Solutions To Soliton Equations And Integrable Systems

The major contents in this paper include: constructing the C-D pair methods and exact solutions of solitary evolution equations with traveling wave solutions, period solutions, the relations between obtaining Backlund transformations and soliton solutions. We discuss the similarity reductions, which involve classical Lie Group and nonclassical Lie group, the generations of soliton hierarchies of equations and the structure equations of Lie group, Hamiltonian structures and Lax representations, the various constructing methods of the integrable couplings etc.In the second chapter, we introdcucing the C-D integrable system and its roles of obtaining exact solutions of solitary equations. Further using the method yields solutions to many nonlinear evolution equations with important physics senses, these solutions include that exact solitary solutions, singular solutions , period solutions and rational fraction solutions. Various methods of generating C-D pair are discussed. We study the methods of constructing Darboux transformations by using C-D pair. As applications, the Darboux transformations of Duffing-form equation are presented, in particular, the method for constructing solitary solutions of Duffing-form equation by the use of Darboux transformations of an isospectral problem with 3 potentials is showed.In the third chapter, major discussions of constructing methods of Backlund transformations (BT), including the BT with no parameter and the BT with some parameters. As example illustration, we obtain superposition formula and infinite conserved laws of the Benjamin equation.In the forth chapter, we discuss the similarity reductions of nonlinear evolution equations. We take the heat-conduction equation for example to illustrate the constructing method of getting nonclassical Lie groups. Using nonlinear function method to solve the similarity reductions of differential equations is presented, and further the similarity reductions of the well-known Boussinesq equation are obtained. Specially, we improve the direct reduction method (CK method) to give rise to the various forms of similarity reductions of the generalized Burgers equation, includingtraveling wave reduction, logarithm reduction, power-form reduction, rational fraction reduction etc..Furthermore these reductions are interpreted by the nonclassical Lie group method.In the fifth chapter, we discuss the methods of generating nonlinear evolution equations and their integrability. A few aspects are included as follows:(1) Expanding the applying scopes of Tu-model, i.e. the pattern in the loop algebraA1 is expanded to one in the loop algebra A2. It follows that the Lax pairtransformations, Hamiltonian structures, Darboux transformations of higher-order symmetry constraint flows are obtained.(2) Constructing a new loop algebra G leads to the integrable couplings of the KN hierarchy etc.(3) A general method of constructing Lax pair transformations, by using loopalgebra G, the integrable couplings of the TD hierarchy , the generalized AKNS hierarchy are obtained.(4) Constructing another new loop algebra G , which is different from the loop algebra G, obtains the integrable coupling of the BPT hierarchy.