Analytic Study On Some Nonlinear Evolution Equations In Several Fields

Many nonlinear phenomena can always be observed to exist in the nature science field, engineering field, and so on. In order to well and thoroughly understand the nonlinear phenomena, more and more people are gradually turning their interest to the nonlinear science. In the past few years, nonlinear science, as one of the newly fundamental subjects, has made remarkable development. Moreover, nonlinear science can be used in the study of the aspects of nonlinear science, for instance, the formation mechanism and motion characteristics. Nonlinear science is mainly divided into three research branches:soliton, fractal and chaos. Soliton theory, as one of three branches in nonlinear science, has acquired appreciably progress. For example, people have observed the bell-shaped solitons in the fluid field, envelope solitons in the nonlinear optical field, and Davydov solitons in the biology field. Besides, to study the soli-ton phenomena in the various fields, in the final analysis, it is necessary to establish and analyze the kinds of the nonlinear evolution equations (such as the nonlinear partial differential equations). Therefore, by virtue of several analytic methods including the Painleve test, Lax pair, Hi-rota bilinear technique and Bell-polynomial approach, and with the aid of the computerized mathematical software, such as Mathematica, this dissertation will aim to analytically study certain types of the nonlin-ear evolution equations from several fields, including variable-coefficient, higher-dimensional, single and coupled nonlinear evolution equations.The main research contents of this dissertation will be organized as follows:(1) Investigation is carried out on the generalized nonlinear Schrodinge (NLS) equation with radial symmetry. Through an appropriate transfor-mation, the original equation is manipulated. By virtue of the Painleve test, the dependent variable transformation (or the bilinear transforma-tion) is obtained. Via the Hirota bilinear technique, the bilinear form is acquired. With symbolic computation, one-and two-soliton solutions are constructed, and N-soliton solutions are deduced as well. Besides, from the bilinear form, the bilinear Backlund transformation (BT) is derived. The dynamics of the one and two solitons is simulated graphically and analyzed.(2) The variable-coefficient Korteweg-de Vries (KdV) and generalized variable-coefficient extended KdV equations are analytically researched.(a) With rational dependent variable transformation and Hirota bilin-ear technique, the bilinear form and soliton solutions for the variable-coefficient KdV equation are obtained. Based on those soliton solutions, soliton amplitude and velocity are both explicitly given, and asymptotic analysis is used to prove that the collisions are elastic. Besides, the elastic collisions between two solitons (or among three solitons) are di-rectly shown in some figures, e.g., the head-on and overtaking collisions;(b) Through the Bell-polynomial approach, binary-Bell-polynomial form for the generalized variable-coefficient extended KdV equation is derived. With the connection between the Bell polynomials and Hirota bilinear operators, a more general bilinear form for this equation is obtained. Symbolic computation is used to explicitly give the one-, two-and N-soliton solutions. Besides, bell-polynomial BT, bilinear BT and Lax pair are constructed.(3) By virtue of the introduction of auxiliary functions and Bell-polynomial approach, the general shallow water wave (GSWW), general Kaup-Kupershmidt (GKK) and (2+1)-dimensional breaking soliton equa-tions are investigated.(a) Bilinear forms, bilinear BT and soliton solu-tions for the GSWW and GKK equations are respectively derived;(b) Bilinear form and soliton solutions for the (2+1)-dimensional breaking soliton equation are given. Moreover, soliton propagation and collision are plotted and discussed, e.g., the collision between two solitons, the col-lision between one soliton and V-type structure, and the collision between two V-type structures.(4) The (2+1)-dimensional Boiti-Leon-Pempinelli (BLP) and mod-ified Kadomtsev-Petviashvili (mKP) equations are analytically investi-gated.(a) With the Bell-polynomial approach, the binary-Bell-polynomial form, bilinear form and soliton solutions for the (2+1)-dimensional BLP equations are obtained. Moreover, the bell-polynomial BT, bilinear BT and Lax pair are given. Soliton propagation, elastic collision and inelastic collision are discussed and simulated.(b) Via the Hirota bilinear tech-nique, a new bilinear form and soliton solutions for the mKP equation is derived through the rational transformation and auxiliary function. Based on the one-soliton solution, the parametric conditions are obtained for the existence of the shock, elevation soliton, and depression soliton, which are shown in some figures. Based on the two-soliton solution, some types of the soliton collisions are analyzed and plotted, e.g., the parallel elastic collision between the shock wave and elevation soliton, the oblique elastic collision between the elevation and depression solitons, and the oblique inelastic collision between two shock waves.(5) On the basis of the Hirota bilinear technique and symbolic com-putation, the bilinear form for the3-coupled NLS equations are obtained. Via two types of the formal parameter expansions, two kinds of the mixed-type (2-bright-1-dark and1-bright-2dark) two-and three-soliton solutions are constructed. In virtue of the asymptotic analysis and graphical sim-ulation on these solutions, it is found that inelastic collisions only exist in the2-bright-1-dark solitons, while they cannot be observed in the1-bright-2-dark ones. Moreover, the evolution of bound solitons and the collision between the. bound solitons and regular one soliton are discussed for two types of mixed-type solitons. For example, in the2-bright-l-dark two solitons, bright solitons show the breather-like structures, while the dark ones keep parallel; in the2-bright-l-dark three solitons, the states of the bound solitons will have a visible change before and after the col-lision with regular one soliton. Above analysis and discussions are also applicable to meet the N-coupled NLS equations.(6) With the Hirota bilinear technique, two types of the bilinear forms for a higher-order NLS equation are derived. Through the two kinds of the formal parameter expansions, two sets of the soliton solutions are constructed, respectively. From those soliton solutions, parametric conditions are respectively given for the existence of (a) bright single-and double-hump solitons,(b) dark single-hump, double-hump and anti-dark solitons. Besides, soliton propagation and collision are discussed and simulated, e.g., the collisions between bright single-hump and double- hump solitons, two dark double-hump solitons and two anti-dark soltions(7) By means of the Hirota bilinear technique, the bilinear forms and mixed-type soliton solutions for the coupled higher-order NLS equations. Based on those solutions, three sets of conditions are obtained for the non-singular solutions, and the condition is derived for the inelastic inter-action through the asymptotic analysis. Besides, the propagation, elastic collisions, inelastic collisions and bound states of the mixed-type solitons are observed in some figures.