HYPERELLIPTIC KLEINIAN FUNCTIONS AND APPLICATIONS

We develop the theory of hyperelliptic Kleinian functions. As applications we consider construction of the explicit matrix realization of the hyperelliptic Kummer varieties, differential operators to have the hyperelliptic curve as spectral variety, solution of the KdV equations by Kleinian functions.

[1]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[2]  H. Baker.  Abel's Theorem and the Allied Theory: Including the Theory of the Theta Functions , 2007 .

[3]  H. F. Baker,et al.  On the hyperelliptic sigma functions , 1898 .

[4]  C. Weierstrass,et al.  Zur Theorie der Abelschen Functionen. , 1854 .

[5]  F. Klein.  Ueber hyperelliptische Sigmafunctionen , 1886 .

[6]  D. V. Leikin.  On the Weierstrass cubic for hyperelliptic functions , 1995 .

[7]  Igor Krichever,et al.  METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS , 1977 .

[8]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[9]  Steven Roman.  The Umbral Calculus , 1984 .

[10]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[11]  Fritz Gesztesy,et al.  Treibich-Verdier potentials and the stationary (m)KDV hierarchy , 1995 .

[12]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[13]  Fritz Gesztesy,et al.  On Picard potentials , 1995, Differential and Integral Equations.

[14]  W. Wirtinger,et al.  Abelsche Funktionen und allgemeine Thetafunktionen , 1921 .

[15]  G. B. Mathews,et al.  Kummer's Quartic Surface , 1990 .

[16]  J. M. Anderson,et al.  RIEMANN SURFACES (Graduate Texts in Mathematics, 71) , 1982 .

[17]  Joe W. Harris,et al.  Principles of Algebraic Geometry , 1978 .

[18]  David Mumford,et al.  Curves and their Jacobians , 1975 .

[19]  V. M. Bukhshtaber,et al.  Abelian Bloch solutions of the two-dimensional Schrödinger equation , 1995 .

[20]  E. Belokolos,et al.  Algebro-geometric approach to nonlinear integrable equations , 1994 .

[21]  V. Buchstaber,et al.  Explicit algebraic description of hyperelliptic Jacobians on the basis of the Klein σ-functions , 1996 .