Akhiezer Function Research Articles

The coupled Sasa–Satsuma hierarchy: Trigonal curve and finite genus solutions

Based on the Lenard recursion equations and the stationary zero-curvature equation, we derive the coupled Sasa–Satsuma hierarchy, in which a typical number is the coupled Sasa–Satsuma equation. The properties of the associated trigonal curve and the meromorphic functions are studied, which naturally give the essential singularities and divisors of the meromorphic functions. By comparing the asymptotic expansions for the Baker–Akhiezer function and its Riemann theta function representation, we arrive at the finite genus solutions of the whole coupled Sasa–Satsuma hierarchy in terms of the Riemann theta function.

Tangential polynomials and matrix KdV elliptic solitons

Let (X, q) be an elliptic curve marked at the origin. Starting from any cover π: Γ → X of an elliptic curve X marked at d points {πi} of the fiber π−1(q) and satisfying a particular criterion, Krichever constructed a family of d × d matrix KP solitons, that is, matrix solutions, doubly periodic in x, of the KP equation. Moreover, if Γ has a meromorphic function f: Γ → P1 with a double pole at each pi, then these solutions are doubly periodic solutions of the matrix KdV equation Ut = 1/4(3UUx + 3UxU + Uxxx). In this article, we restrict ourselves to the case in which there exists a meromorphic function with a unique double pole at each of the d points {pi}; i.e. Γ is hyperelliptic and each pi is a Weierstrass point of Γ. More precisely, our purpose is threefold: (1) present simple polynomial equations defining spectral curves of matrix KP elliptic solitons; (2) construct the corresponding polynomials via the vector Baker–Akhiezer function of X; (3) find arbitrarily high genus spectral curves of matrix KdV elliptic solitons.

Finite-band solutions of the coupled dispersionless hierarchy

The coupled dispersionless hierarchy is derived with the help of the zero curvature equation. Based on the Lax matrix, we introduce an algebraic curve of arithmetic genus n, from which we establish the corresponding meromorphic function ϕ, the Baker–Akhiezer function , and Dubrovin-type equations. The straightening out of all the flows is given under the Abel–Jacobi coordinates. Using the asymptotic properties of ϕ and , we obtain the explicit theta function representations of the meromorphic function ϕ, the Baker–Akhiezer function and of solutions for the whole hierarchy.

The application of trigonal curve to the Mikhailov–Shabat–Sokolov flows

Resorting to the characteristic polynomial of Lax matrix for the Mikhailov–Shabat–Sokolov hierarchy associated with a \({3 \times 3}\) matrix spectral problem, we introduce a trigonal curve, from which we deduce the associated Baker–Akhiezer function, meromorphic functions and Dubrovin-type equations. The straightening out of the Mikhailov–Shabat–Sokolov flows is exactly given through the Abel map. On the basis of these results and the theory of trigonal curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire Mikhailov–Shabat–Sokolov hierarchy.

Quasi-periodic Solutions to the K(−2, −2) Hierarchy

Abstract With the help of the characteristic polynomial of Lax matrix for the K(−2, −2) hierarchy, we define a hyperelliptic curve 𝒦 n+1 of arithmetic genus n+1. By introducing the Baker–Akhiezer function and meromorphic function, the K(−2, −2) hierarchy is decomposed into Dubrovin-type differential equations. Based on the theory of hyperelliptic curve, the explicit Riemann theta function representation of meromorphic function is given, and from which the quasi-periodic solutions to the K(−2, −2) hierarchy are obtained.

Quasi-periodic solutions to the hierarchy of four-component Toda lattices

Starting from a discrete 3×3 matrix spectral problem, the hierarchy of four-component Toda lattices is derived by using the stationary discrete zero-curvature equation. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a trigonal curve Km−2 of genus m−2 and present the related Baker–Akhiezer function and meromorphic function on it. Asymptotic expansions for the Baker–Akhiezer function and meromorphic function are given near three infinite points on the trigonal curve, from which explicit quasi-periodic solutions for the hierarchy of four-component Toda lattices are obtained in terms of the Riemann theta function.

Cauchy–Jost function and hierarchy of integrable equations

Properties of the Cauchy--Jost (known also as Cauchy--Baker--Akhiezer) function of the KPII equation are described. By means of the $\bar\partial$-problem for this function it is shown that all equations of the KPII hierarchy are given in a compact and explicit form, including equations on the Cauchy--Jost function itself, time evolutions of the Jost solutions and evolutions of the potential of the heat equation.

Quasiperiodic solutions of the Kadomtsev–Petviashvili equation via the multidimensional Baker–Akhiezer function generated by the Broer–Kaup hierarchy

The problem of constructing quasiperiodic solutions of the Kadomtsev–Petviashvili equation is revisited. Following the idea of Krichever and the technique developed by Gesztesy and Holden, we shall explicitly construct the Baker–Akhiezer function from the compatible solutions which satisfy infinite numbers of equations in the Broer–Kaup hierarchy. Analytic and asymptotic properties will also be studied by introducing a special meromorphic function ψ1(P)ψ(P⁎), from which we derive theta function representations for the Baker–Akhiezer function and quasiperiodic solutions of the Kadomtsev–Petviashvili equation.

Evolution of Smooth Shapes and Integrable Systems

We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Lowner–Kufarev equation, we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The ‘positive’ Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation, they are conserved along the Hamiltonian flow. The ‘negative’ Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Lowner–Kufarev trajectories into the Segal–Wilson Grassmannian, construct the \(\tau \)-function, and the Baker–Akhiezer function which are related to a class of solutions to the KP hierarchy.

Manifolds of solutions for Hirzebruch functional equations

For the nth Hirzebruch equation we introduce the notion of universal manifold Mn of formal solutions. It is shown that the manifold Mn, where n > 1, is algebraic and its dimension is not greater than n + 1. We give a family of polynomials generating the relation ideal in the polynomial ring on Mn. In the case n = 2 the generators of this ideal are described. As a corollary we obtain an effective description of the manifold M2 and therefore all series determining complex Hirzebruch genera that are fiberwise multiplicative on projectivizations of complex vector bundles. A family of analytic solutions of the second Hirzebruch equation is described in terms of Weierstrass elliptic functions and in terms of Baker–Akhiezer functions of elliptic curves. For this functions the curves differ, yet the series expansions in the vicinity of 0 coincide.

Ghost symmetry of the discrete KP hierarchy

In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction (adjoint eigenfunction) and the spectral representation of its Baker–Akhiezer function and adjoint Baker–Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy.

Algebro-geometric solutions to the Manakov hierarchy

The Manakov hierarchy associated with a matrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Manakov hierarchy, we introduce a trigonal curve of arithmetic genus , from which we construct the related Baker–Akhiezer function, two algebraic functions carrying the data of the divisor and Dubrovin-type equations. Based on the theory of trigonal curves, the explicit theta function representations of the Baker–Akhiezer function, the two algebraic functions, and in particular, that of solutions for the entire Manakov hierarchy are obtained.

Explicit solutions for a hierarchy of differential–difference equations

A new discrete isospectral problem and the corresponding hierarchy of nonlinear differential–difference equations are proposed. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the hierarchy of differential–difference equations are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the corresponding hierarchy of differential–difference equations are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function, the Baker–Akhiezer function and the hyperelliptic curve.

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $G(t)=\alpha_1(t)I+\alpha_2(t)Q(t)$, $\alpha_1(t), \alpha_2(t)\in H(L)$, $Q(t)$ is a $2\times 2$ zero-trace polynomial matrix, and $I$ is the unit matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of the essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires finding of the $\rho$ zeros of the Baker-Akhiezer function ($\rho$ is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree-$\rho$ polynomial and solution of a certain linear algebraic system of $\rho$ equations.

A Two-Component Generalization of Burgers' Equation with Quasi-Periodic Solution

In this paper, we aim for the theta function representation of quasi-periodic solution and related crucial quantities for a two-component generalization of Burgers' equation. Our tools include the theory of algebraic curves , meromorphic functions , Baker—Akhiezer functions and the Dubrovin-type equations for auxiliary divisor. Eith these tools, the explicit representations for above quantities are obtained.

Algebro-geometric solutions for the two-component Camassa–Holm Dym hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Camassa–Holm Dym (CHD2) hierarchy. Our main tools include the polynomial recursive formalism, the hyperelliptic curve with finite number of genus, the Baker–Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the algebro-geometric solutions are obtained for the entire CHD2 hierarchy.

Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy

Based on the stationary zero-curvature equation and the Lenard recursion equations, we derive the coupled modified Korteweg–de Vries (cmKdV) hierarchy associated with a 3×3 matrix spectral problem. Resorting to the Baker–Akhiezer function and the characteristic polynomial of Lax matrix for the cmKdV hierarchy, we introduce a trigonal curve with three infinite points and two algebraic functions carrying the data of the divisor. The asymptotic properties of the Baker–Akhiezer function and the two algebraic functions are studied near three infinite points on the trigonal curve. Algebro-geometric solutions of the cmKdV hierarchy are obtained in terms of the Riemann theta function.

Algebro-Geometric Solution to the Bullough-Dodd-Zhiber-Shabat Equation

Based on solutions of the stationary zero-curvature equation associated with a 3×3 matrix spectral problem, we introduce a trigonal curve related to the Bullough–Dodd–Zhiber–Shabat equation and define Baker–Akhiezer functions and appropriate meromorphic function on the trigonal curve. Resorting to the algebro-geometric characters of the trigonal curve and properties of the three kinds of Abel differentials, we arrive at the explicit Riemann theta function representations of the Baker–Akhiezer functions and the meromorphic function. The algebro-geometric solution for the Bullough–Dodd–Zhiber–Shabat equation is obtained with the help of the asymptotic properties of the meromorphic function.

Algebro-geometric Quasi-periodic Solutions to the Three-Wave Resonant Interaction Hierarchy

Based on two sets of the Lenard recursion gradients, we derive the three-wave resonant interaction (TWRI) hierarchy associated with a $3\times 3$ matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the TWRI hierarchy, we introduce a trigonal curve ${\cal K}_{m-2}$ of arithmetic genus $m-2$ with three infinite points and establish the corresponding Baker--Akhiezer function and meromorphic functions on ${\cal K}_{m-2}$. The TWRI equations are decomposed into a system of Dubrovin-type ordinary differential equations. Using the theory of the trigonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit theta function representations of the Baker--Akhiezer function, of the meromorphic functions, and, in particular, of solutions for the entire TWRI hierarchy.

Connection coefficients for basic Harish-Chandra series

Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednikʼs root system analogs of the quantum Knizhnik–Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang–Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker–Akhiezer functions are discussed.