Theta functions and non-linear equations

We obtain a characterization of theta functions of Jacobian varieties of curves with automorphisms among theta functions of principally polarized abelian varieties (p.p.a.v.). We first give a characterization in terms of finite dimensional orbits for a suitable action in the Sato Grassmannian. Secondly, the introduction of formal Baker-Akhiezer functions and formal T -functions attached to a p.p.a.v. (for the multipuncture case) allows us to characterize, in terms of bilinear identities, those Baker-Akhiezer functions that are Baker-Akhiezer functions of Jacobians of curves with automorphisms. Further, in the case of automorphisms with fixed points, we rewrite the previous result as a hierarchy of partial differential equations for the τ-function of a p.p.a.v. Finally, since Baker-Akhiezer and T functions are written in terms of theta functions, these results give rise to characterizations of p.p.a.v. in terms of their theta functions.

Toda Molecule Equations

In this short note, we discuss a new formula for solving the nonlocal \(\overline{\partial }\)-problem, and discuss application to the Manakov–Zakharov dressing method. We then explicitly apply this formula to solving the complex (2+1)D Kadomtsev–Petviashvili equation and complex (2+1)D completely integrable generalization of the (2+1)D Kaup–Broer (or Kaup–Boussinesq) system. We will also discuss how real (1+1)D solutions are expressed using this formalism. It is simple to express the formalism for finite gap primitive solutions from [9, 11] using the formalism of this note. We also discuss recent results on the infinite soliton limit for the (1+1)D Korteweg–de Vries equation and the (1+1)D Kaup-Broer system. In an appendix, the classical solutions to the 3D Laplace equation (2+1)D d’Alembert wave equation by Whittaker are described. This appendix is included to elucidate an analogy between the dressing method and the Whittaker solutions.

Linear and nonlinear propagation of one- and two-dimensional drift ion acoustic waves with the effects of trapping in electron–positron–ion degenerate plasma has been considered. The characteristics of the dispersion relation in a two-dimensional quantum degenerate plasma has modified significantly due to the inclusion of positrons. The variations of drift KdV and KP equation have been studied with magnetic field, positron concentration and electron degeneracy temperature. The system under consideration admits compressive solitary structures only. The present studies may be valuable to understand the propagation of drift solitary structures with special reference to white dwarf stars.

Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

A class of doubly periodic waves for nonlinear evolution equations

Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

Introduction Riemann surfaces Riemann-Roch theorem Abel-Jacobi theorem and period relations Divisors and theta functions Dimension of the space of theta functions Projective embedding and theta functions (chapter 6) Elliptic curves as the intersection of two quadrics The Fermat curve Discrete subrgroups of SL(sub 2)(R) Riemann surface structure of $\Gamma \setminus \mathcal H^*$ The modular curve X(N) Generalities on abelian varieties The conjecture of Tate Finiteness of isogeny classes (chapter 14) Mordell's conjecture Bibliography Index.

The purpose of this paper is to carry out the abelianization program pro posed by Atiyah [1] and Hitchin [9] for the geometric quantization of SU(2) Wess-Zumino-Witten model. Let C be a Riemann surface of genus g. Let Mg be the moduli space of semi-stable rank 2 holomorphic vector bundles on C with trivial determinant. For a positive integer k, let T(Mg, Ck) be the space of holomorphic sections of the fc-th tensor product of the determinant line bundle C on Mg. An element of T(Mg, Ck) is called a rank 2 theta function of level k. The main result of our abelianization is to give an explicit representation of a base of T{Mg, Ck) as well as its transformation formula in terms of classical Riemann theta functions with automorphic form coefficients defined on the Prym variety P associated with a two-fold branched covering surface C of C. r(Mg,Ck) can be identified with the conformai block of level k of the SU(2) WZW model ([5], [15]). The abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way. Firstly, we construct a projectively flat connection on the vector bundle over the Teichm?ller space with fibre T{Mg,Ck). Secondly, making use of our ex plicit representation of rank 2 theta functions we construct a Hermitian product on the vector bundle preserved by the connection. Also our explicit represen tation enables us to prove that Y(Mg,Ck) has the predicted dimension from the Quantum Clebsh-Gordan conditions. A natural connection on the said vector bundle for the SU(iV) WZW model was first constructed by Hitchin [11]. It will turn out that the connection constructed in this paper coincides with the Hitchin connection. Laszlo [16] showed that the Hitchin connection coincides with the con nection constructed by Tsuchiya, Ueno and Yamada [21] through the above identification. On the other hand Kirillov [13], [14] constructed a Hermitian product on the conformai block compatible with the Tsuchiya-Ueno-Yamada connection using the representation theory of affine Lie algebras together with the theory of hermitian modular tensor categories; cf. [22]. Laszlo's result

One of the most elegant classical results about nonsingular algebraic curves is Riemann’s singularity theorem, which provides a beautiful interpretation of the singularities of Riemann’s theta function in terms of the geometry of special divisors. In recent times, soliton theory has suggested generalizations of the classical geometry. Integrable partial differential equations posed with periodic or scattering boundary conditions lead to Riemann surfaces of infinite, or even “continuously infinite”, genus, and to correspondingly huge Jacobians ([McKT], [EMcK]). More special boundary conditions have been studied via the theory of infinite-dimensional Lie groups and their homogeneous spaces. In this latter approach, one finds a large family of (possibly infinite-dimensional) algebraic varieties that are natural generalizations of the classical Jacobians. The Lie-theoretic aspects of these varieties are, of course, prominent in such work ([SW], [Sch]). In another direction, soliton problems have led naturally to singular limits of smooth Riemann surfaces. While, in the abstract, singular curves and their Jacobians are a subject of current research, soliton theory provides a concrete context in which to study such algebraic varieties.

We give a geometric proof of a formula, due to Segal and Wilson, which describes the order of vanishing of the Riemann theta function in the direction which corresponds to the direction of the tangent space of a Riemann surface at a marked point. While this formula appears in the work of Segal and Wilson as a by-product of some nontrivial constructions from the theory of integrable systems (loop groups, infinite-dimensional Grassmannians, tau functions, Schur polynomials, etc.) our proof only uses the classical theory of linear systems on Riemann surfaces.

Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups

The Riemann’s theta functions associated with a closed Riemann surface are absolutely convergent. In the present paper, we shall show an example of an hyperelliptic Riemann surface of infinite genus such that the Riemann’s theta functions associated with are absolutely convergent.

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

Conserved currents and the energy-momentum tensor in conformally invariant theories for general dimensions