Invariant Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

Algorithmic Construction of Hyperfunction Solutions to Invariant Differential Equations on the Space of Real Symmetric Matrices

This is the second paper on invariant hyperfunction solutions of invariant linear differential equations on the vector space of n × n real symmetric matrices. In the preceding paper [22], we proved that every invariant hyperfunction solution is expressed as a linear combination of Laurent expansion coefficients of the complex power of the determinant function with respect to the parameter. Fund...

Symmetry Groups and Group Invariant Solutions of Partial Differential Equations

Dynamics of Differential Equations on Invariant Manifolds

The simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice. Appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics. This paper develops an approach which avoids the selection of coordinate systems on the man...

Symmetries and Group-invariant Solutions of Nonlinear Fractional Differential Equations

In the paper, methods of Lie group analysis are applied to investigate symmetry properties of some classes of nonlinear fractional differential equations. For class of equations Dy = f(x, y), 0 < α < 1, the problem of group classification is solved. Symmetry properties of equations D t u = (k(u)ux)x, 0 < α ≤ 2 for different orders of α are compared. Obtained symmetries are used to construct exa...

Entire Invariant Solutions to Monge-ampère Equations

We prove existence and regularity of entire solutions to MongeAmpère equations invariant under an irreducible action of a compact Lie group. We consider Monge-Ampère equations of the form f(∇φ) detDijφ = g(x) (0.1) where f and g are nonnegative measurable functions on R. We recall first the concept of a weak solution of (0.1). Let φ be a convex function. Then ∇φ is a well-defined multi-valued m...