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Schur function expansions of KP $$\tau$$-functions associated to algebraic curves. (English. Russian original) Zbl 1231.14025
Russ. Math. Surv. 66, No. 4, 767-807 (2011); translation from Usp. Mat. Nauk 66, No. 4, 137-178 (2011).
The Schur function expansion of Sato-Segal-Wilson KP $$\tau-$$functions is reviewed. The case of $$\tau-$$ functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the Pluecker coordinate coefficients appearing in the expansion are obtained in terms of directional derivatives of the Riemann $$\theta-$$function or Klein $$\sigma-$$function along the KP flow directions. By using the fundamental bi-differential it is shown how the coefficients can be expressed as polynomials in terms of Klein’s higher-genus generalizations of Weierstrass’ $$\zeta-$$ and $$\rho-$$functions. The cases of genus-two hyperelliptic and genus-three trigonal curves are detailed as illustrations of the approach developed here.

##### MSC:
 14H42 Theta functions and curves; Schottky problem 35Q53 KdV equations (Korteweg-de Vries equations) 14H70 Relationships between algebraic curves and integrable systems 14H55 Riemann surfaces; Weierstrass points; gap sequences
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