Coherent Structures And Interactions Of Nonlinear Wave Models

Solitary wave, which is a special coherent structures, describes a kind of stable nature wave phenomena. In this dissertation, the exact solutions of the 2+1-dimensional ( two spatial- dimensions and one time dimension) nonlinear wave models (equations) which originate from practical water wave and other physical problems are investigated by means of symbolic computation and the more abundant localized and non-localized coherent structures in 2+1-dimensional nonlinear wave models are revealed as well as the rich interaction properties for these structures are discussed.Part I devoted to investigated exact solutions of 2+1-dimensional nonlinear wave equations. (1) The Hirota bilinear method is improved and investigated several 2+1 dimensional nonlinear wave models, such as 2+1-dimensional Schrodinger equation, 2+1-dimensional Maccari equation and 2+1-dimensional breaking soliton equation, the generalized dromion structures with arbitrary functions are obtained and some characteristics are found. (2) The variable separation method is established to deal with 2+1-dimensional nonlinear wave models. The Painlevetruncated expansion approach and homogenous balance approach are employed respectively to explore the variables separation solution of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional Nizhnik-Novikov-Veselov equation, 2+1-dimensional dispersive long wave equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional long dispersive wave equation, 2+1-dimensional generalized Nizhnik-Novikov-Veselov equation, and a quite "universal " variable separation solution formula with several arbitrary function which is valid for a large classes of 2+1- dimensional nonlinear wave models is obtained. (3) The direct algebraic methods are generalized to solving 2+1-dimensional nonlinear wave models. The Jacobi elliptic function method and formal mapping method are introduced and discussed respectively and several class of 2+1-dimensional nonlinear wave models , such as 2+1-dimensional breaking soliton equation, 2+1-dimensional resonant interaction between long and short wave equation, 2+1-dimensional Nizhnik-Novikov-Veselov equation^ 2+1-dimensional Burgers equation, 2+1-dimensional Wu-Zhang equation, 2+1-dimensional Schrodinger equation, 2+1-dimensional Boussinesq equa-tion are studied by making use of Maple and mathematica. Their doubly periodic wave structures and line superposition periodic solutions of Jacobi elliptic functions which will change in their amplitudes, shapes and period are obtained.Partll is devoted to reveals the abundant coherent structures and interaction properties contained in 2+1-dimensional nonlinear wave equations. Prom the 'universal'variable separation solution of 2+1-dimensional nonlinear wave models and by introducing suitably these arbitrary functions, we constructed the considerable abundant coherent structures, including multi-line solitary wave solutions, multi-lump solutions, multi-solitoff solutions, multi-dromion solutions, multi-compacton solution, multi-peakon solution, multi-foldon solution, lattice dromiln solution, oscillating dromion solutions, ring-soliton solutions, motive and static breather solutions, instanton solutions, doubly periodic wave solutions, chaos pattern structures, fractal pattern structures and so on. The development of computer algebra and the application of Maple and Mathematica improve our study and enhance efficiency greatly. Based on the plots and mathematical analysis, we explored all this exotic coherent structures. Dromions are localized solutions decaying exponentially in all directions, which can be driven not only by straight line solitons but also by curved line solitons and can be located not only at the cross points of the lines but also at the closed points of the curves. The solitoff solution decays exponentially in all directions except for a preferred one, two straight line soliton become only one half straight line soliton because of the resonance effect. The dromion lattice is...