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Wronskian Solutions to Soliton Equations

Wronskian Solutions to Soliton Equations. Zhang Da-jun Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China email: djzhang@mail.shu.edu.cn www: http://www.scicol.shu.edu.cn/siziduiwu/zdj/index.htm. Menu. Bilinear Derivatives. Hirota method. Wronskian technique.

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Wronskian Solutions to Soliton Equations

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  1. Wronskian Solutions to Soliton Equations Zhang Da-jun Dept. Mathematics, Shanghai Univ., 200444, Shanghai, China email:djzhang@mail.shu.edu.cn www:http://www.scicol.shu.edu.cn/siziduiwu/zdj/index.htm Wronskian Solutions to Soliton Equations

  2. Menu Bilinear Derivatives Hirota method Wronskian technique Classification of Wronskian solutions References Wronskian Solutions to Soliton Equations

  3. 1. Bilinear derivatives or 1.1 Definition [H] examples Wronskian Solutions to Soliton Equations

  4. (1) (2) (3) If then and Menu 1. Bilinear derivatives 1.2 Simple Properties Hirota method Wronskian Solutions to Soliton Equations

  5. Bilinear equation 2. Hirota method [H] Korteweg-de Vries (KdV) equation 2.1 Bilinear equation Wronskian Solutions to Soliton Equations

  6. 2. Hirota method 2.2 Perturbation expansion Wronskian Solutions to Soliton Equations

  7. JUST TAKE! 1-soliton 2. Hirota method 2.3 Truncate the expansion: 1-soliton Wronskian Solutions to Soliton Equations

  8. 2-soliton N-soliton 2. Hirota method 2.4 N-soliton Menu Wronskian Solutions to Soliton Equations

  9. 3. Wronskian technique This technique is developed by Freeman and Nimmo for directly verifying solutions to bilinear equations. [FN] Wronskian Solutions to Soliton Equations

  10. Wronskian Compact form 3. Wronskian technique 3.1 Wronskian Wronskian Solutions to Soliton Equations

  11. jth column is the derivative of (j-1)th column Derivatives of a Wronskian has simple forms 3. Wronskian technique 3.2 Properties Examples Wronskian Solutions to Soliton Equations

  12. (1) if then 3. Wronskian technique Equality (1) 3.3 Needed equalities (I) Example usage of equality (1) Wronskian Solutions to Soliton Equations

  13. If then (2) In fact, using Laplace’s expansion rule, we have 3. Wronskian technique Equality (2) 3.3 Needed equalities (II) Wronskian Solutions to Soliton Equations

  14. 3. Wronskian technique 3.4 Wronskian technique Equality (I) Wronskian Solutions to Soliton Equations

  15. If take then Hirota 3. Wronskian technique Now we have two forms for N-soliton, Hirota form and Wronskian form. Are they same? 3.5 N-soliton in Hirota form and in Wronskian form They are same! Wronskian Solutions to Soliton Equations

  16. generalization diagonal arbitrary same 3. Wronskian technique 3.5 Generalizaion [SHR] Menu Wronskian Solutions to Soliton Equations

  17. (3). A determines kinds of solutions. (1). A and lead to same solution. (2). Consider to be the normal form of A . 4. Classification of solutions in Wronskian form 4.1 Normalization of A Wronskian Solutions to Soliton Equations

  18. Wronskian entries Solutions obtained inCase Iare called negatons. When we get N-soliton solutions. 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.1 Case I, A has N distinct negative eigenvalues: Wronskian Solutions to Soliton Equations

  19. Wronskian entries Solutions obtained inCase IIare called positons. 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.2 Case II, A has N distinct positive eigenvalues: Wronskian Solutions to Soliton Equations

  20. Wronskian entries (*1) (*2) Another choice 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.3 Case III, A has N same negative eigenvalues: Note: (*1) and (*2) lead to same solution due to their coefficient matrixes having same Jordan form, and we call the solution high-order negatons. Wronskian Solutions to Soliton Equations

  21. 4. Classification of solutions in Wronskian form Wronskian entries 4.2 Classification of solutions 4.2.4 Case IV, A has N same positive eigenvalues: or Name: high-order positons Wronskian Solutions to Soliton Equations

  22. Wronskian entries or Note: sink and cosk do not lead new results due to 4. Classification of solutions in Wronskian form 4.2 Classification of solutions 4.2.5 Case V, A has N zero eigenvalues: Name: rational solution Wronskian Solutions to Soliton Equations

  23. If real coefficient matrix A has N=2M distinct complex eigenvalues, then thses eigenvalues appear in conjugate couple, and we can still get real solutions to the KdV equation;[M] Solutions obtained in Case III, IV, and V are called Jordan block solutions or multipoles solutions in IST sense; Jordan block solutions can be obtained from a limit of Case I or II solutions; Wronskian solution can also be derived based on Sato Theory and Darboux transformation. Conditions for Wronskian entries are usually related to Lax pair; 4. Classification of solutions in Wronskian form 4.3 Notes Other examples Menu Wronskian Solutions to Soliton Equations

  24. Equality (1) Usage of equality (1) From the identity [Back to 3.3.2] [N-soliton] Wronskian Solutions to Soliton Equations

  25. KdV equation Lax pair Lax pair (u=0): [negatons] [positons] Name of solution conditions for Wronskian entries [Mat] Wronskian Solutions to Soliton Equations

  26. Limit of solitons [Back to 4.3] Wronskian Solutions to Soliton Equations

  27. Other examples --- Toda lattice 1. Bilinear form Wronskian Solutions to Soliton Equations

  28. Condition: Other examples --- Toda lattice 2. Casoratian solution Wronskian Solutions to Soliton Equations

  29. Other examples --- Schrodinger equation 1. Bilinear form Wronskian Solutions to Soliton Equations

  30. Other examples --- Schrodinger equation (M+N)-order column vectors: 2. Double-Wronskian If M=0, it is an ordinary N -order Wronskian; if N=0, vice versa. Wronskian Solutions to Soliton Equations

  31. Bilinear NLSE complex matrix independent of x Other examples --- Schrodinger equation 3. Double-Wronskian solution to the NLSE Conditions: and [Back to 4.3] Wronskian Solutions to Soliton Equations

  32. [FN] N.C. Freeman, J.J.C. Nimmo, Soliton solutions of the KdV and KP equations: the Wronskian technique, Phys. Lett. A, 95 (1983) 1-3. [H] R. Hirota, The Direct Method in Soliton Theory (in English), Cambridge University Press, 2004. [M] W.Y. Ma, Solving the KdV equation by its bilinear form: Wronskian solutions, Transaction Americ. Math. Soc., 357 (2005) 1753-1778. V.B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A, 166 (1992) 205-208. [Mat] [N] J.J.C. Nimmo, A bilinear Backlund transformation for the nonlinear Schrodinger equation, Phys. Lett. A, 99 (1983) 279-280. [S] J. Satsuma, A Wronskian representation of n-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn., 46 (1979) 359-360. D.J. Zhang, Singular solutions in Casoratian form for two differential-difference equations, Chaos, Solitons and Fractals, 23 (2005) 1333-1350. [Z] References S. Sirianunpiboon, S.D. Howard, S.K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A, 134 (1988) 31-33. [SHR] [Back to 1.1] [Back to 2.1] [Back to 3] [Back to 3.5] [Back to Name of solution] Wronskian Solutions to Soliton Equations

  33. Thank You! Thank You! Wronskian Solutions to Soliton Equations

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