Functional self-similarity in a problem of plasma theory with electron nonlinearity

Functional self-similarity and renormalization group symmetry in mathematical physics

“…Eventually, when we interpret the equation parameters as independent variables, two more operators appear and involve these parameters in the group transformations; these operators correspond to scaling transformations of the respective variables a and ν [19]. Restricting the group admitted by manifold (22) on the solution u = U(t, x, a, ν) of the Cauchy problem, i.e., verifying the FS condition, results in an algebraic relation, which expresses the coordinate of the infinite-dimensional subgroup generator (the function α) through the coordinates of the remaining eight operators at any time t, including t = 0 when this solution U(0, x, a, ν) = f (x) is known from boundary condition (23). Using the standard representation for the solution of a linear equation on the function α with the initial value α(0, x, a, ν) obtained from the FS condition and substituting this representation in the formula that determines the general element of the Lie algebra, we obtain the desired RGS operators.…”

Group analysis and renormgroup symmetries

“…The first application of RG-approach to a particular problem of laser plasma was announced in [15]. This problem, namely the problem of a nonlinear interaction of a powerful laser radiation with inhomogeneous plasma, has been detailed in subsequent publications [16,33,34]. A mathematical model was given by a system of nonlinear DEs for components of electron velocity, electron density and the electric and magnetic fields.…”

“…The desired RG-symmetry appears as Lie point symmetry that takes account of transformations of a boundary parameter (common to (Ia) approach), which is related to the amplitude of the magnetic field at a critical density point. RG-symmetry obtained made it possible to get the exact solution of original equations, that was then used to evaluate the efficiency of harmonics generation in cold and hot plasma (see [16]). …”

“…The group obtained was then utilized to construct the desired nonlinear solution of the BVP. This approach has further been developed (see [16] and references therein) and we shall describe it in some detail in the next Section.…”

The Bogolyubov renormalization group

“…Recently, interesting attempts have been made [45,46] to use the RG concept in classical mathematical physics, in particular, to solve nonlinear differential equations. These articles discuss the possibility of establishing a regular method for finding a special class of symmetries of the equations in modern mathematical physics, namely, RG-type symmetries.…”