Related papers: Contact germs and partial differential equations
We investigate the fractional Schr\"odinger equation with a periodic $\mathcal{PT}$-symmetric potential. In the inverse space, the problem transfers into a first-order nonlocal frequency-delay partial differential equation. We show that at…
An effective method for generating linear equations of maximal symmetry in their much general normal form is obtained. In the said normal form, the coefficients of the equation are differential functions of the coefficient of the term of…
An algorithm for studing the symmetrical properties of the partial differential equation of the type Lu=0 is proposed. By symmetry of this equation we mean the operators Q satisfying commutational relations of order p more than p=1 on the…
The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type…
An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrodinger-like…
The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…
We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra,…
Detailed description of nuclei necessitates model Hamiltonians which break most dynamical symmetries. Nevertheless, generalized notions of partial and quasi dynamical symmetries may still be applicable to selected subsets of states, amidst…
Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the…
We introduce a class of partial differential equations on metric graphs associated with mixed evolution: on some edges we consider diffusion processes, on other ones transport phenomena. This yields a system of equations with possibly…
The method of moments in the context of Nonlinear Schrodinger Equations relies on defining a set of integral quantities, which characterize the solution of this partial differential equation and whose evolution can be obtained from a set of…
We introduce a method of approximate nonclassical Lie-B\"acklund symmetries for partial differential equations with a small parameter and discuss applications of this method to finding of approximate solutions both integrable and…
We present several results on the inverse problem and equivalent contactLagrangian systems. These problems naturally lead to consider smooth transformations on the z variable (i.e., reparametrizations of the action). We present the extended…
The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general…
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
The nonlocal Darboux transformation of the stationary axially symmetric Schr\"odinger equation is considered. It is shown that a special case of the nonlocal Darboux transformation provides the generalization of the Moutard transformation.…
Rational solutions and special polynomials associated with the generalized K_2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations…
The dynamics of distributed sources is described by nonlinear partial differential equations. Lagrangian analytical solutions of these (and associated) equations are obtained and discussed in the context of Lagrangian modeling - from the…
In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups…
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the…