Group and renormgroup symmetry of a simple model for nonlinear phenomena in optics, gas dynamics, and plasma theory

Kovalev, V. F.; Pustovalov, V. V.

doi:10.1016/s0895-7177(97)00067-8

Cited by 14 publications

(22 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will consider here the perturbed system (8.3) and discuss the perturbation of the exact symmetry X 2 from the list (7.3). The results obtained in [16] allow to find the following perturbation of the operator X 2 in the form…”

Section: Hodograph Transformation Of Symmetriesmentioning

confidence: 99%

Invariant solutions and shock atmospheric waves in a thin circular layer

Ibragimov

Kovalev

2018

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

The objective of this paper is to investigate the nonlinear mathematical model describing equatorial waves from Lie group analysis point of view in order to understand the nature of shallow water model theory, which is associated to planetary equatorial waves. Such waves correspond to the Cauchy–Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a solid circle of a large radius.

show abstract

Section: Hodograph Transformation Of Symmetriesmentioning

confidence: 99%

Invariant solutions and shock atmospheric waves in a thin circular layer

Ibragimov

Kovalev

2018

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

show abstract

“…The formula for the recursion operator L 1 is valid for arbitrary nonlinearity function ψ(I), while operators L 2 and L 3 arise for those functions ψ(I) that fulfill the condition Ref. [11]:…”

Section: Recursion Operators and Lie-bäcklund Symmetries Of The Seconmentioning

confidence: 99%

“…As it was formulated in Refs. [11,20], an invariant solution to the boundary value problem, in particular the one given by Eqs. (3.11), must be found from the constructed Lie-Bäcklund symmetries under the invariance conditions Unfortunately, the Lie point group generators (3.20) with s = 0 and s = 1 are not sufficient in order to determine a linear superposition able to satisfy a smooth localized (symmetric in χ) intensity distribution at the boundary.…”

Section: Recursion Operators and Lie-bäcklund Symmetries Of The Seconmentioning

confidence: 99%

Exact and approximate symmetries for light propagation equations with higher order nonlinearity

Garcia

Kovalev²,

Tatarinova

2010

Lobachevskii J Math

View full text Add to dashboard Cite

Abstract-For the first time exact analytical solutions to the eikonal equations in (1 + 1) dimensions with a refractive index being a saturated function of intensity are constructed. It is demonstrated that the solutions exhibit collapse; an explicit analytical expression for the self-focusing position, where the intensity tends to infinity, is found. Based on an approximated Lie symmetry group, solutions to the eikonal equations with arbitrary nonlinear refractive index are constructed. Comparison between exact and approximate solutions is presented. Approximate solutions to the nonlinear Schr¨odinger equation in (1 + 2) dimensions with arbitrary refractive index and initial intensity distribution are obtained. A particular case of refractive index consisting of Kerr refraction and multiphoton ionization is considered. It is demonstrated that the beam collapse can take place not only at the beam axis but also in an off-axis ring region around it. An analytical condition distinguishing these two cases is obtained and explicit formula for the self-focusing position is presented.

show abstract

“…where coordinates f i and g i are linear combinations of τ and χ and their first derivatives τ 1 = (∂τ /∂n) and χ 1 = (∂χ/∂n) with coefficients depending only on v and n [40,41]. For a particular case ϕ = 1 they are…”

Section: Rg As Lie Point Subgroupmentioning

confidence: 99%

“…L-B symmetries admitted by the RG-manifold (10) are characterized by the same canonical infinitesimal operator (22) where additional terms proportional to higher-order derivatives of τ and χ should be added in coordinates f and g. Similarly to first-order symmetries, these terms are linear combinations of τ and χ and their derivatives τ i = (∂ i τ /∂n i ) and χ i = (∂ i χ/∂n i ) with coefficients that depend only on v and n [39][40][41]. For the second-order Lie-Bäcklund symmetries in a particular case ϕ(n) = 1, we have five additional operators X i with i = 7, .…”

Section: Rg As Lie-b äCklund Subgroupmentioning

confidence: 99%