Study on Integrable Properties for Two Kinds of Variable-coefficient Nonlinear Partial Differential Equations
|School||Beijing University of Posts and Telecommunications|
|Keywords||Nonlinear Partial Differential Equations B (a ¨) cklund transform Lax pair Darboux transformation Painlevé nature Soliton Solutions|
In physics and engineering in many mathematical models often ultimately boils down to a nonlinear partial differential equations, these equations possess various properties, such as explicit and exact solutions, conservation laws, Hamilton structure, etc. For some explanation of physical phenomena becomes is very important. However, the number of nonlinear partial differential equations is enormous, and with the rapid development of modern science and technology, a large number of new PDE positive emerged from the various disciplines. Therefore, the properties of these equations is more and more important. Soliton solutions of nonlinear partial differential equations are a special class of solutions. If the nonlinear partial differential equations represent a physical process, then the corresponding soliton solutions also have definite physical meaning. Currently, the soliton as an important branch of nonlinear science is being studied extensively. In this thesis, the theory of nonlinear partial differential equations, based on the use of computer symbolic computation two nonlinear partial differential equations, ie, Boussinesq equations with variable coefficients and variable coefficient fifth-order Korteweg-de Vries (KdV) equation in nature studied. The main contents include the following aspects: (1) Painlevé test. In 1983, Weiss, Tabor and Carnevale Painlevé through the promotion of detection methods for ordinary differential equations, partial differential equations and Painlevé proposed detection algorithm. This method not only provides a nonlinear evolution equation determining whether a completely integrable necessary conditions, but can be studied using the method of nonlinear evolution equations integrable nature, such as B (?) Cklund transformation, Lax pairs, etc., and make that it can effectively be used for many integrable nonlinear system properties. Meanwhile, as a \This chapter focusses Boussinesq equations with variable coefficients Painlevé test proved that the equation under certain constraints is Painlevé integrable, that equation has Painlevé nature. Finally, Painlevé truncated expansion method to construct the equation from B (?) Cklund transformations. (2) In the actual science and engineering, the variable coefficient nonlinear partial differential equations with constant coefficients with respect to, the more effectively describe the actual physical process, therefore, for varying coefficient models of it all the more important. In the third chapter, the paper presents a comparative study effective nonlinear partial differential equations with variable coefficients method. The method is through the establishment of a coordinate transformation relations, making the new coordinate system, will have to study nonlinear partial differential equations with variable coefficients into a standard equation with constant coefficients. Transform using the established relationship between the various properties of the standard equations mapped to equations with variable coefficients, so as to study the nature of nonlinear evolution equations with variable coefficients purpose. With this method, the third chapter, were established through the Boussinesq equations with variable coefficients and fifth order KdV equation with variable coefficients coordinate transformation, they were transformed into the corresponding constant coefficient equations to study, but has been having these two equations product properties constraints. Using the known Boussinesq equations with constant coefficients series of properties obtained by transforming relational mapping Boussinesq equations with variable coefficients B (?) Cklund transform, nonlinear superposition formula, Lax integrable nature of the peer. (3) Darboux transformation method is to construct explicit solutions of nonlinear equations an extremely effective method. Starting from a trivial solution (usually taken zero solution), once or several times in succession through the Darboux transformation equations can get a single or multi-soliton solutions of soliton solutions. Constructing such a transformation is the key to determine a way to maintain the form of equation Lax pair invariant gauge transformations. In this chapter, based on the third chapter of the theoretical basis as well as the conclusions, constructed out of the Boussinesq equations with variable coefficients Darboux transformation and fifth order KdV equation with variable coefficients Lax pair and Darboux transformation. Darboux transformation obtained through fifth order KdV equation with variable coefficients of new single-soliton and two soliton solutions. (4) Finally, the third chapter in the Boussinesq equations with variable coefficients obtained for B (?) Cklund transform the given initial solution thus obtained Boussinesq equations with variable coefficients new soliton solutions. Use Mathematica, when taking different variations coefficients for the Boussinesq equations with variable coefficients obtained and fifth order KdV equation for mapping analysis, and its physical meaning and application may be discussed.