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

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), / Publications-qiao.html

Functional Gradient

CH hierarchy (Qiao, Comm. Math Phys 239(2003), )

Pair of Lenard’s operators: K and J

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

Lenard’s operators for ANKS hierarchy

A 3rd order spectral problem

Hierarchy of nonlinear equations

CH Hierarchy

Integrability

Solution of Matrix equation for the CH hierarchy

Lax Form for the CH hierarchy

Relation to Finite-dimensional Integrable System

Constraint

Canonical Hamiltonian System

Integrability of the Hamiltonian system

Parametric Solutions

Parametric solution for the CH equation

Explicit Solution ( Qiao, Comm. Math Phys 239(2003), )

Thanks for your attention Any questions/comments? Today’s talk is based on Qiao, Comm. Math Phys 239(2003), / Publications-qiao.html