Math 6399 lecture notes how to generate an integrable hierarchy from a spectral problem
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Math 6399 Lecture Notes: How to generate an integrable hierarchy from a spectral problem. Dr. Zhijun Qiao ( [email protected] ) Department of Mathematics The University of Texas – Pan American UTPA, MAGC 1.410. Outline. Functional Gradient Pair of Lenard’s operators

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Math 6399 Lecture Notes: How to generate an integrable hierarchy from a spectral problem

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Math 6399 lecture notes how to generate an integrable hierarchy from a spectral problem

Math 6399 Lecture Notes:How to generate an integrable hierarchy from a spectral problem

Dr. Zhijun Qiao

([email protected])

Department of Mathematics

The University of Texas – Pan American

UTPA, MAGC 1.410


Outline

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


Functional gradient

Functional Gradient


Functional gradient1

Functional Gradient


Functional gradient2

Functional Gradient


Functional gradient3

Functional Gradient


Ch hierarchy qiao comm math phys 239 2003 309 341

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


Pair of lenard s operators k and j

Pair of Lenard’s operators: K and J


Pair of lenard s operators k and j for the ch hierarchy

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


Lenard s operators for anks hierarchy

Lenard’s operators for ANKS hierarchy


Lenard s operators for anks hierarchy1

Lenard’s operators for ANKS hierarchy


A 3rd order spectral problem

A 3rd order spectral problem


Hierarchy of nonlinear equations

Hierarchy of nonlinear equations


Ch hierarchy

CH Hierarchy


Ch hierarchy1

CH Hierarchy


Ch hierarchy2

CH Hierarchy


Integrability

Integrability


Solution of matrix equation for the ch hierarchy

Solution of Matrix equation for the CH hierarchy


Lax form for the ch hierarchy

Lax Form for the CH hierarchy


Relation to finite dimensional integrable system

Relation to Finite-dimensional Integrable System


Constraint

Constraint


Canonical hamiltonian system

Canonical Hamiltonian System


Integrability of the hamiltonian system

Integrability of the Hamiltonian system


Parametric solutions

Parametric Solutions


Parametric solution for the ch equation

Parametric solution for the CH equation


Explicit solution qiao comm math phys 239 2003 309 341

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


Thanks for your attention

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

[email protected]


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