Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions in Mathematics

1. Nondegenerate Solutions of Dispersionless Toda Hierarchy and Tau Functions Teo Lee Peng University of Nottingham Malaysia Campus L.P. Teo, “Conformal Mappings and Dispersionless Toda hierarchy II: General String Equations”, Commun. Math. Phys. 297 (2010), 447-474.

2. Dispersionless Toda Hierarchy Dispersionless Toda hierarchy describes the evolutions of two formal power series: with respect to an infinite set of time variables tn, n Z. The evolutions are determined by the Lax equations:

3. where The Poisson bracket is defined by

4. The corresponding Orlov-Schulman functions are They satisfy the following evolution equations: Moreover, the following canonical relations hold:

5. Generalized Faber polynomials and Grunsky coefficients Given a function univalent in a neighbourhood of the origin: and a function univalent at infinity: The generalized Faber polynomials are defined by

6. They can be compactly written as The generalized Grunsky coefficients are defined by

7. Hence,

8. It follows that

9. Tau Functions Given a solution of the dispersionless Toda hierarchy, there exists a phi function and a tau function such that Identifying then

10. Riemann-Hilbert Data The Riemann-Hilbert data of a solution of the dispersionless Toda hierarchy is a pair of functions U and V such that and the canonical Poisson relation

11. NondegenerateSoltuions If then Hence, and therefore Such a solution is said to be degenerate.

12. If Then

13. Then Hence,

14. We find that and we have the generalized string equation: Such a solution is said to be nondegenerate.

16. Let Define

17. One can show that

18. Define Proposition:

19. Proposition: where

20. is a function such that

21. Hence,

22. Let Then

23. We find that

24. Hence, Similarly,

25. Special Case

26. Generalization to Universal Whitham Hierarchy K. Takasaki, T. Takebe and L. P. Teo, “Non-degenerate solutions of universal Whitham hierarchy”, J. Phys. A 43 (2010), 325205.

27. Universal Whitham Hierarchy Lax equations:

28. Orlov-Schulman functions They satisfy the following Lax equations and the canonical relations

29. where They have Laurent expansions of the form

30. From we have

31. In particular,

32. Hence, and

33. Free energy The free energy F is defined by

34. Generalized Faber polynomials and Grunsky coefficients Notice that

35. The generalized Grunsky coefficients are defined by

36. The definition of the free energy implies that

37. Riemann-Hilbert Data: Nondegeneracy implies that for some function Ha.

38. Nondegenerate solutions

39. One can show that and

40. Construction of a It satisfies

41. Construction of the free energy Then

42. Special case

43. ~ Thank You ~