Solutions of Yang-Mills and Yang-Mills-Higgs equations

V. A. Steklov Mathematics Institute, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 3, pp. 402–412, December, 1991.     

Practical scheme of reduction to gauge-invariant variables

For systems with first-class constraints, the reduction scheme to gauge-invariant variables is considered. The method is based on an analysis of restricted1-forms in gauge-invariant variables. This scheme is applied to the models of electrodynamics and Yang-Mills theory. For the finite-dimensional model with the SU(2) gauge group of symmetry, a possible mechanism of confinement is obtained.Republished from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 90–106, October, 1996.     

Dirichlet problem for Hermitian Yang-Mills-Higgs equations over Hermitian manifolds

In this paper, we investigate the Dirichlet problem for one type of vortex equations, which generalize the well-known Hermitian Yang-Mills equations, over general Hermitian manifolds.     

Darboux transformation and solitons of Yang-Mills-Higgs equations in R2,1

The Darboux transformations for soliton equations are applied to the Yang-Mills-Higgs equations.New solutions can be obtained from a known one via universal and purely algebraic formulas. SU(N) soliton solutions are constructed with explicit formulas. The interaction of solitons is described by the splitting theorem:each p-soliton is splitting into p single solitons asymptotically as t →±∞.     

Darboux transformation and solitons of Yang-Mills-Higgs equations in R2,1

The Darboux transformations for soliton equations are applied to the Yang-Mills-Higgs equations. New solutions can be obtained from a known one via universal and purely algebraic formulas. SU(N) soliton solutions are constructed with explicit formulas. The interaction of solitons is described by the splitting theorem: each p-soliton is splitting into p single solitons asymptotically as t → ±∞. The main contents of this work were reported at the Moshé Flato memorial conference (Dijon, September 2000) and W. L. Chow and K. T. Chen memorial conference (Tianjin, October 2000).     

Diophantine equations in separated variables

We study Diophantine equations of type \(f(x)=g(y)\), where both f and g have at least two distinct critical points (roots of the derivative) and equal critical values at at most two distinct critical points. Various classical families of polynomials \((f_n)_n\) are such that \(f_n\) satisfies these assumptions for all n. Our results cover and generalize several results in the literature on the finiteness of integral solutions to such equations. In doing so, we analyse the properties of the monodromy groups of such polynomials. We show that if f has coefficients in a field K of characteristic zero, and at least two distinct critical points and all distinct critical values, then the monodromy group of f is a doubly transitive permutation group. In particular, f cannot be represented as a composition of lower degree polynomials. Several authors have studied monodromy groups of polynomials with some similar properties. We further show that if f has at least two distinct critical points and equal critical values at at most two of them, and if \(f(x)=g(h(x))\) with \(g, h\in K[x]\) and \(\deg g>1\), then either \(\deg h\le 2\), or f is of special type. In the latter case, in particular, f has no three simple critical points, nor five distinct critical points.     

The Coupled Yang-Mills-Higgs flow

In this paper, we discuss some properties of the Coupled Yang-Mills-Higgs flow, including the energy inequality, Bochner-type inequality and monotonicity of certain quantities; we also discuss the asymptotic behavior of a regular Coupled Yang-Mills-Higgs flow.     

Yang-Mills-Higgs soliton dynamics in 2+1 dimensions

Dimensional reduction of the self-dual Yang-Mills equation in 2+2 dimensions produces an integrable Yang-Mills-Higgs-Bogomolnyi equation in 2+1 dimensions. For theSU(1,1) gauge group, a t'Hooft-like ansatz is used to construct a monopole-like solution and an N-soliton-type solution, which describes both the static deformed monopoles and the exotic monopole dynamics including a transmutation. How the monopole solution results from the twistor formalism is shown. Multimonopole solutions are commented on. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 339–350, December, 1998.     

On generalized quadratic equations in three prime variables

Leta ₁,a ₂ anda ₃ be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp _j. We obtain CardE(X)≤X ^1-ε. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx _j with the above bound for CardE(X) better than that in [2].     

On the suppression of variables in Boolean equations

The resultant of suppression of variables from a Boolean equation is a Boolean equation, derived from the parent equation, whose solutions are exactly those of the parent equation that do not involve the suppressed variables. Two examples in the literature are discussed, in which it is necessary to solve a Boolean equation while excluding solutions involving certain variables. In such cases it would be advantageous to solve the resultant of suppression of those variables rather than solving the original equation and filtering the desired solutions from the results.     

Bi-Hamiltonian ordinary differential equations with matrix variables

We consider a special class of Poisson brackets related to systems of ordinary differential equations with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets, and find the corresponding hierarchy of integrable models, which generalizes the two-component Manakov matrix system to the case of an arbitrary number of matrices.     

Matrix differential equations with separation of variables

We consider matrix differential equations with separation of variables and homogeneous equations reducible to them. For the mentioned equations we obtain sufficient conditions for the solvability in quadratures.     

Separation of variables in two-dimensional wave equations with potential

In the present paper, we solve the problem of separation of variables in the wave equationu _tt –u _xx +V(x)u=0 and give a complete classification of potentialsV(x) forwhich equations of this type admit nontrivial separation of variables. Furthermore, we construct all coordinate systems in which this separation is possible.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1343–1361, October, 1994.     

Meixner polynomials in several variables satisfying bispectral difference equations

We construct a set _Md${\mathfrak{M}}_d$ whose points parametrize families of Meixner polynomials in d   variables. There is a natural bispectral involution b$\mathfrak{b}$ on _Md${\mathfrak{M}}_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of d   commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution b$\mathfrak{b}$.