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

الأصل · EN

We determine by two related methods the invariance algebra of the `pseudo-diffusion equation' (PSDE) L Q ≡ [∂/∂ t - 1 4 (∂²/∂ x² - 1 t² ∂²/∂ p²)] 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=e²ʸ. 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 uₜ-uxx+ b(t) uyy=0 of PSDE has a smaller symmetry algebra than, except for b(t) equals to a constant or it is proportional to t⁻². We apply the group elements Gᵢ():= [Aᵢ] 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 `2-sided kernel', because it has to depend on two times, t₀ < t < t₁. We include a detailed discussion of the identification of the Lie algebraic structure of the symmetry algebra, and its contraction from (1,1)⊕(3,1).