It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions. 0. Idea and Motivation The standard Lie group theory applied to symmetries of solutions of PDEs, Olver [1-3], deals with classical, and specifically, C-smooth solutions of such e...

All known today integrable scalar (1+1)-dimensional evolution equations with time-independent coefficients possess infinite-dimensional Abelian algebras of time-independent higher order symmetries (see e.g. [1, 2]). However, the equations of this kind usually do not have local timedependent higher order symmetries. The only known exceptions from this rule seem to occur [3] for linearizable equa...

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.

We examine some kinds of discrete symmetries which are dynamically preserved, using the (generalized) Gowdy models of the first kind.

Symmetries abound in nature. Observing symmetries often provides the key to discovering internal structure. In problem solving, observing and reasoning about symmetries is a powerful tool for shifting viewpoints on a problem. A calculus for reasoning about problem symmetries has been developed, called Invariant Logic. Invariant Logic is partially implemented in STRATA, a system which synthesize...

We revisit universal features of duality in linear and nonlinear relativistic scalar Abelian 1-form theories with single or multiple fields, which exhibit ordinary generalized global symmetries. show that such symmetries can be interpreted as Killing isometries on a suitable, possibly graded, target space fields its jet when the theory contains higher derivatives. This is realized via sigma mod...

We demonstrate, using the symbolic method together with p-adic and resultant methods, the existence of systems with exactly one or two generalized symmetries. Since the existence of one or two symmetries is often taken as a sure sign (or as the definition) of integrability, that is, the existence of symmetries on infinitely many orders, this shows that such practice is devoid of any mathematica...

A new efficient method for finding the generalized equivalence transformations related to a differential equation system via its extended classical symmetries is presented. This technique applies to the case when the arbitrary functions involve only the independent variables of the system. Intriguingly, for this type of equations, any symbolic manipulation program designed to find classical Lie...

Symmetries of generalized gravitational actions, yielding field equations which typically involve at most second-order derivatives of the metric, are considered. The field equations for several different higher-derivative theories in the first-order formalism are derived, and variations of a generic set of higher-order curvature terms appearing in string effective actions are studied. It is sho...