Introduction to bilinear method

1. Introduction to bilinear method 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 Introduction to bilinear method

2. Menu Part I Basic intrduction Bilinear Derivatives Hirota method Wronskian technique Part II Further discussion Generalization of Wronskian technique Classification of Wronskian solutions Introduction to bilinear method

3. 1. Bilinear derivatives or 1.1 Definition [H] examples Introduction to bilinear method

4. (1) (2) (3) If then and Menu 1. Bilinear derivatives 1.2 Simple Properties Hirota method Introduction to bilinear method

5. Bilinear equation 2. Hirota method [H] Korteweg-de Vries (KdV) equation 2.1 Bilinear equation Introduction to bilinear method

6. 2. Hirota method 2.2 Perturbation expansion Introduction to bilinear method

7. JUST TAKE! 1-soliton 2. Hirota method 2.3 Truncate the expansion: 1-soliton Introduction to bilinear method

8. 2-soliton N-soliton 2. Hirota method 2.4 N-soliton Menu Introduction to bilinear method

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

10. Wronskian Compact form 3. Wronskian technique 3.1 Wronskian Introduction to bilinear method

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 Introduction to bilinear method

12. (1) if then 3. Wronskian technique Equality (1) 3.3 Needed equalities (I) Example usage of equality (1) Introduction to bilinear method

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

14. 3. Wronskian technique 3.4 Wronskian technique Equality (I) Introduction to bilinear method

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! Introduction to bilinear method

16. generalization diagonal arbitrary same 4. Generalization of Wronskian technique 4.1 Generalizaion [SHR] Introduction to bilinear method

17. 4. Generalization of Wronskian technique 4.2 Needed equality Menu Introduction to bilinear method

18. (3). A determines kinds of solutions. (1). A and lead to same solution. (2). Consider to be the normal form of A . 5. Classification of solutions in Wronskian form 5.1 Normalization of A Introduction to bilinear method

19. Wronskian entries Solutions obtained inCase Iare called negatons. When we get N-soliton solutions. 5. Classification of solutions in Wronskian form 5.2 Classification of solutions 5.2.1 Case I, A has N distinct negative eigenvalues: Introduction to bilinear method

20. Wronskian entries Solutions obtained inCase IIare called positons. 5. Classification of solutions in Wronskian form 5.2 Classification of solutions 5.2.2 Case II, A has N distinct positive eigenvalues: Introduction to bilinear method

21. Wronskian entries (*1) (*2) Another choice 5. Classification of solutions in Wronskian form 5.2 Classification of solutions 5.2.3 Case III, A has N same negative eigenvalues: Note: (*1) and (*2) lead to same solution because they are linearly dependent. We call the solution high-order negatons. Introduction to bilinear method

22. 5. Classification of solutions in Wronskian form Wronskian entries 5.2 Classification of solutions 5.2.4 Case IV, A has N same positive eigenvalues: or Name: high-order positons Introduction to bilinear method

23. Wronskian entries or Note: sink and cosk do not lead new results due to 5. Classification of solutions in Wronskian form 5.2 Classification of solutions 5.2.5 Case V, A has N zero eigenvalues: Name: rational solution Introduction to bilinear method

24. 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; 5. Classification of solutions in Wronskian form 5.3 Notes Other examples Menu Introduction to bilinear method

25. Equality (1) Usage of equality (1) From the identity [Back to 3.3.2] [N-soliton] Introduction to bilinear method

26. KdV equation Lax pair Lax pair (u=0): [negatons] [positons] Name of solution conditions for Wronskian entries [Mat] Introduction to bilinear method

27. Limit of solitons [Back to 5.3] Introduction to bilinear method

28. Other examples --- Toda lattice 1. Bilinear form Introduction to bilinear method

29. Condition: Other examples --- Toda lattice 2. Casoratian solution Introduction to bilinear method

30. Other examples --- Schrodinger equation 1. Bilinear form Introduction to bilinear method

31. 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. Introduction to bilinear method

32. Bilinear NLSE complex matrix independent of x Other examples --- Schrodinger equation 3. Double-Wronskian solution to the NLSE Conditions: and [Back to 5.3] Introduction to bilinear method

33. Other e.g. --- Equations with self-consistent terms 1 Bilinear form Introduction to bilinear method

34. Other e.g. --- Equations with self-consistent terms 2 Wronskian solution [Back to 5.3] Introduction to bilinear method

35. [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 4.1] [Back to Name of solution] Introduction to bilinear method

36. Thank You! Thank You! Introduction to bilinear method