Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

Abstract

Within the framework of inverse Lie problem, we give some nontrivial examples of coupled Lie remarkable equations, i.e., classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of R n+m , and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of R 4 .

Thermodynamical analysis and constitutive equations for a mixture of viscous Korteweg fluids

“…Condition (44) has three independent components while liquid-vapor phase equilibria are determined by just one physical variable, namely the mass density, i.e., the equilibrium system ( 44) is overdetermined. A remarkable result obtained in 1983 by Serrin [40] proved that, unless rather special conditions are satisfied, the only geometric phase boundaries which are consistent with relation (44) are either spherical, cylindrical, or planar. Using a general theorem proved in Ref.…”

“…in order to avoid [41] that the solution of (44) possesses solutions described only by level surfaces with constant mean and Gaussian curvature [42,43,44], which are either (pieces of) concentric spheres, or concentric circular cylinders, or parallel planes. From the physical viewpoint, this result reflects the experimental evidence that several (but not all) phase boundaries have constant mean curvature.…”

ReLie: A Reduce Program for Lie Group Analysis of Differential Equations

“…In [119], within the framework of inverse Lie problem, strongly and weakly Lie remarkable differential equations have been defined; relevant examples of such equations have been studied in [120][121][122]. Their analysis involves the study of the rank of the distribution of prolongations of a Lie algebra of generators.…”

“…By using the program, it is possible to compute almost automatically: Lie point symmetries, conditional symmetries, contact symmetries, variational symmetries (all these symmetries may be either exact or approximate) of differential equations, and equivalence transformations for classes of differential equations containing arbitrary elements. Moreover, the program implements functions for computing Lie brackets, the commutator table of a list of Lie generators, and the distribution of an algebra of Lie symmetries (useful in the context of inverse Lie problem [119][120][121][122]). Remarkably, the program can be used interactively in all the cases where the determining equations are not automatically solved (for instance, when one looks for conditional symmetries or in some group classification problems).…”

“…Here we show with a simple example how ReLie can be used for investigating the Lie remarkability of differential equations [119][120][121][122]. The following code shows how one can easily verify that Monge-Ampère equation describing a surface in R 3 with zero Gaussian curvature,…”