and Applied Analysis 3 which as is well known has the seven symmetries (Lie [1] and e.g., [4])

It is well known that x-translation and t-translation invariance of (1) leads to the following symmetries: ux, ut of the KdV equation (1). In order to find more generalized symmetries, the concepts of recursion operators or strong symmetries, and hereditary symmetries were introduced by Olver and Fuchssteiner and used to find these symmetries [1, 2]. Furthermore, Galilean invariance of the KdV ...

New analytical results for two-dimensional force-free fields are presented. First, a number of exact solutions for force-free fields described by Liouville equation are given and their physical relevance for laboratory and astrophysical plasmas is discussed. Subsequently, Lie point symmetries of Liouville equation are reviewed and Lie point symmetries of the equation describing helically symmet...

Asymptotic reductions of the Fanno model for one-dimensional turbulent compressible flow of a perfect gas in a long tube are investigated. Conservation laws are derived using the multiplier method for a nonlinear wave equation and a nonlinear diffusion equation for the mean velocity and a nonlinear diffusion equation for the mean pressure. Two conserved quantities for the mean velocity are obta...

I describe how integrable quantum field theories in 2 spacetime dimensions are characterized by infinite dimensional quantum group symmetries, namely the q-deformations of affine Lie algebras, and their Yangian limit. These symmetries can provide a new non-perturbative formulation of the theories.

Secondly combining group theory and singularity analysis we obtained some information about the similarity solutions of the fourth order generalized Boussinesq equation GBE4. It has been shown that this equation possess a three dimensional Lie algebra of the point symmetries and the similarity reductions due to these symmetries have been obtained.

Lie point symmetries of time-fractional potential Burgers' equation are presented. Using these symmetries fractional potential Burgers' equation has been transformed into an ordinary differential equation of fractional order corresponding to the Erdélyi-Kober fractional derivative. Further, an analytic solution is furnished by means of the invariant subspace method. AMS subject classifications:...