The Symmetry Structure of the Heavenly Equation

We show that excitations of physical interest for the heavenly equation are generated by symmetry operators which yield two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi–Hanson and Gibbons–Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the so-called Schröder equation, which plays a crucial role in the bootstrap model and in renormalization theory.     

Method of group foliation and non-invariant solutions of partial differential equations

Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, i.e. the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: sheftel@gursey.gov.tr     

Anti-self-dual gravitational metrics determined by the modified heavenly equation

In paper Doubrov and Ferapontov (2010) on the classification of integrable complex Monge–Ampère equations, the modified heavenly (MH) equation of Dubrov and Ferapontov is one of canonical equations. It is well known that solutions of the first and second heavenly equations of Plebañski (1975) and those of the Husain equation in Husain (1994) provide potentials for anti-self-dual (ASD) Ricci-flat vacuum metrics. For another canonical equation, the general heavenly equation of Dubrov and Ferapontov (2010), we had constructed in Malykh and Sheftel (2011) ASD Ricci-flat metric governed by this equation. Thus, the modified heavenly equation remains the only one in the list of canonical equations in Doubrov and Ferapontov (2010) for which such a metric is missing so far. Our aim here is to construct null tetrad of vector fields, coframe 1-forms and ASD Ricci-flat metric for the latter equation. We study reality conditions and signature for the resulting metric. As an example, we obtain a multi-parameter cubic solution of the MH equation which yields a family of metrics with the above properties. Riemann curvature 2-forms are also explicitly presented for the cubic solution.     

Twists in Hopf algebras andRT 1 T 2=T 2 T 1 relations

LetR be a matrix unitary quasi-classical solution of the Yang-Baxter equation. Considering an associative algebra defined by the relationRT ₁ T ₂=T ₂ T ₁ we find a universal twistF such thatR is the image ofR=F ₂₁ F ⁻¹ in the vector representation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

De Rham complex with ternary differential on the quantum group SL h (2)

In this work, we construct the de Rham complex with the differential operator d satisfying theQ-Leibniz rule, whereQ is a complex number, and the condition d³=0 (ternary differential) on theh-deformed quantum plane in dimension two following the general formalism elaborated in [Bazunova et al.: Czech. J. Phys.51 (2001) 1226]. Then we construct the de Rham complex on the group SL _h (2) which preserves theh-deformed quantum plane by the differentiation of RTT-relations, and, introducing the Maurer-Cartan forms, get an analog of the Maurer-Cartan equation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Schwinger representation for the symmetric group: Two explicit constructions for the carrier space

We give two explicit constructions for the carrier space for the Schwinger representation of the group Sn. While the first relies on a class of functions consisting of monomials in antisymmetric variables, the second is based on the Fock space associated with the Greenberg algebra.     

Conformal Invariance and Conserved Quantities of General Holonomic Systems

Conformed invariance and conserved quantities of general holonomic systems are studied. A one-parameter infinitesimal transformation group and its infinitesimal transformation vector of generators are described. The definition of conformal invariance and determining equation for the system are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and suttlcient condition, that conformal invariance of the system would be Lie symmetry, is obtained under the infinitesimal one-parameter transformation group. The corresponding conserved quantity is derived with the aid of a structure equation. Lastly, an example is given to demonstrate the application of the result.     

RTT presentation of ortho-symplectic super Yangians. Staggered integrable spin chains

We give a RTT presentation of ortho-symplectic super Yangians that encompasses the orthogonal and symplectic cases. In a second part, we construct an integrable ladder chain model and study its quantum symmetry. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation

We solve the inverse scattering problem for multidimensional vector fields and we use this result to construct the formal solution of the Cauchy problem for the second heavenly equation of Plebanski, a scalar nonlinear partial differential equation in four dimensions underlying self-dual vacuum solutions of the Einstein equations, which arises from the commutation of multidimensional Hamiltonian vector fields.     

The twisted photon associated to hyper-Hermitian four-manifolds

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a pair of potentials that generalises Plebanski's second heavenly equation for hyper-Kähler four-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields.     

Symmetries,integrals, and three-dimensional reductions of Plebañski’s second heavenly equation

We study symmetries and conservation laws for Plebañski’s second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems.     

Nonlocal symmetries of Plebański’s second heavenly equation

We study nonlocal symmetries of Plebański’s second heavenly equation in an infinite-dimensional covering associated to a Lax pair with a non-removable spectral parameter. We show that all local symmetries of the equation admit lifts to full-fledged nonlocal symmetries in the infinite-dimensional covering. Also, we find two new infinite hierarchies of commuting nonlocal symmetries in this covering and describe the structure of the Lie algebra of the obtained nonlocal symmetries.     

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.     

On geometric approach to Lie symmetries of differential-difference equations

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach.     

A novel realization of the Virasoro algebra in number state space

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d?2, accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for su(1,1). As a step towards the solution, a realization of the (d=1) Virasoro algebra Vir∼Diff₊(S(1)) in the universal envelope of su(1,1) (and h(1)) is presented, which is simple in the discrete positive series irreducible unitary representation of su(1,1).     

Stability of Motion of a Nonholonomic System

Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev＇s type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.     

BI-HAMILTONIAN REPRESENTATION,SYMMETRIES AND INTEGRALS OF MIXED HEAVENLY AND HUSAIN SYSTEMS

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.     

The Landau electron problem on a cylinder

We consider the quantum mechanics of an electron confined to move on an infinite cylinder in the presence of a uniform radial magnetic field. This problem is in certain ways very similar to the corresponding problem on the infinite plane. Unlike the plane however the group of symmetries of the magnetic field, namely, rotations about the axis and the axial translations, is not realized by the quantum electron but only a subgroup comprising rotations and discrete translations along the axial direction, is. The basic step size of discrete translations is such that the flux through the ‘unit cylinder cell’ is quantized in units of the flux quantum. The result is derived in two different ways: using the condition of projective realization of symmetry groups and using the more familiar approach of determining the symmetries of a given Hamiltonian.     

Solution of the Cauchy problem for a continuous limit of the Toda lattice and its superextension

A supersymmetric equation associated with a continuum limit of the classical superalgebra sl(n/n+1) is constructed. This equation can be considered as a superextension of a continuous limit of t the Toda lattice with fixed end-points or, in other words, as a supersymmetric version of the heavenly equation. A solution of the Cauchy problem for the continuous limit of the Toda lattice and for its superextension is given using some formal reasonings.     

A New Conserved Quantity Corresponding to Mei Symmetry of Tzenoff Equations for Nonholonomic Systems

A new conserved quantity is investigated by utilizing the definition and discriminant equation of Mei symmetry of Tzénoff equations for nonholonomic systems. In addition, the expression of this conserved quantity, and the determining condition induced new conserved quantity are also presented.