The Wronskian Solutions For Soliton Equations

The dissertation mainly focuses on the solutions in Wronskian form for the KdV-typc equation including: the most possibly wide condition of thc Wronskian condition equation, the cxplict gencral solutions to condition equations and the relations between the related different solutions. Besides, the N-soliton solutions and double Wronskian solutions of the nonisospectral 3rd-order AKNS equation are given, the nonisospectal modified KdV equation and its N-soliton solutions of Hirota's form and double Wronskian form are obtained by reduction. We also derive solutions to two nonisospectral modified KdV equations with self-consistent sources and the nonisospectral sine-Gordon equations with self-consistent sources by means of Hirota method and Wronskian technique, respectively, and study their dynamics also.Concretely, in the third chapter, we first investigate properties of the lower Tocplitz matrices i.e., the matrices commuting with a Jordan block, by which we derive explict general solutions to equations satisfied by Wronskian/Casoratian entry vectors, which we call condition equations. These solutions are given according to the coefficient matrix in the condition equation taking diagonal or Jordan block form. Limit relations between these different solutions are described. Taking KdV equation and the Toda lattice to serve as two examples for solutions in Wronskian form and Casoratian form, respectively.In the fourth chapter, a nonisospectral 3rd-order AKNS equation is generated from the AKNS spectral problem. Bilinear form of this equation is given by introducing an transformation. N-soliton solutions and double Wronskian solutions are obtained by Hirota's method and Wronskian technique, respectively. Reducing this equation to nonisospectal modified KdV equation and the corresponding Hirota and double Wronskian solutions are obtained.In the fifth chapter, two nonisospectral modified KdV equations with self-consistent sources are derived, which correspond to the time-dependent spectral parameterλsat isfyingλ_t=λandλ_t=λ~3, respectively. Gauge transformation between the first nonisospectral equation (corresponding toλ_t=λ) and its isospectral counterpart is given, by which exact solutions and conservation laws for the nonisospectral one are easily listed out. Besides, solutions to the two nonisospectral modified KdV equations with self-consistent sources are derived by means of Hirota method and Wronskian technique, respectively. Nonisospectrai dynamics, including 1-soliton characteristics in non-uniform mediae, two solitons scattering and special behaviors related to sources (for example, the "ghost" solitons in degenerate 2-soliton case), are investigated analytically. With the same method, we give the corresponding conclusion of non-isospectral sine-Gordon equations with selfconsistent sources.