An infinite lie group of symmetry of one-dimensional gas flow, for a class of entropy distributions

Abstract

We have shown in earlier works the existence of three previously unknown symmetries of the equations of one-dimensional gas dynamics, with arbitrary entropy distribution and arbitrary polytropic index γ. These symmetries are seen here to form a group whenever the equation of state is of the form P = ϱ 3( a 0 + a 1 M + a 2 M 2) −2 where M = ∝ ϱd r is the Lagrangian mass coordinate. Introducing the remaining symmetry of space-translation enlarges the group into a Lie group of symmetry of infinite order, from which an infinite number of conservation laws can be deduced by application of Noether's theorem. The Lie group has a finite sub-algebra of order eight, which has SU3 structure; the list of associated conservation laws includes each of the six ones that are derivable from general physical principles, namely: the energy, momentum and the center-of-mass integrals, two integrals expressing scale invariance, and one associated with the virial theorem; the remaining two integrals of the octet are of a new type. Such a situation reminds us of the case of the Korteweg-de Vries equation in the soliton problem, where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse-scattering method. Thus the question is raised of whether the inverse-scattering method also applies to gas-dynamical equations (with the above equation of state), or else whether another method of linearization may be found.