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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…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
A general field theory for classical particle-field systems is developed. Compared with the standard classical field theory, the distinguish feature of a classical particle-field system is that the particles and fields reside on different…
We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Lambda symmetry under some Lie point vector field. After a brief…
Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamilton's variational principle. In particular, the discussion focuses on the Euler-Poincar\'e formulation of three major hybrid MHD models, which…
The concept of symmetries in physics is briefly reviewed. In the first part of these lecture notes, some of the basic mathematical tools needed for the understanding of symmetries in nature are presented, namely group theory, Lie groups and…
A family of nonlinear partial differential equations of divergence form is considered. Each one is the Euler-Lagrange equation of a natural Riemaniann variational problem of geometric interest. New uniqueness results for the entire…
We consider the Dirichlet problem for the nonlinear $p(x)$-Laplacian equation. For axially symmetric domains we prove that, under suitable assumptions, there exist Mountain-pass solutions which exhibit partial symmetry. Furthermore, we show…
We apply Lie symmetry analysis of partial differential equations (PDEs) to the Euler-Lagrange equations of the two-Higgs-doublet model (2HDM), to determine its scalar Lie point symmetries. A Lie point symmetry is a structure-preserving…
Hydrodynamic equations for a one-component plasma are derived as a generalization of the Euler equations to include the effects of the long-range Coulomb interaction. By using a variational principle, these equations self-consistently unify…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
We analyse the underlying nonlinear partial differential equation which arises in the study of gravitating flat fluid plates of embedding class one. Our interest in this equation lies in discussing new solutions that can be found by means…
Symmetry properties of conservation laws of partial differential equations are developed by using the general method of conservation law multipliers. As main results, simple conditions are given for characterizing when a conservation law…
The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived…
Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related…
The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to 10 dimensional. The…
With using the algebraic approach Lie symmetries of Schr\"odinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, the admissible…
Plasma supports collective modes and particle-wave interactions that leads to complex behavior in inertial fusion energy applications. While plasma can sometimes be modeled as a charged fluid, a kinetic description is useful towards the…
This paper recalls some classical motivations in fluid dynamics leading to a partial differential equation which is prescribed on a domain whose boundary possesses two connected components, one endowed with a Dirichlet datum, and the other…
We study the separability of permutationally symmetric quantum states. We show that for bipartite symmetric systems most of the relevant entanglement criteria coincide. However, we provide a method to generate examples of bound entangled…