A Note on Schrödinger Operator Relations and Power-Law Energies

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-momentum relation to determine new families of nonlinear partial differential relations. The Planck–de Broglie duality principle arises from Planck’s energy expression e=hν, de Broglie’s equation for momentum p=h/λ, and Einstein’s special relativity energy, where h is the Planck constant, ν and λ are the frequency and wavelength, respectively, of an associated wave having a wave speed w=νλ. The author has extended these relations to a family that is characterised by a second fundamental constant h′ and underpinned by Lorentz invariant power-law particle energy-momentum expressions. In this note, we apply generalized Schrödinger operator relations and the power-law relations to generate a new family of nonlinear partial differential equations that are characterised by the constant κ=h′/h such that κ=0 corresponds to the Klein–Gordon equation. The resulting partial differential equation is unusual in the sense that it admits a stretching symmetry giving rise to both similarity solutions and simple harmonic travelling waves. Three simple solutions of the partial differential equation are examined including a separable solution, a travelling wave solution, and a similarity solution. A special case of the similarity solution admits zeroth-order Bessel functions as solutions while generally, it reduces to solving a nonlinear first-order ordinary differential equation.