Let G = SpecA be an affine R-monoid scheme. We prove that the category of dual functors (over the category of commutative R-algebras) of G-modules is equivalent to the category of dual functors of A∗-modules. We prove that G is invariant exact if and only if A∗ = R × B∗ as R-algebras and the first projection A∗ → R is the unit of A. If M is a dual functor of G-modules and wG := (1, 0) ∈ R×B ∗ =...

Dual integral equations with trigonometric kernel are reinvestigated here for a solution. The behaviour of one of the integrals at the end points of the interval complementary to the one in which it is defined plays the key role in determining the solution of the dual integral equations. The solution of the dual integral equations is then applied to find an exact solution of the water wave scat...

We construct the most general reducible connection that satisfies the self-dual Yang–Mills equations on a simply-connected, open subset of flat R. We show how all such connections lie in the orbit of the flat connection on R under the action of non-local symmetries of the self-dual Yang–Mills equations. Such connections fit naturally inside a larger class of solutions to the self-dual Yang–Mill...

It is known that many integrable systems can be reduced from self-dual Yang-Mills equations. The formal solution space to the self-dual Yang-Mills equations is given by the so called ADHM construction, in which the solution space are graded by vector spaces with dimensionality concerning topological index. When we consider a reduced self-dual system such as the Bogomol’nyi equations, in terms o...

A concept of strong necessary conditions for extremum of functional has been applied for analysis an existence of dual equations for a system of two nonlinear Partial Differential Equations (PDE) in 1+1 dimensions. We consider two types of the dual equations: the Bäcklund transformations and the Bogomolny equations. A general form of the second order PDE with a derivative-less non-linear term h...

Recently Kovner and Lublinsky proposed a set of equations which can be viewed as dual to JIMWLK evolution. We show that these dual equations have a natural dipole-like structure, as conjectured by Kovner and Lublin-sky. In the high energy large N c limit these evolution equations reduce to equations previously derived in the dipole model. We also show that the dual evolution kernel is scheme de...