In this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u‖∇u)). A special class of solutions is discussed for which (∇u‖∇u) is constant on each level of the function u. It is proved that levels of such solutions form in the space of independent variable’s hypersurfaces with all principal curvatures constant. The general form of such hypersurfaces is given. Then it is proved that via the method of characteristics it is possible to construct (in principle) all the solutions of the discussed class. They may be obtained by integration of an ODE of second order using a special class of the polynomial functions. Some new solutions are given for equations⧠v=4Av3+3Bv2+2cv+D, ⧠v=μ exp v, ⧠v=sin v, ⧠v=cosh v, and ⧠v=sinh v.

Let a given particle symmetry be described by a reductive Lie group G. It is proved that the corresponding Weyl group W(R) acts canonically in all zero-weight spaces of G and hence, in particular, on observables. Moreover, it is shown how this W(R) action provides many physical relations, including those believed to be implied by G-transformation properties of observables. The results simplify a testing of symmetries based on various Lie groups (algebras). Their use extends beyond particle physics, e.g., to nuclear physics.

Let a particle symmetry be described by a simple Lie algrebra 𝔤 of the type An−1, i.e., 𝔤=sl(n,C) or 𝔤 is a real form of sl(n,C). For 𝔤 representations describing three-particle or particle–antiparticle states, relationships between two actions, the action of 𝔤 and the action of the corresponding Weyl group Sn, on observables are analyzed. It is shown, in particular, how impossible physical relations depend on these two actions. The results enable one to verify quickly if given experimental data can be fitted by means of a 𝔤-symmetry theory.

It is shown how outer automorphisms of a semisimple Lie algebra and automorphisms of the corresponding root system describe conjugations of general charges. New results about representations of semidirect products of non-Abelian Lie groups and about projective representations of finite groups are derived.

Let E be n-dimensional (n≧2) real vector space with a nondegenerate symmetric scalar product (.|.):E × E → R1 with an arbitrary signature (p, n–p). Let us consider a second order partial differential equation (P.D.E.) of the form:where φ is a given function of two variables, v is an unknown function (defined on an open subset 0 ⊂E), |∇ν|2: =(∇ν|∇ν) is the square of the gradient ∇ν of the function ν and ∇2, denotes the Laplace-Beltrami operator.