Dispersionless KdV Research Articles

Bihamiltonian Cohomology of KdV Brackets

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.

Symplectic Field Theory of a Disk, Quantum Integrable Systems, and Schur Polynomials

We consider commuting operators obtained by quantization of Hamiltonians of the Hopf (aka dispersionless KdV) hierarchy. Such operators naturally arise in the setting of Symplectic Field Theory (SFT). A complete set of common eigenvectors of these operators is given by Schur polynomials. We use this result for computing the SFT potential of a disk.

Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case

We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.

Infinite-dimensional 3-algebra and integrable system

The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV hierarchy follows from the Nambu-Poisson evolution equation given the suitable Hamiltonians. We find that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. Due to the Nambu-Poisson evolution equation involving two Hamiltonians, more intriguing relationships between these Hamiltonians are revealed. As an application, we investigate the system of polytropic gas equations and derive an integrable gas dynamics system in the framework of Nambu mechanics.

Non-Abelian gauge theories, prepotentials, and Abelian differentials

I discuss particular solutions of the integrable systems, starting from well-known dispersionless KdV and Toda hierarchies, which define in most straightforward way the generating functions for the Gromov-Witten classes in terms of the rational complex curve. On the ``mirror'' side these generating functions can be identified with the simplest prepotentials of complex manifolds, and I present few more exactly calculable examples of them. For the higher genus curves, corresponding in this context to the non Abelian gauge theories via the topological gauge/string duality, similar solutions are constructed using extended basis of Abelian differentials, generally with extra singularities at the branching points of the curve.

From Geometry of Jets to Quasiclassical Hierarchies

I consider quasiclassical integrable systems, starting from the well-known dispersionless KdV and Toda hierarchies, which can be totally understood in terms of jet spaces over the rational curves with one or two punctures. For the nontrivial geometry of the higher genus curves, the same approach leads to construction of quasiclassical tau-functions or prepotentials, using the period integrals for Abelian differentials. I discuss also some physical applications of this construction.

On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type

We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivative ∂ u of the field is inverted. We reformulate the question about deformations as a question about the cohomology of a certain double complex, and calculate the appropriate cohomology group.

Gromov–Witten invariants of target curves via Symplectic Field Theory

We compute the Gromov–Witten potential at all genera of target smooth Riemann surfaces using Symplectic Field Theory techniques and establish differential equations for the full descendant potential. We need to impose (and possibly solve) different kinds of Schrödinger equations related to some quantization of the dispersionless KdV hierarchy. In particular, we find explicit formulas for the Gromov–Witten invariants of low degree of P 1 with descendants of the Kähler class.

Lagrangian Approach to Dispersionless KdV Hierarchy

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamil- tonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.

Integrable dispersionless KdV hierarchy with sources

An integrable dispersionless KdV hierarchy with sources (dKdVHWS) is derived. Lax pair equations and bi-Hamiltonian formulation for dKdVHWS are formulated. Hodograph solution for the dispersionless KdV equation with sources (dKdVWS) is obtained via hodograph transformation. Furthermore, the dispersionless Gelfand-Dickey hierarchy with sources (dGDHWS) is presented.

Unitized Jordan algebras and dispersionless KdV equations

Multicomponent KdV systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied. Recursion laws for conserved densities are given under the assumption that the algebra possesses a unity element. Sufficient conditions are given for the unitized counterpart of a diagonalizable non-unital system to be diagonalizable. Hamiltonian structure is discussed within the context of DN Jordan algebras and N scattering problems.

Deformations of dispersionless KdV hierarchies

The obstructions to the existence of a hierarchy of hydrodynamic conservation laws are studied for a multicomponent dispersionless KdV system. It is shown that if an underlying algebra is Jordan, then the lowest obstruction vanishes and that all higher obstructions automatically vanish. Deformations of these multicomponent dispersionless KdV-type equations are also studied. No new obstructions appear, and hence the existence of a fully deformed hierarchy depends on the existence of a single purely hydrodynamic conservation law.

Quantizing the KdV Equation

We consider the quantization procedure for the Gardner–Zakharov–Faddeev and Magri brackets using the fermionic representation for the KdV field. In both cases, the corresponding Hamiltonians are sums of two well-defined operators. Each operator is bilinear and diagonal with respect to either fermion or boson (current) creation/annihilation operators. As a result, the quantization procedure needs no space cutoff and can be performed on the entire axis. In this approach, solitonic states appear in the Hilbert space, and soliton parameters become quantized. We also demonstrate that the dispersionless KdV equation is uniquely and explicitly solvable in the quantum case.

Jordan Manifolds and Dispersionless KdV Equations

Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied, under the assumptions that the Jordan algebra has a unity element and a compatible non-degenerate inner product. Much of this structure may be encoded in a so-called Jordan manifold, akin to a Frobenius manifold. In particular the Hamiltonian properties of these systems are investigated.

HAMILTONIAN STRUCTURES FOR THE GENERALIZED DISPERSIONLESS KdV HIERARCHY

We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so-called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and third Hamiltonian structures are calculated directly from the r-matrix approach. Since the third structure is not related recursively with the first two the generalized dispersionless KdV hierarchy can be characterized as a truly tri-Hamiltonian system.

Remarks on dispersionless KP, KdV, and 2D gravity

We use the SDiff(2) framework of Takasaki and Takebe and the (L, M) program (L is the Lax operator andMω=ωλ) to show that\(\mathfrak{M}\)=semiclassical limit ofM is\(\hat \xi + \sum\nolimits_2^\infty {T'_n } \lambda ^{n - 1} \), where (\(\lambda , - \hat \xi \)) are action angle variables in the Gibbons-Kodama theory of Hamilton-Jacobi type for dispersionless KP. We also show\(\hat \xi \) is the semiclassical limit ofWxW −1 (W is the gauge operator), whereG=WxW −1 is a quantity studied by the author in an earlier paper in connection with symmetries. We give then a semiclassical version of the Jevicki-Yoneya action principle for 2D gravity, where again\(\hat \xi \) arises in calculations, and this yields directly the Landau-Ginsburg equation that corresponds to the semiclassical limit of an integrated string equation. For KdV we also show how inverse scattering data are connected to Hamiltonians for dispersionless KdV. We also discuss Hirota bilinear formulas relative to the dispersionless hierarchies and establish various limiting formulas.

ADDITIONAL SYMMETRIES OF KP, GRASSMANNIAN, AND THE STRING EQUATION

It is well-known that solutions to the string equation are generated by elements of Sato's Grassmannian which are invariant under action of some differential operator. Here it is shown that this operator is nothing else than the infinitesimal operator of the group of additional symmetries of the KdV flow. This is done for KdV hierarchies of arbitrary orders. Virasoro constraints are obtained in a slightly more general form than they are usually written.