Quasi-invariance (rescaling) in nonlinear physics

Consider the following phase space-time transformation q/sup i/ = xi/sup i/C(t), p/sub i/ = π/sub i//C(t), THETA = THETA(t) where q/sup i/, p/sub i/, t and xi/sup i/, π/sub i/, THETA are the coordinate, momentum and time variables in the old and new space respectively, and where C(t) and THETA(t) are arbitrary functions of time. These transformations are shown to be generalized canonical transformations (GCT). The new variables reduce to the usual invariants of similarity solutions when such a transformation is found that leaves the governing equations strictly invariant. In most cases, for equations not completely integrable, no more than one group may usually be found. The invariants of the group are then used to absorb one independent variable, decreasing by one their number, but at the price of specializing initial conditions. For second order ordinary differential equations (including systems with friction), the method can be used to find strict or asymptotic invariants. We give two examples. The first deals with a one species plasma in cylindrical geometry embedded in a time-varying magnetic field for which the Brillouin flow is shown to be a general attractor. The second deals with the nonlinear heat-diffusion equation for which we compare the asymptotic solution to those obtained by self-similarity arguments