Baker-Akhiezer Function Research Articles

Algebro-geometric solutions for the Hunter–Saxton hierarchy

This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through studying an algebro-geometric initial value problem. 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 HS2 hierarchy.

Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

For the rational Baker-Akhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the self-duality property of the usual Gaussian function with respect to the Fourier transformation. We show that the value of properly normalised Baker-Akhiezer function at the origin can be given by an integral of Macdonald-Mehta type and explicitly compute these integrals for all known Baker-Akhiezer arrangements. We use the Dotsenko-Fateev integrals to extend this calculation to all deformed root systems, related to the non-exceptional basic classical Lie superalgebras.

Orthogonality relations and Cherednik identities for multivariable Baker–Akhiezer functions

We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type.

A Riemann–Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution

We derive a Riemann–Hilbert problem satisfied by the Baker–Akhiezer function for the finite-gap solutions of the Korteweg–de Vries (KdV) equation. As usual for Riemann–Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi-periodic finite-genus solutions of the KdV equation.

QUASI-PERIODIC SOLUTIONS OF THE DISCRETE mKdV HIERARCHY

On the basis of the theory of algebraic curves, the continuous flow related to the discrete mKdV hierarchy is straightened using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete mKdV hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

Quasi-periodic Solutions of the Kaup–Kupershmidt Hierarchy

Based on solving the Lenard recursion equations and the zero-curvature equation, we derive the Kaup–Kupershmidt hierarchy associated with a 3×3 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the Kaup–Kupershmidt hierarchy, we introduce a trigonal curve \(\mathcal {K}_{m-1}\) and present the corresponding Baker–Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the Kaup–Kupershmidt flows. With the aid of the properties of the Baker–Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. The Riemann–Jacobi inversion problem is achieved by comparing the asymptotic expansion of the Baker–Akhiezer function and its Riemann theta function representation, from which quasi-periodic solutions of the entire Kaup–Kupershmidt hierarchy are obtained in terms of the Riemann theta functions.

Algebro-geometric Solutions for the Degasperis--Procesi Hierarchy

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy.

Quasi-periodic solutions to the two-component nonlinear Klein–Gordon equation

Based on solutions of the stationary zero-curvature equation associated with the 3×3 matrix spectral problem, we introduce a trigonal curve related to the two-component nonlinear Klein–Gordon equation. Resorting to the theory of trigonal curves and properties of the three kinds of Abel differentials, we deduce the explicit theta function representations of the Baker–Akhiezer function and two meromorphic functions. The two-component nonlinear Klein–Gordon flows are straightened using the Abel map and the Lagrange interpolation formula under certain conditions. The explicit theta function representations of solutions for the two-component nonlinear Klein–Gordon equation are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the two meromorphic functions.

Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Algebro-geometric solutions to the modified Sawada-Kotera hierarchy

Based on solving the Lenard recursion equation and the zero-curvature equation, we derive the modified Sawada-Kotera (SK) hierarchy associated with a 3 × 3 matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix for the modified SK hierarchy, we introduce a trigonal curve \documentclass[12pt]{minimal}\begin{document}$\mathcal {K}_{m-1}$\end{document}Km−1 and present the corresponding Baker-Akhiezer function and meromorphic function on it. The Abel map is introduced to straighten out the modified SK flows. With the aid of the property of the Baker-Akhiezer function and the meromorphic function and their asymptotic expansions, we arrive at their explicit Riemann theta function representations. Algebro-geometric solutions of the entire modified SK hierarchy are obtained by using asymptotic expansion of the meromorphic function and its Riemann theta function representation. As an application, we give two simple examples.

Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.

混合Boussinesq 方程的有限亏格解

The mixed Boussinesq hierarchy associated with a 3×3 matrix spectral problem is proposed in view of Lenard recursion equations and the stationary zero-curvature equation. A trigonal curve K m -1 is introduced with the help of the characteristic polynomial of the Lax matrix, from which we construct closely related Baker- Akhiezer function, the meromorphic function and Dubrovin-type equations. Moreover, the flows are straighten out under the Abel map. Based on the basic knowledge of the trigonal curves and three kinds of Abelian differential forms, the Riemann θ function representations of the Baker-Akhiezer function, the meromorphic function, and in particular, that of finite genus solutions for the mixed Boussinesq equation are obtained.

Some Remarks on Very-Well-Poised8φ7Series

Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised 8phi7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new derivation of a known quadratic transformation formula for very-well-poised 8phi7 series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.

C-Map as [formula omitted] string

We show the existence of a duality between the c-map space describing the universal hypermultiplet at tree level and the matrix model description of two-dimensional string theory compactified at a self-dual radius and perturbed by a sine-Liouville potential. It appears as a particular case of a general relation between the twistor description of four-dimensional quaternionic geometries and the Lax formalism for Toda hierarchy. Furthermore, we give an evidence that the instanton corrections to the c-map metric coming from NS5-branes can be encoded into the Baker–Akhiezer function of the integrable hierarchy.

QUASI-PERIODIC SOLUTIONS OF THE RELATIVISTIC TODA HIERARCHY

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy 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 relativistic Toda hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve K m − 1 of arithmetic genus m − 1 , from which we establish the associated Baker–Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.

Quasi‐Periodic Solutions of Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with three potentials is proposed. With the aid of the characteristic polynomial of Lax matrix for the hierarchy, we introduce an algebraic curve of arithmetic genus m − 1, and discuss in detail the properties of the associated Baker–Akhiezer function and meromorphic function. On the basis of the theory of algebraic curves, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and, in particular, that of solutions for the entire hierarchy of nonlinear evolution equations.

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ℂ2 with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ℂ*: the charged trigonometric Calogero-Moser system. When the particles lie on S1 ⊂ ℂ*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.

Minimal Lagrangian submanifolds in ℂP n with diagonal metric

We propose a method for constructing minimal Lagrangian submanifolds in ℂPn using the Baker-Akhiezer functions of algebraic spectral curves.

The relative frame bundle of an infinite-dimensional flag variety and solutions of integrable hierarchies

We develop a group theory approach for constructing solutions of integrable hierarchies corresponding to the deformation of a collection of commuting directions inside the Lie algebra of upper-triangular ZxZ matrices. Depending on the choice of the set of commuting directions, the homogeneous space from which these solutions are constructed is the relative frame bundle of an infinite-dimensional flag variety or the infinite-dimensional flag variety itself. We give the evolution equations for the perturbations of the basic directions in the Lax form, and they reduce to a tower of differential and difference equations for the coefficients of these perturbed matrices. The Lax equations follow from the linearization of the hierarchy and require introducing a proper analogue of the Baker—Akhiezer function.