Abstract
The most general second order irrotational vector field evolution equation is constructed, that can be transformed to a single equation for the Cole–Hopf potential. The exact solution to the radial Burgers equation, with constant mass influx through a spherical supply surface, is constructed. The complex linear Schrödinger equation is equivalent to an integrable system of two coupled real vector equations of Burgers type. The first velocity field is the particle current divided by particle probability density. The second vector field gives a complex valued correction to the velocity that results in the correct quantum mechanical correction to the kinetic energy density of the Madelung fluid. It is proposed how to use symmetry analysis to systematically search for other constrained potential systems that generate a closed system of vector component evolution equations with constraints other than irrotationality.
Highlights
Speaking, integrable equations are related to linear equations either by a classical Darboux transformation (c-integrable) or an inverse scattering transform (s-integrable) [1]
The first is the detection of extended symmetries of order three or higher
Unlike first-order contact symmetries and their equivalent second-order “vertical” symmetries, higher-order symmetry transformations cannot be closed at some finite order [2]
Summary
Integrable equations are related to linear equations either by a classical Darboux transformation (c-integrable) or an inverse scattering transform (s-integrable) [1]. It remains challenging to apply symmetry methods to classify integrable systems of parabolic evolution equations in more than one space dimension. There is the integrable transport model, the Burgers equation ut + uux = νuxx (1). This equation resembles the momentum transport equation of incompressible Newtonian fluid but one-dimensional incompressible flow is trivial. The outcome was that on a manifold with constant non-zero Ricci curvature scalar, Burgers’ equation transforms to a reaction-diffusion equation for scalar Cole–Hopf potential ψ, with linear diffusion term but nonlinear reaction term proportional to ψ log ψ. General form of the integrable nonlinear vector equation that results from the Cole–Hopf transformation in the reverse direction ?. In higher dimensions it is more convenient to use index notation, especially when the coordinates are allowed to be non-Cartesian
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