Localized Excitations And Related Fractal And Chaotic Behaviors In (2+1)-dimensional Nonlinear Systems

Chaos, fractals and solitons are three important parts of nonlinearity. Conventionally, these three aspects are treated independently since one often considers solitons are basic excitations of an integrable model while chaos and fractals are elementary behaviors of non-integrable systems. In other words, one does not analyze the possibility of existence of chaos and fractals in a soliton system. However, the above consideration may not be complete, or even should be modified, especially in some higher dimensions.In this dissertation, we will discuss the localized excitations and related fractal and chaotic behaviors in (2+1)-dimensional (two spatial-dimensions and one time dimension) nonlinear systems, which were originated from many natural sciences, such as fluid dynamics, plasma physics, field theory, condensed matter physics, mechanical and optical problems. With help of variable separation approach in linear physics and symmetry reduction theory in nonlinear physics, the multilinear variable separation approach and the direct algebra method were extended to nonlinear physics successfully, then a new algorithm, a general extended mapping approach, was proposed and applied to various (2+1)-dimensional nonlinear systems. Based on multilinear variable separation solutions and general mapping solutions respectively, abundant localized excitations and related fractal and chaotic behaviors for (2+1)-dimensional nonlinear models are investigated as well as rich evolution properties for these localized structures are discussed. The research results indicate that fractals and chaos in higher-dimensional soliton systems are quite universal phenomena. Meanwhile, it is also shown that one can establish the relationship between multilinear variable separation approach and extended mapping approach, and even Charkson-Kruskal reduction method. The main contents are summarized as follows.In the first chapter, we outline a brief history and the current state on studying solitary waves and solitons, as well as review the traditional theoretical relations among solitons, chaos and fractals and list some new or typical (2+1)-dimensional nonlinear systems. The research arrangements of the dissertation are also given out in the end of the chapter.In the second chapter, the multilinear variable separation approach is extended andapplied to several (2+l)-dimensional nonlinear models, such as generalized Ablowitz-Kaup-Newell-Segur system, generalized Broer-Kaup system, generalized Nizhnik-Novikov-Veselov model, general perturbed nonlinear Schrodinger equation, Boiti-Leon-Pempinelli system, and new dispersive long water wave system etc. A quite "universal" variable separation formula with several arbitrary function which is valid for a large classes of (2+l)-dimensional nonlinear models is obtained. In terms of the "universal" formula, various localized excitations, such as multi-dromion solutions, multi-lump solutions, multi-compacton solutions, multi-peakon solutions, multi-foldon solutions, lattice dromion solutions, oscillating dromion solutions, ring-soliton solutions, motiving or static breather solutions, instanton solutions, periodic wave solutions, chaotic pattern structures and fractal pattern structures for (2+l)-dimensional nonlinear systems are revealed by selecting appropriate initial and/or boundary conditions. Based on the plots and theoretical analysis, we explored some typical localized excitaions. Dromions are localized solutions decaying exponentially in all directions, which can be driven not only by straight line solitons but also driven 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. Dromion lattice is a special type of multi-dromion solution. The oscillating dromion solution is a dromion oscillating in special dimensional direction. Ring solitons are not the point-like localized excitations, which are not equal to zero identically at some closed curves and decay exponentially away from the closed curves. The breathers may breath in their amplitudes, shapes, distances among the peaks and even the number of the peaks. The amplitudes of instantons will change fleetly with the time. Peakons are peaked-like solitons at their wave crests in which one-order derivatives are not continuous. Compactons with finite wavelengths are a class of solitary waves with compact supports. Foldons are a class of multi-valued solitary waves, which can be folded in all directions. The fractal solitions and chaotic solitons reveal fractal characteristic and chaotic dynamic behaviors in solitary waves, respectively.In the third chapter, the direct algebraic method based on traveling wave reduction is generalized to solve nonlinear partial differential systems and (2+l)-dimensional nonlinear models with constants and variable coefficients respectively. The tanh function approach, Ja-cobi elliptic function method and deformation mapping approach are introduced and extended respectively, then applied to several class of nonlinear models, such as Ablowitz-Ladik-Lattice system, Hybrid-Lattice system, Toda Lattices system, relativity Toda Lattices system, discrete mKdV system and variable coefficient KdV system etc. by making use of computer algebra. Rich solitary wave solutions and Jacobian doubly periodic wave structures for theabove mentioned nonlinear partial differential systems are obtained, as well as abundant solitary waves, periodic waves, Jacobian doubly periodic waves and Weierstrass doubly periodic waves, rational function solutions and exponential function solutions to (2+l)-dimensional nonlinear models with constants and variable coefficients are derived.In the forth chapter, a new algorithm, i.e. a general extended mapping approach, was proposed and applied to various (2+l)-dimensional nonlinear systems, such as Broer-Kaup-Kupershmidt system, Boiti-Leon-Pempinelli system, generalized Broer-Kaup system and dispersive long water-wave model. A new type of variable separation solution (also named extended mapping solution) with two arbitrary functions, which is valid for all the above-mentioned nonlinear systems, is derived. Then making further the new mapping approach in a symmetric form, we find abundant mapping solutions to above-mentioned (2+l)-dimensional nonlinear systems. In terms of the new type of mapping solution, we can find rich localized excitations. Actually, all the localized excitations based on the multilinear variable separation solutions can be re-derived from the new mapping solutions.Based on a new universal extended mapping solution derived from (2+l)-dimensional nonlinear systems in chapter 4, chapter 5 is devoted to revealing some new or typical localized coherent excitations and their evolution properties contained in (2+l)-dimensional nonlinear systems. By introducing suitably these arbitrary functions, we constructed considerably novel localized structures, such as solitons with and without propagating properties, some semi-folded localized structures with and without phase shafts, and certain localized excitations with fission and fusion behaviors. Some typical localized excitations with fractal properties and chaotic behaviors are also discussed. Why the localized excitations possess such kinds of chaotic behaviors and fractal properties? If one considers the boundary or initial conditions of the chaotic and fractal solutions obtained here, one can straightforwardly find that the initial or boundary conditions possess chaotic and fractal properties. These chaotic and fractal properties of the localized excitations for an integrable model essentially come from certain "nonintegrable" chaotic and fractal boundary or initial conditions. From these theoretical results, one may interpret that chaos and fractals in higher-dimensional integrable physical models would be a quite universal phenomenon. Meanwhile, we have established a simple relation between the multilinear variable separation solutions and the universal extended mapping solutions, which are essentially equivalent by taking certain variable transformation. Therefore, all the localized excitations based on the multilinear variable separation solution can be re-derived by the universal extended mapping solution. The general extended map-...