Related papers: Superposition rules, Lie theorem and partial diffe…
We consider in the plane the problem of reconstructing a domain from the normal derivative of its Green's function with pole at a fixed point in the domain. By means of the theory of conformal mappings, we obtain existence, uniqueness,…
An analogue of Serre's theorem is established for finite dimensional simple Lie superalgebras, which describes presentations in terms of Chevalley generators and Serre type relations relative to all possible choices of Borel subalgebras.…
We initiate the representation theory of restricted Lie superalgebras over an algebraically closed field of characteristic p>2. A superalgebra generalization of the celebrated Kac-Weisfeiler Conjecture is formulated, which exhibits a…
We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrodinger (NLS) equation,…
This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear…
The need for modification of the Huygens-Fresnel superposition principle arises even in the description of the free fields of massive particles and, more extensively, in nonlinear field theories. A wide range of formulations and…
The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by…
The study of the relation between Lie algebras and groups, and especially the derivation of new algebras from them, is a problem of great interest in mathematics and physics, because finding a new Lie group from an already known one also…
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
After reviewing the basic role of Lie's theory for the mathematics and physics of this century, we identify its limitations for the treatment of systems beyond the local-differential, Hamiltonian and canonical-unitary conditions of the…
We establish the existence of multiple solutions for a nonlinear problem of critical type. The problem considered is fractional in nature, since it is obtained by the superposition of $(s,p)$-fractional Laplacians of different orders. The…
We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof…
Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of…
Lie symmetries for ordinary differential equations are studied. In systems of ordinary differential equations, there do not always exist non-trivial Lie symmetries around equilibrium points. We present a necessary condition for existence of…
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular,…
In this note, we formulate and prove branching rules of simple polynomial modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. Our branching rules depend on the conjugacy class of the Borel subalgebra. A Gelfand-Tsetlin basis of a…