wronskian技巧.ppt

Wronskian Solutions to Soliton Equations Wronskian Solutions to Soliton Equations Menu 1. Bilinear derivatives 1.1 Definition [H] 1. Bilinear derivatives 1.2 Simple Properties 2. Hirota method [H] 2.1 Bilinear equation 2. Hirota method 2.2 Perturbation expansion 2. Hirota method 2.3 Truncate the expansion: 1-soliton 2. Hirota method 2.4 N-soliton 3. Wronskian technique This technique is developed by Freeman and Nimmo for directly verifying solutions to bilinear equations. [FN] 3. Wronskian technique 3.1 Wronskian 3. Wronskian technique 3.2 Properties 3. Wronskian technique 3.3 Needed equalities (I) 3. Wronskian technique 3.3 Needed equalities (II) 3. Wronskian technique 3.4 Wronskian technique 3. Wronskian technique 3.5 N-soliton in Hirota form and in Wronskian form 3. Wronskian technique 3.5 Generalizaion [SHR] 4. Classification of solutions in Wronskian form 4.1 Normalization of A 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.1 Case I, A has N distinct negative eigenvalues: 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.2 Case II, A has N distinct positive eigenvalues: 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.3 Case III, A has N same negative eigenvalues: 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.4 Case IV, A has N same positive eigenvalues: 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.5 Case V, A has N zero eigenvalues: 4. Classification of solutions in Wronskian form 4.3 Notes Usage of equality (1) Name of solution Limit of solitons Other examples --- Toda lattice 1. Bilinear form Other examples --- Toda lattice 2. Casoratian solution Other examples --- Schrodinger equation 1. Bilinear form Other examples --- Schrodinger equation 2. Double-Wronskian Other examples --- Schrodinger equat