Zheng Chun-Long

Localized Coherent Structures with Chaotic and Fractal Behaviors in a (2+1)-Dimensional Modified Dispersive Water-Wave System*

In this work, we reveal a novel phenomenon that the localized coherent structures of some (2+1)-dimensional physical models possess chaotic and fractal behaviors. To clarify these interesting phenomena, we take the (2+1)-dimensional modified dispersive water-wave system as a concrete example. Starting from a variable separation approach, a general variable separation solution of this system is derived. Besides the stable localized coherent soliton excitations like dromions, lumps, rings, peakons, and oscillating soliton excitations, some new excitations with chaotic and fractal behaviors are derived by introducing some types of lower dimensional chaotic and fractal patterns.

Variable Separation Solutions of Generalized Broer Kaup System via a Projective Method

Using an extended projective method, a new type of variable separation solution with two arbitrary functions of the (2+1)-dimensional generalized Broer-Kaup system (GBK) is derived. Based on the derived variable separation solution, some special localized coherent soliton excitations with or without elastic behaviors such as dromions, peakons, and foldons etc. are revealed by selecting appropriate functions in this paper.

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2 + 1)-Dimensional Long Dispersive Wave System*

By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.

Localized Structures on Periodic Background Wave of (2+1)-Dimensional Boiti Leon Pempinelli System via an Object Reduction

In this paper, we present an object reduction for nonlinear partial differential equations. As a concrete example of its applications in physical problems, this method is applied to the (2+1)-dimensional Boiti–Leon–Pempinelli system, which has the extensive physics background, and an abundance of exact solutions is derived from some reduction equations. Based on the derived solutions, the localized structures under the periodic wave background are obtained.

Two classes of fractal structures for the (2+1)-dimensional dispersive long wave equation

Using the mapping approach via a Riccati equation, a series of variable separation excitations with three arbitrary functions for the (2+1)-dimensional dispersive long wave (DLW) equation are derived. In addition to the usual localized coherent soliton excitations like dromions, rings, peakons and compactions, etc, some new types of excitations that possess fractal behaviour are obtained by introducing appropriate lower-dimensional localized patterns and Jacobian elliptic functions.

General Jacobian elliptic function expansion method and its applications

An extended Jacobian elliptic function expansion method is presented and successfully applied to the nonlinear Schrodinger (NLS) equation and Zakharov equation. We obtain some new solutions besides Fu et als results. The results show that our method is more powerful to construct Jacobian elliptic function and can be applied to other nonlinear physics systems.

Chaotic Dynamical Behaviour in Soliton Solutions for a New (2+1)-Dimensional Long Dispersive Wave System

With the help of variable separation approach, a quite general excitation of a new (2+1)-dimensional long dispersive wave system is derived. The chaotic behaviour, such as chaotic line soliton patterns, chaotic dromion patterns, chaotic-period patterns, and chaotic-chaotic patterns, in some new localized excitations are found by selecting appropriate functions.

Complex wave excitations and chaotic patterns for a general (2+1)-dimensional Korteweg–de Vries system

Starting from an improved mapping approach and a linear variable separation approach, a new family of exact solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for a general (2+1)-dimensional Korteweg de Vries system (GKdV) is derived. According to the derived solutions, we obtain some novel dromion-lattice solitons, complex wave excitations and chaotic patterns for the GKdV system.

Variable Separation Approach to Solve (2+1)-Dimensional Generalized Burgers System: Solitary Wave and Jacobi Periodic Wave Excitations*

By means of the standard truncated Painleve expansion and a variable separation approach, a general variable separation solution of the generalized Burgers system is derived. In addition to the usual localized coherent soliton excitations like dromions, lumps, rings, breathers, instantons, oscillating soliton excitations, peakons, foldons, and previously revealed chaotic and fractal localized solutions, some new types of excitations — compacton and Jacobi periodic wave solutions are obtained by introducing appropriate lower dimensional piecewise smooth functions and Jacobi elliptic functions.

Standing, Periodic and Solitary Waves in (1 + 1)-Dimensional Caudry—Dodd—Gibbon—Sawada—Kortera System

In the paper, the variable separation approach, homoclinic test technique and bilinear method are successfully extended to a (1 + 1)-dimensional Caudry—Dodd—Gibbon—Sawada—Kortera (CDGSK) system, respectively. Based on the derived exact solutions, some significant types of localized excitations such as standing waves, periodic waves, solitary waves are simultaneously derived from the (1 + 1)-dimensional Caudry—Dodd—Gibbon—Sawada—Kortera system by entrancing appropriate parameters.