Gelfand-Dickey Hierarchy Research Articles

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

Higher-order heat equation and the Gelfand-Dickey hierarchy

In this paper we analyze the heat kernel of the equation ∂tv=±Lv, where L=∂xN+uN−2(x)∂xN−2+⋯+u0(x) is an N-th order differential operator and the ± sign on the right-hand side is chosen appropriately. Using formal pseudo-differential operators, we derive an explicit formula for Hadamard's coefficients in the expansion of the heat kernel in terms of the resolvent of L. Combining this formula with soliton techniques and Sato's Grassmannian, we establish different properties of Hadamard's coefficients and relate them to the Gelfand-Dickey hierarchy. In particular, using the correspondence between commutative rings of differential operators and algebraic curves due to Burchnall-Chaundy and Krichever, we prove that the heat kernel consists of finitely many terms if and only if the operator L belongs to a rank-one commutative ring of differential operators whose spectral curve is rational with only one cusp-like singular point, and the coefficients uj(x) vanish at x=∞. We also characterize these operators L as the rational solutions of the Gelfand-Dickey hierarchy with coefficients uj vanishing at x=∞, or as the rank-one solutions of the bispectral problem vanishing at ∞.

Geometrical correspondence of the Miura transformation induced from affine Kac–Moody algebras

AbstractThe Miura transformation plays a crucial role in the study of integrable systems. There have been various extensions of the Miura transformation, which have been used to relate different kinds of integrable equations and to classify the bi‐Hamiltonian structures. In this paper, we are mainly concerned with the geometric aspects of the Miura transformation. The generalized Miura transformations from the mKdV‐type hierarchies to the KdV‐type hierarchies are constructed under both algebraic and geometric settings. It is shown that the Miura transformations not only relate integrable curve flows in different geometries but also induce the transition between different moving frames. Moreover, the Miura transformation gives the factorization of generating operators of constraint Gelfand–Dickey hierarchy. Other geometric formulations are also investigated.

Generalized conditional symmetries and pre-Hamiltonian operators

In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand–Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs.

Open r-spin theory III: A prediction for higher genus

In our previous two papers, we constructed an r-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's r-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open r-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand–Dickey hierarchy, and we conjectured that the whole solution controls the open r-spin intersection numbers in all genera, which do not yet have a geometric definition. In this paper, we provide geometric and algebraic evidence for the correctness of this conjecture.

Extensions and Generalizations of Lattice Gelfand–Dickey Hierarchy

Extensions and Generalizations of Lattice Gelfand–Dickey Hierarchy

Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchy

The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer N, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande’s dressing operators for the equivariant Gromov-Witten theory of . This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to 0. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Gelfand–Dickey hierarchy, generalized BGW tau-function, and W-constraints

Let be an integer. The generalized Brezin–Gross–Witten (BGW) tau-function for the Gelfand–Dickey hierarchy of dependent variables (aka the r-reduced Kadomtsev–Petviashvili hierarchy) is defined as a particular tau-function that depends on constant parameters . In this paper we show that this tau-function satisfies a family of linear equations, called the W-constraints of the second kind. The operators giving rise to the linear equations also depend on constant parameters. We show that there is a one-to-one correspondence between the two sets of parameters.

Lagrangian Multiforms for Kadomtsev–Petviashvili (KP) and the Gelfand–Dickey Hierarchy

Abstract We present, for the first time, a Lagrangian multiform for the complete Kadomtsev–Petviashvili hierarchy—a single variational object that generates the whole hierarchy and encapsulates its integrability. By performing a reduction on this Lagrangian multiform, we also obtain Lagrangian multiforms for the Gelfand–Dickey hierarchy of hierarchies, comprising, among others, the Korteweg–de Vries and Boussinesq hierarchies.

Additional symmetries of the dispersionless extended noncommutative Gelfand–Dickey hierarchy

This paper aims at additional symmetries of the unextended and extended, commutative and noncommutative dispersionless Gelfand–Dickey (dGD) hierarchies. Being similar to the Lax formalism of the Gelfand–Dickey (GD) hierarchy, we construct the function [Formula: see text] and Orlov–Schulman function [Formula: see text] of the hierarchies. Meanwhile, the additional symmetry will be studied with the infinite flows of [Formula: see text] and [Formula: see text] function of the dGD hierarchy and one can find that only a part of additional flows can survive under the GD constraints with the corresponding string equation. Furthermore, we pay attention to the additional symmetries of the dispersionless extended Gelfand–Dickey (dEGD) hierarchy which has a quantum torus algebraic structure and show the flows in detail. The additional symmetry of dispersionless noncommutative Gelfand–Dickey (dNCGD) hierarchy and dispersionless extended noncommutative Gelfand–Dickey (dENCGD) hierarchy are studied.

Open 𝑟-Spin Theory I: Foundations

Abstract We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten bundle; we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically relatively oriented relative to the moduli space. In the sequel to this paper, we use these constructions to define open $r$-spin intersection theory and relate it to the Gelfand–Dickey hierarchy, thus providing an analog of Witten’s $r$-spin conjecture in the open setting.

Virasoro symmetries of multicomponent Gelfand–Dickey systems

In this paper, we mainly study the additional symmetry and $\tau$ functions of a multi-component Gelfand-Dickey hierarchy which includes many classical integrable systems, such as the multi-component KdV hierarchy and the multi-component Boussinesq hierarchy. With other kinds of reductions, we can derive a B type multi-component Gelfand-Dickey hierarchy and a C type multi-component Gelfand-Dickey hierarchy. In our research, the additional flows of the additional symmetries can not all survive. By calculating, we find that the generator of the additional symmetry of the C type multi-component Gelfand-Dickey hierarchy is different from that of the B type multi-component Gelfand-Dickey hierarchy, while the forms of their surviving additional flows are the same.

Additional symmetries for the supersymmetric Gelfand–Dickey hierarchy

In this paper, we first define a supersymmetric Gelfand–Dickey hierarchy using the Sato equation. Then we introduce some time variables and derivations involving those that from a non-abelian Lie superalgebra. Next, we construct additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the aid of the Orlov–Schulman operators which involve the above time variables and dressing operators. We give the additional flow equations of the supersymmetric Gelfand–Dickey hierarchy, and prove that the additional flows are symmetries of the supersymmetric Gelfand–Dickey hierarchy. In this way, an isomorphism between the additional symmetries of the supersymmetric Gelfand–Dickey hierarchy and the super [Formula: see text] algebra is constructed. Finally, the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy is constructed and we consider the additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the [Formula: see text] type condition which is compatible with the flows of the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy.

A Variational Perspective on Continuum Limits of ABS and Lattice GD Equations

A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures where previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.

Closed extended [formula omitted]-spin theory and the Gelfand–Dickey wave function

We study a generalization of genus-zero r-spin theory in which exactly one insertion has a negative-one twist, which we refer to as the “closed extended” theory, and which is closely related to the open r-spin theory of Riemann surfaces with boundary. We prove that the generating function of genus-zero closed extended intersection numbers coincides with the genus-zero part of a special solution to the system of differential equations for the wave function of the rth Gelfand–Dickey hierarchy. This parallels an analogous result for the open r-spin generating function in the companion paper Buryak et al. (2018) to this work.

Symmetries and Reductions on the noncommutative Kadomtsev-Petviashvili and Gelfand-Dickey hierarchies

In this paper, we construct the additional flows of the noncommutative Kadomtsev-Petviashvili (KP) hierarchy and the additional symmetry flows constitute an infinite dimensional Lie algebra W1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions and this generating symmetry is used to construct the noncommutative KP hierarchy with self-consistent sources and the constrained noncommutative KP hierarchy. The above results will be further generalized to the noncommutative Gelfand-Dickey hierarchies which contain many interesting noncommutative integrable systems such as the noncommutative KdV hierarchy and noncommutative Boussinesq hierarchy. Meanwhile, we construct two new noncommutative systems including odd noncommutative C type Gelfand-Dickey and even noncommutative C type Gelfand-Dickey hierarchies. Also using the symmetry, we can construct a new noncommutative Gelfand-Dickey hierarchy with self-consistent sources. Based on the natural differential Lax operator of the noncommutative Gelfand-Dickey hierarchy, the string equations of the noncommutative Gelfand-Dickey hierarchy are also derived.

Rational reductions of the 2D-Toda hierarchy and mirror symmetry

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz–Ladik hierarchy and E. Frenkel's q -deformed Gelfand–Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki–Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz–Ladik hierarchy to Gromov–Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi–Yau threefolds obtained from minimal resolutions of the local orbifold line.

Towards a description of the double ramification hierarchy for Witten's r-spin class

The double ramification hierarchy is a new integrable hierarchy of Hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for r=3,4,5 and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin–Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the r-th Gelfand–Dickey hierarchy for r=3,4,5.

Tau Function and Virasoro Action for the n × n KdV Hierarchy

This is the third in a series of papers attempting to describe a uniform geometric framework in which many integrable systems can be placed. A soliton hierarchy can be constructed from a splitting of an infinite dimensional group L as positive and negative subgroups \({L_\pm}\) and a commuting sequence in the Lie algebra \({\mathcal{L}_+}\) of \({L_+}\). Given \({f\in L_-}\), there is a formal inverse scattering solution u f of the hierarchy. When there is a 2 co-cycle on \({\mathcal{L}}\) that vanishes on both \({\mathcal{L}_+ {\rm and} \mathcal{L}_-}\), Wilson constructed for each \({f\in L_-}\) a tau function \({\tau_f}\) for the hierarchy. In this third paper, we prove the following results for the n × n KdV hierarchy: (1) The second partials of \({\ln\tau_f}\) are differential polynomials of the formal inverse scattering solution u f . Moreover, \({u_f}\) can be recovered from the second partials of \({\ln\tau_f}\). (2) The natural Virasoro action on \({\ln\tau_f}\) constructed in the second paper is given by partial differential operators in \({\ln\tau_f}\). (3) There is a bijection between phase spaces of n × n KdV hierarchy and Gelfand-Dickey (GD n ) hierarchy on the space of order n linear differential operators on the line so that the flows in these two hierarchies correspond under the bijection. (4) Our Virasoro action on the n × n KdV hierarchy is constructed from a simple Virasoro action on the negative group. We show that it corresponds to the known Virasoro action on the GD n hierarchy under the bijection.

Frobenius Structures on Double Hurwitz Spaces

We present a construction Frobenius structures of “dual type” on the moduli space of ramified coverings of with given ramification type over two points, generalizing Dubrovin's construction of [4, 5]. A complete hierarchy of hydrodynamic type is obtained from the corresponding deformed flat connection. This provides a suitable framework for the theory of weakly deformed soliton lattices of an enlarged class of integrable equations; as applications we consider the q-deformed Gelfand–Dickey hierarchy and the sine-Gordon equation, and explicitly compute the corresponding solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations.