Symmetries of the pseudo-diffusion equation, and its unconventional 2-sided kernel

Abstract

We determine by two related methods the invariance algebra $\g$ of the \emph{`pseudo-diffusion equation'} (PSDE) $$LQ\equiv \left[\frac{\partial }{\partial t}-\frac{1}{4}\left(\frac{{\partial }^2}{\partial x^2}-\frac{1}{t^2}\frac{{\partial }^2}{\partial p^2}\right)\right]Q(x,p,t)=0,$$ which describes the behavior of the $Q$ functions in the $(x,p)$-phase space as a function of a squeeze parameter $y$, where $t=e^{2y}$. The algebra turns out to be isomorphic to that of its constant coefficient version. Relying on this isomorphism we construct a local point transformation which maps the factor $t^{-2}$ to 1. We show that any generalized version $u_t-u_{xx}+b(t)u_{yy}=0$ of PSDE has a smaller symmetry algebra than $\g$, except for $b(t)$ equals to a constant or it is proportional to $t^{-2}$. We apply the group elements $G_i(\ga) := \exp[\ga A_i]$ and obtain new solutions of the PSDE from simple ones, and interpret the physics of some of the results. We make use of the `factorization property' of the PSDE to construct its \textit{`2-sided kernel'}, because it has to depend on two times, $t_0<t<t_1$. We include a detailed discussion of the identification of the Lie algebraic structure of the symmetry algebra $\g$, and its contraction from $\su(1,1)\oplus\so(3,1)$.

25 pages

References in corpus (1)

• Equivalence and Symmetries for Linear Parabolic Equations and Applications Revisited