Existence of additional conservation laws for the Kambapalli and Magan cylindrical shear-wave models

Determine whether the cylindrical PDE systems for shear-wave propagation admit any additional local conservation laws beyond the three presented. Specifically, for the Kambapalli model given by σ_r + (2σ)/r = δ u_tt and u_r − (u/r) = (1/δ) σ (β + σ^2)^n, and for the Magan model given by σ_r + (σ)/r = δ u_tt and u_r = (1/δ) σ (β + σ^2)^n, ascertain the existence and form of any further conserved densities and associated fluxes for arbitrary exponent n and constants δ and β.

Background

The paper computes three conservation laws for each of two cylindrical-coordinate models of shear-wave propagation in elastic media: the Kambapalli model and the Magan model. Using a scaling-homogeneity approach implemented in ConservationLawsMD.m, the authors searched for densities with coefficients up to third degree in r and t and did not find any beyond the ones listed.

This lack of additional conservation laws contrasts with a related Cartesian-coordinate model where seven conservation laws were found and the system is conjectured to have infinitely many conservation laws. The explicit statement of non-discovery for the cylindrical models suggests an unresolved question about whether more conservation laws exist and, if so, how to characterize them.