S. Y. Lou M. Jia and X. Y Tang

1. Conservation Laws Related to Kac-Moody-Virasoro Symmetries S. Y. Lou M. Jia and X. Y Tang

2. Introduction • KMV Symmetry Group • KMV Symmetry Algebra • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

3. Introduction • KMV Symmetry Group • KMV Symmetry Algebra • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

4. Introduction • A well known fact: Importance • Complicated problem • to • Simple one • always? • Symmetry method: G.C.Sym.

5. Introduction • An interesting fact: KMV • Various well known 2+1 dim. integrable systems possesses an infinite dimensional (KMV) • Kac-Moody-Virasoro symmetry algebra

6. Introduction • Methods to find KMV • Standard prolongation meth. • Classical sym. Appro. • Formal series sym. Approach • Group to algebra, method I • Group to algebra, method II

7. Introduction • KMV Symmetry Group • KMV Symmetry Algebra • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

8. KMV Symm. Group • Group to algebra, method I (KP) • Ref. J. Phys. A. 38 (2005) L129 • Ansatz

9. KMV Symm. Group

10. Introduction • KMV Symmetry Group • KMV Symmetry Algebra • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

11. KMV Symmetry Algebra • Limit case of the KMV Group • Generators in symmetry form

12. KMV Symmetry Algebra • Generators in vector form

13. Introduction • KMV Symmetry Algebra • KMV Symmetry Group • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

14. Systems with Same KMV • Ref: J.Math.Phys.45(04) 1020

15. Systems with Same KMV • Arbitrary f(t) • independence

16. Systems with Same KMV

17. Systems with Same KMV

18. Introduction • KMV Symmetry Algebra • KMV Symmetry Group • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

19. ConsLaws w.r.t. same KMV • For the KMV vector • Calculate the kth prolongation • in standard way

20. ConsLaws w.r.t. same KMV • The KMV symmetry is said to be asscociated with a conserved vector if • (*) • where .

21. ConsLaws w.r.t. same KMV • A conjecture on the solution of (*) • are arb. group inv. functions

22. ConsLaws w.r.t. same KMV • Further work one has to do

23. ConsL up to 2nd order group inv. • Group invariants ( ):

27. ConsLaws w.r.t. same KMV • Theorem on the solution of (*) • are arb. functions of

28. The further constraint becomes

29. Eq. (55) can be divided to 15 Eqs. because J_i are only functions of 2nd derivatives of u.

31. Now and are funs. t1-t6.

32. Result: (15 Eqs. Can be solved.)

35. Introduction • KMV Symmetry Algebra • KMV Symmetry Group • Systems with Same KMV • ConsLaws w.r.t. same KMV • Summary and discussions

36. Summary • One fact: 2+1 D int. sys.  KMV • Two Meths. KMV GroupAlgebra • A family: Systems with same KMV • ∞ ConsLaws: CLs with same KMV • 2nd order CLs: 4 (6D) + 2 (4D) Funcs. • A conjecture? • CLs with KMV valid for a fixed Eq.? • Physical meanings? • More simple methods?

37. Thanks