1. Math 6399 Lecture Notes: How to generate an integrable hierarchy from a spectral problem Dr. Zhijun Qiao (qiao@utpa.edu)qiao@utpa.edu Department of Mathematics The University of Texas – Pan American UTPA, MAGC 1.410

2. Outline Functional Gradient Pair of Lenard’s operators Hierarchy of nonlinear equations Lax pair and integrability Relation to finite-dimensional integrable system Conclusions Today’s talk is based on Qiao, Comm. Math Phys 239(2003), 309-341 www.math.panam.edu/~qiao / Publications-qiao.html

3. Functional Gradient

7. CH hierarchy (Qiao, Comm. Math Phys 239(2003), 309-341)

8. Pair of Lenard’s operators: K and J

9. Pair of Lenard’s operators: K and J For the CH hierarchy

10. Lenard’s operators for ANKS hierarchy

12. A 3rd order spectral problem

13. Hierarchy of nonlinear equations

14. CH Hierarchy

17. Integrability

19. Solution of Matrix equation for the CH hierarchy

20. Lax Form for the CH hierarchy

21. Relation to Finite-dimensional Integrable System

22. Constraint

23. Canonical Hamiltonian System

24. Integrability of the Hamiltonian system

25. Parametric Solutions

26. Parametric solution for the CH equation

27. Explicit Solution ( Qiao, Comm. Math Phys 239(2003), 309-341 )

30. Thanks for your attention Any questions/comments? Today’s talk is based on Qiao, Comm. Math Phys 239(2003), 309-341 www.math.panam.edu/~qiao / Publications-qiao.html qiao@utpa.edu