Symbolic Computation On The Soliton Solutions Of Some Nonlinear Models

With the quick development of information technology, symbolic computation, as a new branch of artificial intelligence, has become perfect and ripe gradually, and has been applied to the research of nonlinear science. Nowadays, because of its powerful ability in exactly dealing with the complicated and tedious calculations, and its convenient and direct practicability, symbolic computation has grown into a necessary and wonderful assistant tool for the study of nonlinear science. Especially in the latest decade, much progress has already been made in the further development of soliton theory based on symbolic computation. As a core issue of nonlinear science, the soliton theory has always been the hot research topic, and absorbed the close attention of numerous researchers in the world. Both in theoretical research and in pratical application, the study on soliton solutions of nonlinear models has been an important and difficult issue, which may be encountered in every field of science.Based on symbolic computation and the development status of this area, the present dissertation is carried out to analytically investigate the soliton solutions of some physically important nonlinear models, such as generalizing the known methods that are used to construct the exact soliton solutions, analyzing the stabilities and structures of solitons, studying the interactions between solitonic waves, including elastic and inelastic ones. The research work of the dissertation mainly includes the following aspects:①With the help of symbolic computation, the traditional bilinear method is extended to dealing with the more complicated higher-order and higher-dimensional nonlinear models. The main idea of our extended bilinear method is that, through the bilinear transformation derived by employing the balancing-act method, the original model is transformed into a homogeneous equation, which is then solved directly by the format-parameter expansion method. One key point of the extended method is that the unimaginable complicated calculations involved in the solving process with the increase of the order of solution, are easily done via symbolic computation. This extended procedure skips the process of equation splitting and bilinearization so as to avoid the possibility of introducing additional limitation on the structure of the soliton solutions. Besides, the computerized function of bilinear operation is defined in software Mathematica, and this greatly conveniences our calculation.②Generally speaking, most of the generalized variable-coefficient models are not completely integrable. But when some constraints on the variable-coefficient functions are satisfied, the variable-coefficient models may possess some integrabel properties, such as multi-soliton solutions, Backlund transformation, and Lax pair. With the help of symbolic computation, the multi-soliton solutions of a nonisospectral and variable-coefficient Korteweg-de Vries model and a generalized Kadomtsev-Petviashvili model with variable coefficients are investigated by using the extended bilinear method. Based on their bilinear forms and by means of properties of the bilinear operator, the Backlund transformations in both bilinear form and Lax pair form are derived. Furthermore, starting from the Backlund transformations, the Lax pairs and the nonlinear superposition formula are obtained. Finally, some figures are plotted to analyze the effects of the variable coefficients on the stabilities and propagation characteristics of the solitonic waves.③In recent years, more and more interests have been attracted on the research of partially coherent interactions of multi-structures for nonlinear Schrodinger models. Hereby, via symbolic computation, a generalized (1+1)-dimensional coupled nonlinear Schrodinger model is investigated. With the obtained multi-soliton solutions, some propagation and interaction properties of the solitons are discussed simultaneously. Moreover, some figures are plotted to graphically analyze the pairwise collisions and partially coherent interactions among three solitons.④Recent developments have already revealed that soliton collisions encountered in some coupled nonlinear Schrodinger models possess novel features, such as the shape changes with intensity redistributions or energy exchanges. It is a desirable feature of such collisions that solitons during interaction can transform energy, in other words, solitons can be amplified without using other techniques. This type of collision-based amplification has aroused increasing interests because it does not require any external amplification medium, neither does it induce any noise. Motivated by this, the shape-changing collision of solitons in a generalized coupled higher-order nonlinear Schrodinger model is investigated through asymptotic analysis on the various features underlying the desirable collision between two solitons.⑤It is of great significance to study localized coherent structures and their interaction behavior in (2+1)-dimensional integrable nonlinear systems. With the aid of symbolic computation, under investigation is the (2+1)-dimensional dispersive long wave equations. Based on the analytic multi-soliton solutions derived by the extended bilinear method, some novel soliton interaction behaviors for this model are revealed. At the same time, the main propagation properties are studied through the graphical illustration. Our analysis shows that the novel interactions include the fission phenomenon, which is a kind of non-elastic behavior. Compared with the completely elastic interaction of solitons, a simple answer to why the fission phenomenon exists is given from the viewpoint of mathematics.In conclusion, combined with the technique of symbolic computation, the bilinear method is extended to investigate some complicated higher-order or higher-dimensional models. These models are of practical importance in various branches of physics like fluid dynamics, fiber communications, superconductors, Bose-Einstein condensates, plasma physics, atmosphere and oceans. The study of the models includes deriving soliton solutions, B(a|¨)cklund transformations and Lax pairs, where symbolic computation becomes a necessary and helpful implement. Furthermore, through computer simulation in software Mathematica, the stabilities, propagation characteristics, and the elastic and inelastic interaction properties of the solitonic waves are all graphically analyzed. Moreover, their underlying physical applications are discussed in detail. It is expected that the analytic results and relevant discussions in this dissertation will be observed in the future laboratory experiments, and will be helpful to the future studies.