Akhiezer Function Research Articles

Baker–Akhiezer function for the deformed root system BC(l,1) and bispectrality

Baker–Akhiezer function for the deformed root system BC(l,1) and bispectrality

A KP-mKP hierarchy via pseudo-differential operators with two derivations

By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.

A Riemann–Hilbert method to algebro-geometric solutions of the Korteweg–de Vries equation

A proof without using Dubrovin’s equations is given for the Its–Matveev formula for algebro-geometric solutions of the Korteweg–de Vries (KdV) equation. It is shown that the Baker–Akhiezer (BA) function of the KdV equation can be described in terms of two types of holomorphic vector Riemann–Hilbert problems with σ1-symmetry condition in the complex k-plane. Further, a sufficient condition on determining whether the algebro-geometric KdV solution has mKdV-relevant property is presented. This provides an explanation for the classical Miura transformation in terms of the deformations of associated algebro-geometric datum. Our main tools include algebraic curve and Riemann surface, Riemann theta function, Riemann–Hilbert problem and associated deformation procedure.

Algebro-geometric quasi-periodic solutions to the Satsuma–Hirota hierarchy

In this paper, we study the theory of the resulting tetragonal curve and derive three kinds of Abel differentials, Baker–Akhiezer function and meromorphic function and so on, from which a systematic method is developed to construct algebro-geometric quasi-periodic solutions of soliton equations associated with the 4 × 4 matrix spectral problems. As an illustrative example, the Satsuma–Hirota hierarchy related to the 4 × 4 matrix spectral problem is obtained by utilizing the Lenard recursion equation and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the Satsuma–Hirota hierarchy, we introduce a tetragonal curve Kg of genus g and study the asymptotic properties of the Baker–Akhiezer function ψ1 and the meromorphic function ϕ near the infinite points on the tetragonal curve. The straightening out of various flows is exactly given through the Abel map and Abel–Jacobi coordinates. Using the theory of the tetragonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit Riemann theta function representations of the Baker–Akhiezer function and the meromorphic function, and in particular, that of solutions for the entire Satsuma–Hirota hierarchy.

Rogue-wave and breather solutions of the Fokas–Lenells equation on theta-function backgrounds

A systematic method is developed to obtain localized wave solutions of the Fokas–Lenells equation on theta-function backgrounds. First, using the properties of the theta-functions, we find a theta-function seed solution of the Fokas–Lenells equation. Next, based on the Riccati equation of Lax pair, we construct a Darboux transformation of the Fokas–Lenells equation. Then, the Kaup–Newell type spectral problem with theta-function potentials is solved by using the Baker–Akhiezer functions in the algebro-geometric method. Finally, Exact rogue-wave and breather solutions of the Fokas–Lenells equation on theta-function backgrounds are constructed by using the derived Darboux transformation and Baker–Akhiezer functions. In addition, the interaction dynamics of various localized wave solutions are analyzed by choosing appropriate parameters.

Quasi‐periodic solutions to a hierarchy of integrable nonlinear differential–difference equations

Using the discrete zero‐curvature equation, we derive a hierarchy of new nonlinear differential–difference equations associated with a discrete matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix, we introduce a trigonal curve and the associated three‐sheeted Riemann surface, from which we derive the Baker–Akhiezer function and meromorphic function. By comparing the asymptotic expansions of the meromorphic function and its Riemann theta function representations, we obtain quasi‐periodic solutions for a hierarchy of nonlinear differential–difference equations.

Bispectrality of AG_2 Calogero–Moser–Sutherland System

We consider the generalised Calogero–Moser–Sutherland quantum integrable system associated to the configuration of vectors AG_2, which is a union of the root systems A_2 and G_2. We establish the existence of and construct a suitably defined Baker–Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald–Ruijsenaars type in an explicit form.

Algebro-geometric Constructions of a Hierarchy of Integrable Semi-discrete Equations

A hierarchy of integrable semi-discrete equations is deduced in terms of the discrete zero curvature equation as well as its bi-Hamiltonian structure is gotten through the trace identity. The above hierarchy is separated into soluble ordinary differential equations according to the relationship between the elliptic variables and the potentials, from which the continuous flow is straightened out via the Abel–Jacobi coordinates resorting to the algebraic curves theory. Eventually, the meromorphic function and the Baker–Akhiezer function are introduced successively on the hyperelliptic curve and the algebro-geometric solutions which are expressed as Riemann theta function can be obtained through the two functions mentioned above.

Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

Algebro-geometric solutions of the lattice potential modified Kadomtsev–Petviashvili (lpmKP) equation are constructed. A Darboux transformation of the Kaup–Newell spectral problem is employed to generate a Lax triad for the lpmKP equation, as well as to define commutative integrable symplectic maps which generate discrete flows of eigenfunctions. These maps share the same integrals with the finite-dimensional Hamiltonian system associated to the Kaup–Newell spectral problem. We investigate asymptotic behaviors of the Baker–Akhiezer functions and obtain their expression in terms of Riemann theta function. Finally, algebro-geometric solutions for the lpmKP equation are reconstructed from these Baker–Akhiezer functions.

Finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy

Resorting to the zero-curvature equation and the Lenard recursion equations, the generalized Merola–Ragnisco–Tu lattice hierarchy associated with a 3 × 3 discrete matrix spectral problem is derived. With the aid of the characteristic polynomial of the Lax matrix for the generalized Merola–Ragnisco–Tu lattice hierarchy, a trigonal curve is defined, on which we construct the Baker–Akhiezer function, two meromorphic functions, three kinds of Abelian differentials, and Riemann theta function. By analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions, especially their asymptotic expansions near three infinite points and three zero points, we obtain their essential singularities and divisors. Finally, we obtain the finite genus solutions of the generalized Merola–Ragnisco–Tu lattice hierarchy in terms of the Riemann theta function.

Reduction of Divisors and the Clebsch System

There are a few Lax matrices of the Clebsch system. Poles of the Baker – Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann – Roch theorem, each class has a unique reduced representative. We discuss properties of such a reduced divisor on the spectral curve of $$3\times 3$$ Lax matrix having a natural generalization to $$gl^{*}(n)$$ case.

A vector Geng–Li model: New nonlinear phenomena and breathers on periodic background waves

Based on an introduced (n+2)×(n+2) matrix spectral problem, a vector Geng–Li model is proposed, which is a vector generalization of the Geng–Li model. Employing the Riccati equations derived from the spectral problems and the gauge transformations between the spectral problems, a general N-fold Darboux transformation of the vector Geng–Li model is constructed. As applications of the Darboux transformations, some exact solutions for the Geng–Li model are obtained, which reveal two novel nonlinear phenomena. The first one is that all the solutions of the Geng–Li equation found in this paper have the characteristics of square-root functions. The second one is the interaction of two waves: as time goes on, the amplitudes of the two waves get higher and the distances between the two waves get narrower. And then, a general approach which deals with periodic seed solutions is given by using algebraic curves and Baker–Akhiezer functions, from which breathers of the Geng–Li equation on periodic background waves are obtained.

Three-sheeted Riemann surface and solutions of the Itoh–Narita–Bogoyavlensky lattice hierarchy

The Itoh–Narita–Bogoyavlensky lattice hierarchy associated with a discrete [Formula: see text] matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface [Formula: see text] of arithmetic genus [Formula: see text] and construct the corresponding Baker–Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface.

On a Harry–Dym-Type Hierarchy: Trigonal Curve and Quasi-Periodic Solutions

Resorting to the characteristic polynomial of Lax matrix for a Harry–Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker–Akhiezer function and meromorphic function are introduced. With the help of the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we obtain the quasi-periodic solutions for the entire Harry–Dym-type hierarchy.

Algebro-geometric integration of a modified shallow wave hierarchy

Abstract By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

Bilinear Equation and Additional Symmetries for an Extension of the Kadomtsev–Petviashvili Hierarchy

An extension of the Kadomtsev–Petviashvili (KP) hierarchy defined via scalar pseudo-differential operators was studied in Szablikowski and Blaszak (J. Math. Phys. 49(8), 082701, 20, 2008) and Wu and Zhou (J. Geom. Phys. 106, 327–341, 2016). In this paper, we represent the extended KP hierarchy into the form of bilinear equation of (adjoint) Baker–Akhiezer functions, and construct its additional symmetries. As a byproduct, we derive the Virasoro symmetries for the constrained KP hierarchies.

Elliptic solutions to the KP hierarchy and elliptic Calogero–Moser model

We consider solutions of the Kadomtsev–Petviashvili hierarchy which are elliptic functions of x = t 1. It is known that their poles as functions of t 2 move as particles of the elliptic Calogero–Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian H k of the elliptic Calogero–Moser model which governs the dynamics of poles with respect to the kth hierarchical time. The Hamiltonians H k are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker–Akhiezer function has essential singularity.

Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps

The Q1 lattice equation, a member in the Adler–Bobenko–Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearized to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker–Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps.

Explicit Solutions of the Coupled Bogoyavlensky Lattice 1(2) Hierarchy

By means of the discrete zero curvature representation, the coupled Bogoyavlensky lattice 1(2) hierarchy related to a $$3\times 3$$ matrix problem is derived. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a trigonal curve $$\mathcal {K}_{m-1}$$ of arithmetic genus $$m-1$$ and construct the related Baker–Akhiezer function and meromorphic function. By analyzing the asymptotic expansion of the meromorphic function, the algebro-geometric solutions to the stationary coupled Bogoyavlensky lattice 1(2) are obtained. Moreover, the explicit theta function representations for the coupled Bogoyavlensky lattice hierarchy are given with the help of the Abelian differential.

Riemann theta function solutions to the coupled long wave–short wave resonance equations

Based on the Lenard recursion equations, we derive the Lax pair for the hierarchy of coupled long wave–short wave resonance equations, in which the first nontrivial member is the coupled long wave–short wave resonance equations. Resorting to the characteristic polynomial of Lax matrix for the hierarchy of coupled long wave–short wave resonance equations, we introduce a trigonal curve and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker–Akhiezer function, and an algebraic function carrying the data of the divisor. By comparing the asymptotic expansions for the Baker–Akhiezer function and its Riemann theta function representation, we obtain the explicit Riemann theta function solutions for the entire hierarchy of coupled long wave–short wave resonance equations.