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.     

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.     

Algebraic study on the super-KP hierarchy and the ortho-symplectic super-KP hierarchy

Bilinear residue formulas are established for the super-KP hierarchy and the ortho-symplectic super-KP hierarchy. Furthermore, superframes corresponding to the ortho-symplectic super-KP hierarchy are completely characterized. Soliton solutions to the super-KP hierarchy are given.     

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebro-geometric characters of the meromorphic function.     

Higher order supersymmetries and fermionic conservation laws of the supersymmetric extension of the KdV equation

By the introduction of nonlocal basonic and fermionic variables we construct a recursion symmetry of the super KdV equation, leading to a hierarchy of bosonic symmetries and one of fermionic symmetries. The hierarchies of bosonic and fermionic conservation laws arise in a natural way in the construction.     

Algebro-geometric Constructions of Quasi Periodic Flows of the Discrete Self-dual Network Hierarchy and Applicationsℰ

In this paper we obtain the discrete integrable self-dual network hierarchy associated with a discrete spectral problem. On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the discrete self-dual network hierarchy are 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 self-dual network hierarchy are constructed with the help of the asymptotic properties and the algebra-geometric characters of the meromorphic function, the Baker-Akhiezer function and the hyperelliptic curve.     

Algebro-geometric solutions for the two-component 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 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.     

Repeated application of the recursion operator for a new hierarchy of negative-order integrable KdV equations

ABSTRACT

In this work we use the repeated application of the recursion operator to establish a new hierarchy of negative-order integrable KdV equations of higher orders. The concept of the inverse recursion operator allows us to develop this new hierarchy. The complete integrability of each equation is guaranteed via the use of the recursion operator. We show that the dispersion relations of this hierarchy follow an infinite geometric series. Multiple soliton solutions for each equation of the hierarchy are obtained.     

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K?hler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K?hler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling–Tod (Eguchi–Hansen) solution. An extended space-time ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? is shown to be foliated by four dimensional hyper-K?hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator. Received: 27 January 2000 / Accepted: 20 March 2000     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP(dKP)hierarchy and -dependent KP ( KP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding KP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP (dKP) hierarchy and h-dependent KP (hKP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding hKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

Linear integral transformations and hierarchies of integrable nonlinear evolution equations

Integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations. These are (Riemann-Hilbert type of) integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an (indefinite) parameter k. The potential matrices in these equations obey a set of nonlinear recursion relations, leading to a heirarchy of nonlinear partial differential equations. In decreasing order the same equations give rise to a “reciprocal” hierarchy, associated with Heisenberg ferromagnet type of equations.Central in the treatment is an embedding of the hierarchy into an infinite-matrix structure, which is constructed on the basis of the integral equations. In terms of this infinite-matrix structure the equations governing the hierarchies become quite simple. Furthermore, it leads in a straightforward way to various generalizations, such as to other types of linear spectral problems, multicomponent system and lattice equations. Generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment. Finally, some results on conserved densities and the Hamiltonian structure are briefly discussed.     

The Hamiltonian structure of the N = 2 supersymmetric GNLS hierarchy

The first two Hamiltonian structures and the recursion operator connecting all evolution systems and Hamiltonian structures of the N = 2 supersymmetric (n, m)-GNLS hierarchy are constructed in terms of N = 2 superfields in two different superfield bases with local evolution equations. Their bosonic limits are studied in detail. New local and nonlocal bosonic and fermionic integrals both for the N = 2 supersymmetric (n, m)-GNLS hierarchy and its bosonic counterparts are derived. As an example, in the n = 1, m = 1 case, the algebra and the symmetry transformations for some of them are worked out and a rich N = 4 supersymmetry structure is uncovered.     

On the super-KP hierachy

We obtain super-integrable systems from the super-KP hierarchy of Kac and van de Leur and give a method to construct solutions. In particular, we apply the method to get the super-version of the KP 1-soliton solution.     

Fermionic flows and tau function of the N = (1|1) superconformal Toda lattice hierarchy

An infinite class of fermionic flows of the N = (1|1) superconformal Toda lattice hierarchy is constructed and their algebraic structure is studied. We completely solve the semi-infinite N = (1|1) Toda lattice and chain hierarchies and derive their tau functions, which may be relevant for building supersymmetric matrix models. Their bosonic limit is also discussed.     

Algebro-Geometric Solutions of the TD Hierarchy

Resorting to the Lenard recursion scheme, we derive the TD hierarchy associated with a 2?×?2 matrix spectral problem and establish Dubrovin-type equation in terms of the introduced elliptic variables. Based on the theory of algebraic curve, all the flows associated with the TD hierarchy are straightened under the Abel-Jacobi coordinates. An algebraic function ?, also called the meromorphic function, carrying the data of the divisor is introduced on the underlying hyperelliptic curve $\mathcal {K}_{n}$ . The known zeros and poles of ? allow to find theta function representations for ? by referring to Riemann’s vanishing theorem, from which we obtain algebro-geometric solutions for the entire TD hierarchy with the help of asymptotic expansion of ? and its theta function representation.     

Domain wall partition function of the eight-vertex model with a non-diagonal reflecting end

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex SOS model, we derive the recursion relations of the partition function for the eight-vertex model with a generic non-diagonal reflecting end and domain wall boundary condition. Solving the recursion relations, we obtain the explicit determinant expression of the partition function. Our result shows that, contrary to the eight-vertex model without a reflecting end, the partition function can be expressed as a single determinant.     

An introduction to on-shell recursion relations

This article provides an introduction to on-shell recursion relations for calculations of tree-level amplitudes. Starting with the basics, such as spinor notations and color decompositions, we expose analytic properties of gauge-boson amplitudes, BCFW-deformations, the large z-behavior of amplitudes, and on-shell recursion relations of gluons. We discuss further developments of on-shell recursion relations, including generalization to other quantum field theories, supersymmetric theories in particular, recursion relations for off-shell currents, recursion relation with nonzero boundary contributions, bonus relations, relations for rational parts of one-loop amplitudes, recursion relations in 3D and a proof of CSW rules. Finally, we present samples of applications, including solutions of split helicity amplitudes and of N = 4 SYM theories, consequences of consistent conditions under recursion relation, Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations for color-ordered gluon tree amplitudes, Kawai-Lewellen-Tye (KLT) relations.     

A note on the super Krichever map

We consider the geometrical aspects of the Krichever map in the context of Jacobian super KP hierarchy. We use the representation of the hierarchy based on the Faà di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.