Study On Exact Solutions And Integrability Of Differential Equations With Symbolic Computation

In the middle age of the twentieth century, the most important progress in the fields of physics and mathematics is the descovery of the Invers Scattering Transformation (IST) method. It is the IST method that arouses great interesting to the study of integrability system. Soliton theory, symmetry group theory and the study of infinity conservation laws are always the hotspot problems in nonlinear science. With the rapid development of computer technology and symbolic computation systems, researchers recognize the integrability systems again at a new level and have achieved great success recently. Then a new subject named nonlinear dynamics on the study of common rules of nonlinear phenomena is produced, which includes six aspects: Fractals, Chaos, Solitons, Graphs, Cell Automatic and Complex System. The analysis and understanding to these nonlinear phenomena are reduced to the solve of nonlinear equations such as nonlinear ordinary differential equations (ODEs), nonlinear partial differential equations (PDEs), nonlinear differential-difference equations (DDEs) and functional equations. Hence, in the contemporary study of nonlinear science, the study of exact solutions and integrability of nonlinear systems is the main and hot topic for researches.In this dissertation, the theory of mathematics mechanization proposed by famous mathematician Wu Wentsun is widely used to the constructive study of nonlinear science. Speak concretely, exact solutions, the integrability of symmetry and conservation law are studied with the aid of computer algebra system Maple. Our work is somewhere between the following four aspects:Part I is devoted to build a relationship organically between differential geometry and differential equations (DEs). It is shown that two-component Wadati-Konno-Ichikawa (WKI) equation, i.e. a generalization of the wellknown WKI equation is obtained from the motion of space curves in Euclidean geometry, and it is exactly a system for the graph of the curves when the curve motion is governed by the two-component modified Korteweg-de Vries (mKdV) flow. At the same time, a n-component generalization to the WKI equation is obtained. Also, starting from the motion of curves, mKdV and its symmetry recursion operator is exhibited explicitly; two- and n-component mKdV systems are obtained. It is shown that WKI systems are gauge equivalent to mKdV systems. The two-component WKI equation admits an infinity number of conservation laws and a recursion formula for the conserved densities is given by considering an eigenvalue problem together with introducing an appropriate transformation.Part II is devoted to study exact solutions for DEs. Nowadays, there exist many approaches for obtaining exact solutions of nonlinear NDEs, such as the IST method, the Backlund and Darboux transformations, symmetry group and differential reduction method, Painleve singular analysis method, Hirota method, the homogeneous balance method, the tanh-function and the extended tanh-function method. However, these methods are only an amalgamation of some techniques, which are dispersive, unsystematically and not general. Lie symmetry group method provides a widely applicable technique to find closed form solutions of ODEs. Applied to PDEs, Lie's method can lead to symmetries. Once the symmetries admitted by the PDEs are found, the orginal PDEs can be reducted and then similarity solutions can be obtained. In this dissertation, much attention is focused on symmetry reductions and similarity solutions for several PDEs. As for as the two-component WKI equation, symmetry reductions and similarity solutions which correspond to an optimal system of its Lie point symmetry groups are given systematically. Potential symmetry is one of the most effective method for finding invariant solutions for DEs. By it, the Fokker-Planck eqution is studied. Several new potential symmetries and the corresponding group invariant solutions are obtained. Apart from Lie symmetry method, exact solutions such as solitary wave solutions and loop solitons for WKI systems and mKdV systems are constructed by using REQs method.Part III is devoted to study the integrability of nonlinear evolution equations from the conservation law view of point starting from scaling symmetry. The property of possessing infinity conservation laws of the classical KdV equation attracts the attention and stimulates the curiosity of many physicists and mathematicians throughout the world to study the additional properties of other integrability systems. The existence of an infinite number of conservation laws results in a chain of discovery such as the famous IST method, the Miura transformation and the Lax pair. The stability and particle-like behavior of the soliton solutions could only be explained by the existence of many conservation laws. Possessing infinity conservation laws is one of the most effective manner to judge whether a NDE is integrable or not. However, it is not trivial to construct the conserved quantities of a nonlinear system only by hand. More conservation laws are await the appearance of computers. In this dissertation, starting from scaling symmetry, a modified and expanded approach to construct polynomial conservation laws which depend only on the dependent variables and their derivatives and not on the independent variables explicitly is proposed. The problem of "intermediate expresssion swell" resulting from large amount integral and differential computation is dealt with efficiently. At the same time, Ritt-Wu's method is used correctly in some key steps while designing the algorithm.Then a Maple software package CONSLAW is developed. For a parameterized nonlinear evolution system, CONSLAW can output not only all the possible conservation laws but also all the parameters constraints which guarantee the existence of conservation laws automatically. The filtered parameters constraints are the necessary conditions for a parameterized equation to pass the Painleve test. Also, new integrable systems, as byproducts, may thus be obtained. Furthermore, together with the package CONSLAW, another algorithm to compute conservation laws which depend explicitly on independent variables for some type of nonlinear evolution equations is introduced.Part IV is devoted to study the integrability of nonlinear evolution equations from the symmetry view of point also starting from scaling symmetry. The conserved geometric features of solitons are intimately bound up with notion of symmetry. The symmetry groups of DEs were first studied by Sohpus Lie. In 1918, Emmy Noether proved the remarkable theorem which gives a one-to-one correspondence between symmetry groups and convervation laws for the Euler-Lagrange equations. Possessing infinity symmetries is also one of the most effective manner to judge whether a NDE is integrable or not. Symmetries of DEs can be obtained from Lie symmetry group method. However, because the determining system of DE's symmetry is a linear or nonlinear overdetermined PDEs, which is usually difficult to be solved completely. Hence, it is necessary to design a direct algebraic method to construct generalized symmetries for NDEs. It is lucky that one can directly construct the polynomial generalized symmetries for nonlinear evolution equations starting from scaling symmetry and in view of the concepts of generalized symmetry and Frechet derivative. Based on the algorithm, three symbolic packages GSymmel, GSymme2 and GSymme3 are provided, which are used to compute generalized symmetries which depend only on dependent variables and their derivatives with respect to x for nonlinear evolution equation as well as the system of nonlinear evolution equations, and also the ones which depend not only on dependent variables and their derivatives but also on independent variables explicitly for nonlinear evolution equations respectively.