Trigonometric Solutions of the WDVV Equations from Root Systems

By introduction of an additional variable and addition of a Weyl invariant correction term to the perturbative prepotential in five-dimensional Seiberg-Witten theory we construct solutions of the Witten–Dijkgraaf–Verlinde–Verlinde equations of trigonometric type for all crystallographic root systems.     

Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions

For any classical Lie algebra $mathfrak{g}$,we construct a family ofintegrable generalizations of Toda mechanics labeled a pair of orderedintegers $(m,n)$. The universal form of the Lax pair, equations of motion,Hamiltonian as well as Poisson brackets are provided, and explicit examplesfor $mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,nleq3$ are also given. For all$m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Todachains are natural reductions of that of the $(m,n)$-chain, and for $m=n$,there is also a family of symmetrically reduced Toda systems, the$(m,m)_{mathrm{Sym}}$-Toda systems, which are also integrable. In the quantumcase, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics ofstandard Toda variables coupled to noncommutative variables. Except for thesymmetrically reduced cases, the integrability for all $(m,n)$-Toda systemssurvive after quantization.     

Formulas for An- and Bn-solutions of WDVV equations

The simplest non-trivial solutions of WDVV equations are A_n-and B_n-potentials, which describe metrics of Saito on spaces of versal deformation of A_n-and B_n-singularities. These are some polynomials, which were known for n≤4. In this paper, we find the potentials for all A_n-and B_n-singularities. We give a recurrence formula for coefficients of KP and n-KdV hierarchy.     

Quasigraded Lie Algebras and Hierarchies of Integrable Equations

Using special quasigraded Lie algebras we obtain new hierarchies of integrable equations in partial derivatives admitting zero-curvature representations. In particular, we obtain new type of so(3) anisotropic chiral-field equation along with its higher rank generalization.     

Modular Forms and Second Order Ordinary Differential Equations: Applications to Vertex Operator Algebras

We study the relation between the Kaneko–Zagier equation and the Mathur–Mukhi–Sen classification, and extend it to the case of solutions with logarithmic terms, which correspond to pseudo-characters of non-rational vertex operator algebras. As an application, we prove a non-existence theorem of rational vertex operator algebras.     

Lie Symmetries,Infinite-Dimensional Lie Algebras and Similarity Reductions of Certain (2+1)-Dimensional Nonlinear Evolution Equations

Abstract

The Lie point symmetries associated with a number of (2 + 1)-dimensional generalizations of soliton equations are investigated. These include the Niznik – Novikov – Veselov equation and the breaking soliton equation, which are symmetric and asymmetric generalizations respectively of the KDV equation, the (2+1)-dimensional generalization of the nonlinear Schrödinger equation by Fokas as well as the (2+1)-dimensional generalized sine-Gordon equation of Konopelchenko and Rogers. We show that in all these cases the Lie symmetry algebra is infinite-dimensional; however, in the case of the breaking soliton equation they do not possess a centerless Virasorotype subalgebra as in the case of other typical integrable (2+1)-dimensional evolution equations. We work out the similarity variables and special similarity reductions and investigate them.     

Conditional Similarity Reductions of Jimbo-Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

On a Poisson–Lie Analogue of the Classical Dynamical Yang–Baxter Equation for Self-dual Lie Algebras

We derive a generalization of the classical dynamical Yang–Baxter equation (CDYBE) on a self-dual Lie algebra G by replacing the cotangent bundle T*G in a geometric interpretation of this equation by its Poisson–Lie (PL) analogue associated with a factorizable constant r-matrix on G. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.     

A Homotopy in the Usual Cochain Complex of Free Lie Algebras

In this Letter, we construct a natural contracting homotopy in the usual cochain complex of free Lie algebras. As a consequence, we prove that the triple cohomology of Lie algebras coincides with a slightly different form of the standard cohomology theory.     

Lie Point Symmetries and Exact Solutions of Couple KdV Equations

The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.     

Multiple Soliton Solutions of the High Order Broer-Kaup Equations

Using the extended homogeneous balance method, which is very concise and primary, we find the multiple soliton solutions of the high order Broer-Kaup equations. The method can be generalized to dealing with high-dimensional Broer-Kaup equations and other class of nonlinear equations.     

Application of the cohomology of graded Lie algebras to formal deformations of Lie Algebras

The purpose of the Letter is to show how to use the cohomology of the Nijenhuis-Richardson graded Lie algebra of a vector space to construct formal deformations of each Lie algebra structure of that space. One then shows that the de Rham cohomology of a smooth manifold produces a family of cohomology classes of the graded Lie algebra of the space of smooth functions on the manifold. One uses these classes and the general construction above to provide one-differential formal deformations of the Poisson Lie algebra of the Poisson manifolds and to classify all these deformations in the symplectic case.     

Lie Reduction and Conditional Symmetries of Some Variable Coefficient Nonlinear Wave Equations＊

Lie symmetry reduction of some truly ＂variable coefficient＂ wave equations which are singled out from a class of （1 ＋ 1）-dimensional variable coefficient nonlinear wave equations with respect to one and two-dimensional algebras is carried out. Some classes of exact solutions of the investigated equations are found by means of both the reductions and some modern techniques such as additional equivalent transformations and hidden symmetries and so on. Conditional symmetries are also discussed.     

Two Hierarchies of New Differential-Difference Equations Related to the Darboux Transformations of the Kaup–Newell Hierarchy

Two hierarchies of new nonlinear differential-difference equations with one continuous variable and one discrete variable are constructed from the Darboux transformations of the Kaup–Newell hierarchy of equations. Their integrable properties such as recursion operator, zero-curvature representations, and bi-Hamiltonian structures are studied. In addition, the hierarchy of equations obtained by Wu and Geng is identified with the hierarchy of two-component modified Volterra lattice equations.     

Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

From any given Frobenius manifold one may construct a so-called ’dual’ structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the Witten– Dijkgraaf–Verlinde–Verlinde equations of associativity. Jacobi group orbit spaces naturally carry the structure of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.     

Moufang Loops and Integrability of Generalized Lie Equations

Generalized Lie equations (GLE) for linear birepresentations of the analytic Moufang loops are considered. Integrability conditions of GLE are found and presented in closed Lie algebra form.     

Generation of Nonlinear Evolution Equations by Reductions of the Self-Dual Yang–Mills Equations

With the help of some reductions of the self-dual Yang Mills （briefly written as sdYM） equations, we introduce a Lax pair whose compatibility condition leads to a set of （2 ＋ 1）-dimensional equations. Its first reduction gives rise to a generalized variable-coefficient Burgers equation with a forced term. Furthermore, the Burgers equation again reduces to a forced Burgers equation with constant coefficients, the standard Burgers equation, the heat equation, the Fisher equation, and the Huxley equation, respectively. The second reduction generates a few new （2 ＋ 1）-dimensional nonlinear integrable systems, in particular, obtains a kind of （2 ＋ 1）-dimensional integrable couplings of a new （2 ＋ 1）- dimensional integrable nonlinear equation.     

Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP Equations

Abstract

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with m arbitrary time-dependent coefficients are obtained possessing symmetries involving m arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.

Dedicated to Prof. W.I. Fushchych on the occasion of his 60th birthday     

Primary Branch Solutions of First Order Autonomous Scalar Partial Differential Equations via Lie Symmetry Approach

A primary branch solution (PBS) is defined as a solution with m independent n ? 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction of auxiliary fields. Because of the intrusion of the arbitrary function, the PBSs have abundant and complicated structure. Usually, PBSs are implicit solutions. In some special cases, explicit solutions such as the instanton (rogue wave like) solutions may be obtained by suitably fixing the arbitrary function of the PBS.