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1.
We consider the associativity (or WDVV) equations as they appear in the Seiberg–Witten theory and prove that they are covariant under general electric–magnetic duality transformations. We discuss the consequences of this covariance from various perspectives. 相似文献
2.
B. A. Dubrovin M. V. Pavlov S. A. Zykov 《Functional Analysis and Its Applications》2011,45(4):278-290
We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one
of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such
solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive
at a complete classification of irreducible solutions. 相似文献
3.
We prove that the equations describing compatible N×N metrics of constant Riemannian curvature define a special class of integrable N-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open–closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open–closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras. 相似文献
4.
We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t1, ..., tn) of WDVV equations of associativity polynomial in t1, ..., tn-1, exp tn. 相似文献
5.
Using analytic methods of finite-gap integration, we construct quasihomogeneous algebraic solutions of the WDVV associativity
equations and the nonsemisimple Frobenius manifolds associated with them.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 195–206, May, 2007. 相似文献
6.
Robert Carroll 《Acta Appl Math》2000,60(3):225-316
We sketch some of the different roles played by Whitham times in connection with averaging, adiabatic invariants, soliton theory, Hamiltonian structures, topological field theory (TFT), Seiberg–Witten (SW) theory, isomonodromy problems, Hitchin systems, WDVV and Picard–Fuchs equations, renormalization, soft supersymmetry breaking, etc. 相似文献
7.
An integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group
. The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the
hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations. 相似文献
8.
A. Ya. Kazakov 《Theoretical and Mathematical Physics》1995,102(3):257-264
A modification of the Riccati system that makes it possible to reduce to linear problems the initial-value problem for systems of ordinary differential equations with bilinear nonlinearity is discussed. It is shown that from the algebraic point of view it is natural, in the framework of the scheme, to consider functions that take values in an algebra with two multiplications related by a condition of the type of associativity.State University, St. Petersburg. Translated from Teoretcheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 352–363, March, 1995. 相似文献
9.
M.L. Gandarias 《Journal of Mathematical Analysis and Applications》2008,348(2):752-759
The Type-II hidden symmetries are extra symmetries in addition to the inherited symmetries of the differential equations when the number of independent and dependent variables is reduced by a Lie point symmetry. In [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622] Abraham-Shrauner and Govinder have analyzed the provenance of this kind of symmetries and they developed two methods for determining the source of these hidden symmetries. The Lie point symmetries of a model equation and the two-dimensional Burgers' equation and their descendants were used to identify the hidden symmetries. In this paper we analyze the connection between one of their methods and the weak symmetries of the partial differential equation in order to determine the source of these hidden symmetries. We have considered the same models presented in [B. Abraham-Shrauner, K.S. Govinder, Provenance of Type II hidden symmetries from nonlinear partial differential equations, J. Nonlinear Math. Phys. 13 (2006) 612-622], as well as the WDVV equations of associativity in two-dimensional topological field theory which reduces, in the case of three fields, to a single third order equation of Monge-Ampère type. We have also studied a second order linear partial differential equation in which the number of independent variables cannot be reduced by using Lie symmetries, however when is reduced by using nonclassical symmetries the reduced partial differential equation gains Lie symmetries. 相似文献
10.
Yu-Tung Chen Niann-Chern Lee Ming-Hsien Tu 《Theoretical and Mathematical Physics》2009,161(3):1634-1646
From the bi-Hamiltonian standpoint, we investigate symmetries of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations proposed
by Dubrovin. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-Hamiltonian systems of
hydrodynamic type. In particular, we show that the moduli space of two-primary models under symmetries of the WDVV equations
can be parameterized by the polytropic exponent h. We discuss the transformation properties of the free energy at the genus-one
level. 相似文献
11.
We study several functional equations in two variables that are closely related to the functional equation of associativity. We analyze the hierarchy among these equations, paying a particular attention to selections, semi-lattices and weakenings of associativity. 相似文献
12.
A. V. Marshakov 《Theoretical and Mathematical Physics》2005,142(2):222-236
We discuss the semiclassical geometry and integrable systems related to the gauge—string duality. We analyze semiclassical solutions of the Bethe ansatz equations arising in the context of the AdS/CFT correspondence, comparing them to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma models of the dual string theory and investigate some details of this correspondence.Translated from Teoreticheskaya Matematicheskaya Fizika, Vol. 142, No. 2, pp. 265–283, February, 2005. 相似文献
13.
We discuss the difference equations on a linear lattice for polynomials associated with the classical Hahn, Kravchuk, Meixner, and Charlier polynomials.Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 76–83, January, 1996. 相似文献
14.
In this paper, the parabolic–hyperbolic system of linear thermoelasticity with variable coefficients is transformed into a system of two coupled equations. We discuss first the conditions which govern this separation in the case of a system of two coupled equations for which a general result on the separability is formulated. It is then shown that the explicit traveling wave solutions are obtained in the exact form. 相似文献
15.
We analyze the structure of the Föppl–von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl–von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl–von Kármán equations on the buckling configurations of cylinders. 相似文献
16.
O. I. Mokhov 《Theoretical and Mathematical Physics》2007,152(2):1183-1190
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions
of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of
flat torsionless potential submanifolds. We show that all flat torsionless potential submanifolds in pseudo-Euclidean spaces bear natural structures
of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental
form and the set of the second fundamental forms of the submanifolds (in fact, the structural constants are given by the set
of the Weingarten operators of the submanifolds). We prove that each N-dimensional Frobenius manifold can be locally represented
as a flat torsionless potential submanifold in a 2N-dimensional pseudo-Euclidean space. By our construction, this submanifold
is uniquely determined up to motions. Moreover, we consider a nonlinear system that is a natural generalization of the associativity
equations, namely, the system describing all flat torsionless submanifolds in pseudo-Euclidean spaces, and prove that this
system is integrable by the inverse scattering method.
To the memory of my wonderful mother Maya Nikolayevna Mokhova (4 May 1926–12 September 2006)
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 368–376, August, 2007. 相似文献
17.
We study the dual integral equations related to the Kontorovich-Lebedev integral transforms arising in the course of solution of the problems of mathematical physics, in particular of the mixed boundary value problems for the wedge-shaped regions. We show that the solutions of these equations can be expressed in quadratures, using the auxilliary functions satisfying the integral Fredholm equation of second kind with a symmetric kernel.At present, the dual equations investigated in most detail are those connected with the Fourier and Hankel integral transforms. The results obtained and their applications are given in [1–3]. A large number of papers also deal with the theory and applications of the dual integral equations connected with the Mehler-Fock integral transform and its generalizations [4–11]., The dual integral transforms considered in the present paper belong to a more complex class than those listed above, and so far, no effective solution has been obtained for them. The only relevant results known to the authors are those in [12, 13]. In [12] a method of solving the equations (1.2) is given for a single particular value of the parameter γ = π/2, while in [13] the dual equations of the type under consideration are reduced to a solution of an infinite system of linear algebraic equations. 相似文献
18.
A class of integrable differential–difference systems is constructed based on auxiliary linear equations defined on sequences of Zakharov–Shabat formal dressing operators. We show that the auxiliary equations are compatible with the evolution equations for the Kadomtsev–Petviashvili (KP) hierarchy. The investigation results are used to elaborate a modified version of Krichever rational reductions for KP hierarchies. 相似文献
19.
We study multiple bifurcations in a system of reaction–diffusion equations defined on a unit square with Robin boundary conditions. First we investigate linear stabilities of the system at the uniform steady state solution. Then we discuss how multiple bifurcations can be generated by mode interactions of the system, and how these multiple bifurcations can be preserved in the associated discrete system. A continuation-unsymmetric Lanczos algorithm is described to trace discrete solution curves. Numerical experiments on the Brusselator equations are reported. 相似文献
20.
We prove a convergence theorem for a family of value functions associated with
stochastic control problems whose cost functions are defined by backward stochastic
differential equations. The limit function is characterized as a viscosity solution
to a fully nonlinear partial differential equation of second order. The key
assumption we use in our approach is shown to be a necessary and sufficient assumption
for the homogenizability of the control problem. The results generalize partially
homogenization problems for Hamilton–Jacobi–Bellman equations treated recently by
Alvarez and Bardi by viscosity solution methods. In contrast to their approach, we
use mainly probabilistic arguments, and discuss a stochastic control interpretation
for the limit equation. 相似文献