On the Conservation Laws and Traveling Wave Solutions of a Nonlinear Evolution Equation that Accounts for Shear Strain Waves in the Growth Plate of a Long Bone

Abstract

A nonlinear partial differential equation that accounts for shear strain waves in a long bone’s development plate is investigated. Lie group classification is performed on the underlying equation and it is found that the principal Lie algebra consists of a single time translation symmetry. The underlying analysis prompts that the principal algebra is extended by a single space translation and consequentially it is found the Lie algebra is two dimensional. A linear combination of the translation symmetries results in highly fourth-order nonlinear ordinary differential. Courtesy of an ansatz method this highly fourth-order nonlinear ordinary differential results in obtaining series of hyperbolic and trigonometric traveling wave solutions. Finally, we compute conservation laws courtesy of the invariance and multiplier technique. The findings can well mimic complex waves and their dealing dynamics in a long bone’s development.