Gelfand-Dikii analysis forN=2 supersymmetric KdV equations

We generalize the resolvent approach of Gelfand and Dikii to the KdV equation to study theN=2 supersymmetric KdV equations of Laberge and Mathieu. For the associated Lax operators, we study the coincidence limits of the resolvent kernel and its derivatives, and obtain differential equations which they satisfy. These allow us to obtain recursion relations for the analogues of the Gelfand-Dikii polynomials and to obtain a proof of Hamiltonian integrability of the supersymmetric KdV equations. We are also able to write the Lax equations for the corresponding hierarchies in terms of these polynomials.Address after January 1, 1993: Department of Physics, University of Western Australia, Nedlands, Australia 6009     

ClassicalN=1W-superalgebras from Hamiltonian reduction

A combinatorial proof is presented of the fact that the space of supersymmetric Lax operators admits a Poisson structure analogous to the second Gel'fand-Dickey bracket of the generalized KdV hierarchies. This allows us to prove that the space of Lax operators of odd order has a symplectic submanifold-defined by (anty)symmetric operators-which inherits a Poisson structure defining classicalW-superalgebras extending theN=1 supervirasoro algebra. This construction thus yields an infinite series of extended superconformal algebras.Address after October 1991: Physikalisches Institut der Universität Bonn, FRG     

Towards the construction of N = 2 supersymmetric integrable hierarchies

We formulate a conjecture for the three different Lax operators that describe the bosonic sectors of the three possible N = 2 supersymmetric integrable hierarchies with N = 2 super-W_n second Hamiltonian structure. We check this conjecture in the simplest cases, then we verify it in general in one of the three possible supersymmetric extensions. To this end we construct the N = 2 supersymmetric extensions of the Generalized Non-Linear Schrödinger hierarchy by exhibiting the corresponding super Lax operator. To find the correct Hamiltonians we are led to a new definition of super-residues for degenerate N = 2 supersymmetric pseudo-differential operators. We have found a new non-polynomials Miura-like realization for N = 2 superconformal algebra in terms of two bosonic chiral-anti-chiral free superfields.     

Towards Lax Formulation of Integrable Hierarchies of Topological Type

To each partition function of cohomological field theory one can associate an Hamiltonian integrable hierarchy of topological type. The Givental group acts on such partition functions and consequently on the associated integrable hierarchies. We consider the Hirota and Lax formulations of the deformation of the hierarchy of N copies of KdV obtained by an infinitesimal action of the Givental group. By first deforming the Hirota quadratic equations and then applying a fundamental lemma to express it in terms of pseudo-differential operators, we show that such deformed hierarchy admits an explicit Lax formulation. We then compare the deformed Hamiltonians obtained from the Lax equations with the analogous formulas obtained in Buryak et al. (J Differ Geom 92:153–185, 2012), Buryak et al. (J Geom Phys 62:1639–1651, 2012) to find that they agree. We finally comment on the possibility of extending the Hirota and Lax formulation on the whole orbit of the Givental group action.     

Integrable extensions of N = 2 supersymmetric KdV hierarchy associated with the nonuniqueness of the roots of the Lax operator

We present new supersymmetric integrable extensions of the a = 4, N = 2 KdV hierarchy. The root of the supersymmetric Lax operator of the KdV equation is generalized, by including additional fields. This generalized root generates a new hierarchy of integrable equations, for which we investigate the Hamiltonian structure. In a special case our system describes the interaction of the KdV equation with the two MKdV equations.     

N = 2 Supercomplexification of the Korteweg-de Vries,Sawada-Kotera and Kaup-Kupershmidt Equations

The supercomplexification is a special method of N = 2 supersymmetrization of the integrable equations in which the bosonic sector can be reduced to the complex version of these equations. The N = 2 supercomplex Korteweg-de Vries, Sawada-Kotera and Kaup-Kupershmidt equations are defined and investigated. The common attribute of the supercomplex equations is appearance of the odd Hamiltonian structures and superfermionic conservation laws. The odd bi-Hamiltonian structure, Lax representation and superfermionic conserved currents for new N = 2 supersymmetric Korteweg-de Vries equation and for Sawada-Kotera one, are given.     

Extended super-Kač-Moody algebras and their super-derivation algebras

We study theN-extended super-Ka-Moody algebras, i.e. extensions of the Lie algebra of the loop group over the super-circleA _N . The extensions are characterized by 2-cocycles which are computed in terms of the cyclic cohomology of the Grassmann algebra withN generators. The graded algebra of super-derivations compatible with each extension is determined. The casesN=1,2,3 are examined in detail and their relation with the Ademollo et al. superconformal algebras is discussed. We examine the possibility of defining new superconformal algebras which, forN>1, generalize theN=1 Ramond-Neveu-Schwarz algebra.     

Painlevé analysis,conservation laws,and symmetry of perturbed nonlinear equations

We consider the Lie-Backlund symmetries and conservation laws of a perturbed KdV equation and NLS equation. The arbitrary coefficients of the perturbing terms can be related to the condition of existence of nontrivial LB symmetry generator. When the perturbed KdV equation is subjected to Painlevé analysisa la Weiss, it is found that the resonance position changes compared to the unperturbed one. We prove the compatibility of the overdetermined set of equations obtained at the different stages of recursion relations, at least for one branch. All other branches are also indicated and difficulties associated them are discussed considering the perturbation parameter to be small. We determine the Lax pair for the aforesaid branch through the use of Schwarzian derivative. For the perturbed NLS equation we determine the conservation laws following the approach of Chen and Liu. From the recurrence of these conservation laws a Lax pair is constructed. But the Painlevé analysis does not produce a positive answer for the perturbed NLS equation. So here we have two contrasting examples of perturbed nonlinear equations: one passes the Painlevé test and its Lax pair can be found from the analysis itself, but the other equation does not meet the criterion of the Painlevé test, though its Lax pair is found in another way.     

Blow-up criteria and periodic peakons for a two-component extension of the μ-version modified Camassa-Holm equation

Some two-component extensions of the modifiedμ-Camassa-Holm equation are proposed.We show that these systems admit Lax pairs and bi-Hamiltonian structures.Furthermore,we consider the blow-up phenomena for one of these extensions(2μmCH),and the periodic peakons of this system are derived.     

Dual formulation of classicalW-algebras

By extending the concept of Maurer-Cartan equations, a dual formulation of (classical) nonlinear extensions of the Virasoro algebra is introduced. This dual formulation is closely related to three-dimensional actions which are analogous to a Chern-Simons action. An explicit construction in terms of superfields of theN = 2 superW ₄-algebra is presented.     

On the integrable hierarchies associated with the N = 2 super Wn algebra

A new Lax operator is proposed from the viewpoint of constructing the integrable hierarchies related with the N = 2 super W_n algebra. It is shown that the Poisson algebra associated to the second Hamiltonian structure for the resulting hierarchy contains the N = 2 super Virasoro algebra as a proper subalgebra. The simplest cases are discussed in detail. In particular, it is proved that the supersymmetric two-boson hierarchy is one of the N = 2 supersymmetric KdV hierarchies. Also, a Lax operator is supplied for one of the N = 2 supersymmetric Boussinesq hierarchies.     

On the problem of spacetime symmetries in the theory of supergravity. part II:N=2 supergravity and spinorial Lie derivatives

The definition of a spacetime symmetry, developed in a previous paper in the framework of simple (N=1) supergravity, is extended to theN=2 theory. As an application, the properties of theN=2 plane wave are studied. The mathematically related question of defining the Lie derivative of a spinor is also considered.     

The Riemann-Hilbert Formalism for Certain Linear and Nonlinear Integrable PDEs

Abstract

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations.

In the usual Dressing Method, one first postulates a matrix RH problem and then constructs dressing operators. Here we present an algorithmic construction of matrix Riemann-Hilbert (RH) problems appropriate for the dressing method as opposed to postulating them ad hoc. Furthermore, we introduce two mechanisms for the construction of the relevant dressing operators: The first uses operators with the same dispersive part, but with different decay at infinity, while the second uses pairs of operators corresponding to different Lax pairs of the same linear equation. As an application of our approach, we derive the NLS, derivative NLS, KdV, modified KdV and sine-Gordon equations.     

CompositeZ-boson and CP violation in the processe + e -→Zγ⋆

QCD duality sum rules are applied to the calculation of theB-parameter forN=1 supergravity-induced local operators. Using the chiral realization of local operators, the duality region has been found and the tree-level coupling constant determined. The result is in full accordance with a related calculation of theB-parameter in the standard model by Pich and de Rafael and support it. The result is an order of magnitude smaller than the one obtained by using vacuum saturation. Consequences of softened constraints are discussed.     

Soliton Solutions of the N=2 Supersymmetric KP Equation

Abstract

The N=2 super-KP equation associated with nonstandard flows is bilinearized using the Hirota method and soliton solutions are obtained. The bilinearization has been done for component fields and its KdV limit is discussed by comparing the soliton solutions obtained by this procedure with those found from the N=1 superspace formalism. The equivalence of these two procedures in the KdV limit is observed.     

Fermionic strings on the compactified moduli space

On the compactified moduli space we consider theN=2,N=4 local supersymmetric string theories. It would be proven that theN=2,N=4 fermionic string theories might not develop any tachyon pole, which might imply theg-loop partition functions forN=2,N=4 fermionic string would be finite.     

On Integrability of a (2+1)-Dimensional Perturbed KdV Equation

Abstract

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlevé; test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painlevé; classification of coupled KdV equations by A. Karasu should be revised.