Research On Algebro-geometric Solutions And Wronskian Technique

This thesis can be divided into two parts:one is the construction of algebro-geometric solutions of soliton equations, and the other is the application of Wron-skian technique in soliton equations.Algebro-geometric solutions of soliton equations can not only reveal inherent structure mechanism of solutions, describe the quasi-periodic behavior of nonlinear phenomena and characteristic for the integrability of soliton equations, but also can be reduced to multi-soliton solutions, elliptic function solutions and so on. So the research on the algebro-geometric solutions of soliton equations is very important. In chapter two, we deal with the algebro-geometric solutions of the two component generalized Burgers hierarchy associated with 2×2 spectral problem. Resorting to the characteristic polynomial of Lax matrix, we introduce an algebraic curve KN of arithmetic genus N, on which we establish the meromorphic function φ and investigate its properties. The corresponding flows are straightened out under the Abel-Jacobi coordinates. Based on these preparations, the explicit Riemann theta function representations of the entire two component generalized Burgers hierarchy are derived. In chapter three, we concentrate on algebro-geometric solutions of a new generalized Burgers hierarchy associated with a 3 x 3 matrix spectral problem. With the aid of Lenard recursion equations and the zero-curvature equation, we derive the generalized Burgers hierarchy. Based on the characteristic polynomial of Lax matrix for the hierarchy, a third order algebraic curve Km-1 with genus m-1 is introduced, on which we establish the associated meromorphic function, Baker-Akhiezer functions and Dubrovin-type equations. The Abel map is introduced to straighten out the corresponding flows. Then by employing the properties of the meromorphic function and Baker-Akhiezer function, we obtain their Riemann theta function representations with the help of the second and third Abel differentials and Riernann-Roch theorem. From which we finally obtain the algebro-geometric solutions of soliton equations.In chapter four, the generalized Wronskian solutions of modified Boussinesq equation are obtained by generalizing the equation satisfied by Wronskian entries to the matrix equation through Hirota bilinear derivative and Wronskian technique. Furthermore, soliton solutions, rational solutions, Matveev solutions, complexiton solutions and interaction solutions are derived by taking special cases in general solutions.In chapter five, we generalize the Wronskian technique. Usually we let poten-tial functions in Wronskian conditions be zero when we find Wronskian solutions of soliton equations. However, in this chapter we will take three-order AKNS equa-tion for an example to study the condition that potential functions are not equal to zero. Finally, generalized double Wronskian solutions for the three-order AKNS equation are successfully obtained through Wronskian technique, N-dark soliton solutions are derived by taking special cases for potential functions. In particular, we give its 1-soliton solution. And a defocusing mKdV equation and its generalized double Wronskian solutions and N-dark soliton solutions are given by reducing.