Dispersionless Hierarchy Research Articles

The constrained dispersionless BKP hierarchy and its additional symmetries

In this paper, the constrained dispersionless BKP hierarchy are given by restricting the formal Laurent series [Formula: see text] of the dispersionless BKP hierarchy to [Formula: see text]. Its additional symmetries are given by introducing vital formal Laurent series [Formula: see text]. The additional flows form the nonabelian [Formula: see text]-grading Lie algebra. Furthermore, the additional flows acting on [Formula: see text] and [Formula: see text] are presented.

Virasoro symmetries of the constrained dispersionless mKP hierarchy

This paper aims at the auto-Bäcklund transformations for the dispersionless mKP hierarchy, dispersionless Miura map between the constrained dispersionless mKP hierarchy and the constrained dispersionless Harry Dym hierarchy and Virasoro symmetries of the constrained dispersionless mKP hierarchy. The additional symmetries of the constrained dispersionless mKP hierarchy are constructed in a manner similar to that of the dispersionless mKP hierarchy. By comparison, a new Laurent series Yl is introduced. The additional flows form a subalgebra of the Virasoro algebra.

Decompositions and linear superpositions of B‐type Kadomtsev‐Petviashvili equations

The existence of decomposition solutions of the well‐known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

Additional symmetries of the dispersionless cKP hierarchy

In this paper, the additional symmetries of the dispersionless cKP hierarchy are given by introducing vital formal Laurent series Yl. The additional flows form a subalgebra of the Virasoro algebra. Furthermore, the additional flows acting on μ(t,z) and ρ(t,z) of the dispersionless cKP hierarchy are presented.

Integrable hierarchies associated to infinite families of Frobenius manifolds

We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to A N , D N and B N singularities. In the case of A N Frobenius manifolds our hierarchy turns out to coincide with the dispersionless KP hierarchy; for B N Frobenius manifolds it coincides with the dispersionless BKP hierarchy; and for D N hierarchy it is a certain reduction of the dispersionless 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain coefficients of A N , D N and B N Frobenius potentials.

Six-dimensional heavenly equation. Dressing scheme and the hierarchy

We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation, Lax–Sato equations and develop the dressing scheme for the reduced hierarchy. Using the dressing scheme, we construct a class of solutions for six-dimensional heavenly equation in terms of implicit functions.

Multi-variable reductions of the dispersionless DKP hierarchy

We consider multi-variable reductions of the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) in the elliptic parametrization. The reduction is given by a system of elliptic Löwner equations supplemented by a system of partial differential equations of hydrodynamic type. The compatibility conditions for the elliptic Löwner equations are derived. They are elliptic analogues of the Gibbons–Tsarev equations. We prove solvability of the hydrodynamic type system by means of the generalized hodograph method. The associated diagonal metric is proved to be of the Egorov type.

Linearly degenerate hierarchies of quasiclassical SDYM type

We demonstrate that SDYM (self-dual Yang-Mills) equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of a general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions and introduce a generating relation, Lax-Sato equations, and the dressing scheme for the reduced hierarchy. A multidimensional case is also discussed.

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.

Quantum-Classical Correspondence of Shortcuts to Adiabaticity

We formulate the theory of shortcuts to adiabaticity in classical mechanics. For a reference Hamiltonian, the counterdiabatic term is constructed from the dispersionless Korteweg-de Vries (KdV) hierarchy. Then the adiabatic theorem holds exactly for an arbitrary choice of time-dependent parameters. We use the Hamilton-Jacobi theory to define the generalized action. The action is independent of the history of the parameters and is directly related to the adiabatic invariant. The dispersionless KdV hierarchy is obtained from the classical limit of the KdV hierarchy for the quantum shortcuts to adiabaticity. This correspondence suggests some relation between the quantum and classical adiabatic theorems.

Dispersionless Pfaff-Toda hierarchy and elliptic Löwner equation

We show that one-variable reductions of the Pfaff-Toda integrable hierarchy in the dispersionless limit are described by a system of coupled elliptic Löwner (Komatu-Goluzin) equations.

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.

Dispersionless DKP hierarchy and the elliptic Löwner equation

We show that the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) admits a suggestive reformulation through elliptic functions. We also consider one-variable reductions of the dispersionless DKP hierarchy and show that they are described by an elliptic version of the Löwner equation. With a particular choice of the driving function, the latter appears to be closely related to the Painlevé VI equation with a special choice of parameters.

Integrability properties of the dispersionless Kadomtsev-Petviashvili hierarchy

In the paper, we investigate integrability characteristics for the dispersionless Kadomtsev-Petviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures, and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the master symmetry, symmetries, Noether operator, and conserved covariants, Hamiltonian structures are constructed for both isospectral and non-isospectral dispersionless Kadomtsev-Petviashvili hierarchies. Finally, two sets of conserved quantities and their Lie algebra are derived for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy.

Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy

In this paper, the dispersionless D-type Drinfeld–Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.

Dispersionless BKP Hierarchy and Quadrant Löwner Equation

We show that $N$-variable reduction of the dispersionless BKP hierarchy is described by a L\"owner type equation for the quadrant.

Dunajski–Tod equation and reductions of the generalized dispersionless 2DTL hierarchy

We transfer the scheme for constructing differential reductions recently developed for the Manakov–Santini hierarchy to the case of the two-component generalization of dispersionless 2DTL hierarchy. We demonstrate that the equation arising as a result of the simplest reduction is equivalent (up to a Legendre type transformation) to the Dunajski–Tod equation, locally describing general ASD vacuum metric with conformal symmetry. We consider higher reductions and corresponding reduced hierarchies also.

The Local Gromov–Witten Theory of $${\mathbb{C}\mathbb{P}^1}$$ and Integrable Hierarchies

In this paper we begin the study of the relationship between the local Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large number of cases, the Hamiltonian dispersionless hierarchies that govern the full descendent genus zero theory. Our main tool is the application of Dubrovin's formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of the resolved conifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For this example the relevant dispersionless system turns out to be related to the long-wave limit of the Ablowitz-Ladik lattice. This identification provides us with a complete procedure to reconstruct the dispersive hierarchy which should conjecturally be related to the higher genus theory of the resolved conifold. We give a complete proof of this conjecture for genus g<=1; our methods are based on establishing, analogously to the case of KdV, a "quasi-triviality" property for the Ablowitz-Ladik hierarchy at the leading order of the dispersive expansion. We furthermore provide compelling evidence in favour of the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing it successfully in the primary sector for g=2.

A Class of Infinite-dimensional Frobenius Manifolds and their Submanifolds

We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionless two-component BKP hierarchy. Moreover, we show that these infinite-dimensional Frobenius manifolds contain finite-dimensional Frobenius submanifolds as defined on the orbit space of Coxeter groups of types B and D.

Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

We consider two-component integrable generalizations of the dispersionless two-dimensional Toda lattice (2DTL) hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov–Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph-type transformations. Generating equations and Lax–Sato equations are introduced, and a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, and its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.