Turbulence-bearing analytical solution of Madelung equations for arbitrary domain shape, dimensionality, and Cauchy data by modified Cole-Hopf solution of Schrödinger equation

Abstract

In light of a recently discovered general solution to the Schrödinger equation in an arbitrary integer-dimensional real domain with arbitrary initial conditions, boundary conditions, and quantum mechanical potential via a modified Cole-Hopf transform, a solution of the Madelung equations of equal generality is derived in this work. After a sufficient degree of mathematical context is established by a brief recapitulation of the solution of the Schrödinger equation, the derivation of the Madelung equations from first principles proves that a general Madelung flow field is implied by a general Schrödinger wave function through illustration of the mathematical essence of the quantities in the Madelung equations. The solution follows by a decomposition of the newly found general wave function into its polar form and its substitution into the definitions of the Madelung flow velocity and density, and a transfer from wave function initial and boundary conditions to density and velocity boundary conditions. The transfer between the complex scalar-valued wave function and the real vector-valued velocity field is achieved by a conditional inversion of the gradient operator. Plots of the velocity and vorticity take mere milliseconds to generate on consumer hardware via an enclosed GNU Octave script and show highly turbulent patterns consistent with the laws of quantum mechanics, showing that this analytical Madelung solution possesses unprecedented ability to deterministically replicate quantum turbulence without any numerical methods at all, effectively democratizing the study of quantum turbulence with its internal structure and mathematical origin. This Madelung solution paves the way to equally general solutions of unsolved problems in classical fluid mechanics including the Euler and Navier-Stokes equations, whose methods and analyses including existence and smoothness are expounded in works currently in production by the author.