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Exploring Integrable Hierarchy via Spectral Problems

Learn how to generate integrable hierarchy from spectral problems, featuring functional gradient, Lenard’s operators, nonlinear equations, Lax pair, integrability, and more. Based on Qiao's work in Comm. Math. Phys. (2003).

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Exploring Integrable Hierarchy via Spectral Problems

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  1. Math 6399 Lecture Notes:How to generate an integrable hierarchy from a spectral problem Dr. Zhijun Qiao (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

  4. Functional Gradient

  5. Functional Gradient

  6. 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

  11. Lenard’s operators for ANKS hierarchy

  12. A 3rd order spectral problem

  13. Hierarchy of nonlinear equations

  14. CH Hierarchy

  15. CH Hierarchy

  16. CH Hierarchy

  17. Integrability

  18. Solution of Matrix equation for the CH hierarchy

  19. Lax Form for the CH hierarchy

  20. Relation to Finite-dimensional Integrable System

  21. Constraint

  22. Canonical Hamiltonian System

  23. Integrability of the Hamiltonian system

  24. Parametric Solutions

  25. Parametric solution for the CH equation

  26. Explicit Solution (Qiao, Comm. Math Phys239(2003), 309-341)

  27. 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

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