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14
Abelian Functions for Trigonal Curves of Genus Three
, 2007
"... We develop the theory of generalized Weierstrass σand℘functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘functions, a proof that the coefficients of the power series expansion of the σfunc ..."
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Cited by 19 (11 self)
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We develop the theory of generalized Weierstrass σand℘functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘functions, a proof that the coefficients of the power series expansion of the σfunction are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae.
Abelian functions for cyclic trigonal curves of genus four
 J. GEOM. PHYS
, 2006
"... We discuss the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the “purely trigonal ” (or “cyclic trigonal”) curve y 3 = x 5 + λ4x 4 + λ3x 3 + λ2x 2 + λ1x + λ0 is discussed in det ..."
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Cited by 9 (4 self)
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We discuss the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the “purely trigonal ” (or “cyclic trigonal”) curve y 3 = x 5 + λ4x 4 + λ3x 3 + λ2x 2 + λ1x + λ0 is discussed in detail, including a list of some of the associated partial differential equations satisfied by the ℘ functions, and the derivation of an addition formulae.
Hyperelliptic solutions of KdV and KP equations: Reevaluation of Baker’s study on hyperelliptic sigma functions
 J. Phys. A: Math. Gen
"... ..."
Explicit Hyperelliptic Solutions of Modified Kortewegde Vries Equation: Essentials of Miura Transformation
"... Explicit hyperelliptic solutions of the modified Kortewegde Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic alfunctions over nondegenerated hyperelliptic curve y 2 = f(x) of arbitrary genus g. In the derivation, any θfunctions or BakerAkhieze ..."
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Cited by 3 (3 self)
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Explicit hyperelliptic solutions of the modified Kortewegde Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic alfunctions over nondegenerated hyperelliptic curve y 2 = f(x) of arbitrary genus g. In the derivation, any θfunctions or BakerAkhiezer functions were not essentially used. Then the Miura transformation naturally appears as the connections between the hyperelliptic ℘functions and hyperelliptic alfunctions.
IDENTITIES FOR THE CLASSICAL GENUS TWO ℘ FUNCTION.
"... We present a simple method that allows one to generate and classify identities for genus two # functions for generic algebraic curves of type (2,6). We discuss the relation of these identities to the Boussinesq equation for shallow water waves and show, in particular, that these # functions give ris ..."
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Cited by 2 (1 self)
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We present a simple method that allows one to generate and classify identities for genus two # functions for generic algebraic curves of type (2,6). We discuss the relation of these identities to the Boussinesq equation for shallow water waves and show, in particular, that these # functions give rise to a family of solutions to Boussinesq. 1.
A Sl(2) Covariant Theory Of Genus 2 Hyperelliptic Functions
, 2002
"... We present an algebraic formulation of genus 2 hyperelliptic functions which exploits the underlying covariance of the family of genus 2 curves. This allows a simple interpretation of all identities in representation theoretic terms. We show how the classical theory is recovered when one branch poin ..."
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Cited by 2 (2 self)
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We present an algebraic formulation of genus 2 hyperelliptic functions which exploits the underlying covariance of the family of genus 2 curves. This allows a simple interpretation of all identities in representation theoretic terms. We show how the classical theory is recovered when one branch point is moved to infinity. 1.
Abelian functions for purely trigonal curves of genus three
 IN PREPARATION
, 2006
"... We develop the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus three. The specific example of the “purely trigonal”curve y 3 = x 4 + λ3x 3 + λ2x 2 + λ1x + λ0 is discussed in detail, including a list of the associated partial differential equations satisfied ..."
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Cited by 2 (2 self)
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We develop the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus three. The specific example of the “purely trigonal”curve y 3 = x 4 + λ3x 3 + λ2x 2 + λ1x + λ0 is discussed in detail, including a list of the associated partial differential equations satisfied by the ℘ functions, and the derivation of two addition formulae.
Multidimensional Schrödinger Equations With Abelian Potentials
"... We consider the 2D complex Schrodinger equation with an Abelian potential and a fixed energy level. The potential wave function and the spectral Bloch variety are calculated in terms of the Kleinian hyperelliptic functions associated with a genus two hyperelliptic curve. In the special case when the ..."
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We consider the 2D complex Schrodinger equation with an Abelian potential and a fixed energy level. The potential wave function and the spectral Bloch variety are calculated in terms of the Kleinian hyperelliptic functions associated with a genus two hyperelliptic curve. In the special case when the curve covers two elliptic curves, an intrinsic elliptic 2D exactly solvable Schrodinger equation is constructed. The solutions obtained are illustrated by a number of plots. Brief details of the 3D case are also given. 1.
Soliton Solutions of Kortewegde Vries Equations and Hyperelliptic Sigma Functions
, 2001
"... Soliton Solutions of Kortewegde Vries (KdV) were constructed for given degenerate curves y 2 = (x − c)P(x) 2 in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were also presented. The modern soliton theories [DJKM, SN, SS], ..."
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Soliton Solutions of Kortewegde Vries (KdV) were constructed for given degenerate curves y 2 = (x − c)P(x) 2 in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were also presented. The modern soliton theories [DJKM, SN, SS], which were developed in ending of last century, are known as the infinite dimensional analysis and gave us fruitful and beautiful results, e.g., relations of soliton equations to universal Grassmannian manifold, Plücker embedding, infinite dimensional Lie algebra, loop algebra, representation theories, Schur functions, Young diagram, and so on. They