed by Z.J. Qiao [J. Math. Phys. 48 (2007) 082701]. Two potentials and two pseudopotentials are obtained, from which a new type of inverse scattering problem, Lax equations, and infinite number of conservation laws are obtained. (general)
[en] Based on the method developed by Nucci, the pseudopotentials, Lax pairs and the singularity manifold equations of the generalized fifth-order KdV equation are derived. By choosing different coefficient, the corresponding results and the Baecklund transformations can be obtained on three conditioners which include Caudrey-Dodd-Gibbon-Sawada-Kotera equation, the Lax equation and the Kaup-kupershmidt equation. (general)
[en] In this paper, we investigate higher-order rogue wave solutions of the Kundu–Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu–Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu–Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu–Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged. (paper)
[en] In this paper, we investigate the generalized Q-S synchronization between the generalized Lorenz canonical form and the Roessler system. Firstly, we transform an arbitrary generalized Lorenz system to the generalized Lorenz canonical form, and the relation between the parameter of the generalized Lorenz system and the parameter of the generalized Lorenz canonical form are shown. Secondly, we extend the scheme present by [Yan ZY. Chaos 2005;15:023902] to study the generalized Q-S synchronization between the generalized Lorenz canonical form and the Roessler system, the more general controller is obtained. By choosing different parameter in the generalized controller obtained here, without much extra effort, we can get the controller of synchronization between the Chen system and the Roessler system, the Lue system and the Roessler system, the classic Lorenz system and the Roessler system, the Hyperbolic Lorenz system and the Roessler system, respectively. Finally, numerical simulations are used to perform such synchronization and verify the effectiveness of the controller.