Symmetries and conservation laws of Kadomtsev-Pogutse equations

Abstract

Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials 'in action'. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical interpretation are demonstrated.

Generalized self-similarity of intermittent plasma turbulence in space and laboratory plasmas

“…At the same time, all these governing equations remain formally invariant (as with the NSE) under any affine contraction of the space-time x → x/λ, t → t/λ 1−h , (of scale ratios λ), see discussion in Cartes et al (2009). There are no limitations to suggest that hidden symmetry, similar to NSE, can regulate plasma dynamics: symmetry transforms for MHD equations and other equations used for a description of plasma dynamics are considered in Gridnev (1968), Nucci (1984), Samokhin (1985), Gusyatnikova et al (1989), Bogoyavlenskij (2002). Thus, observation of the ESS property for plasma data is an argument to examine the scalings proposed by the log-Poisson models in the same way as in hydrodynamics.…”

Symmetries and conservation laws of partial differential equations: Basic notions and results

“…Finally, these reduced systems are to be solved. It is clear that all this can be carried out in detail only with aid of the appropriate computer machinery (see [7] for an illustration).…”

“…Also, some general views and perspectives will be discussed. The material of this paper is used in other papers [7]- [12] of this volume in which the general theory is applied to concrete equations of mathematical physics. This demonstrates the theory in action.…”

Generalized Self‐Similarity of Edge Plasma Turbulence in Fusion Devices

“…When the Lie transformations are found, they can be used to build particular solutions of the system under consideration, to reduce the order and to obtain invariants. Considering a symmetry classification problem, the symmetry properties of the MHD equations [19], [20], [21] have general peculiarities equivalent to the symmetries of the Navier-Stokes equations, namely, scale invariance under any contraction of the time-space x → x/λ, t → t/λ 1−h and velocity υ → λ −h υ , of scale ratios l, and other symmetries, see, e.g., [20], [21], [22]. Such similarity is originated from the fact that generalized equations of motion for the Weber-Clebsh potentials (in the Eulerian-Lagrangian formulation) describes both the Navier-Stockes and MHD dynamics [23].…”