A systematic and unified approach to transformations and symmetries of general second order linear parabolic partial differential equations is presented. Equivalence group is used to derive the Appell type transformations, specifically Mehlers kernel
in any dimension. The complete symmetry group classification is re-performed. A new criterion which is necessary and sufficient for reduction to the standard heat equation by point transformations is established. A similar criterion is also valid for the equations to have a four- or six-dimensional symmetry group (nontrivial symmetry groups). In this situation, the basis elements are listed in terms of coefficients. A number of illustrative examples are given. In particular, some applications from the recent literature are re-examined in our new approach. Applications include a comparative discussion of heat kernels based on group-invariant solutions and the idea of connecting Lie symmetries and classical integral transforms introduced by Craddock and his coworkers. Multidimensional parabolic PDEs of heat and Schrodinger type are also considered.
Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but he
re they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $hbar to0$ limit.
We consider a class of generalized Kuznetsov--Zabolotskaya--Khokhlov (gKZK) equations and determine its equivalence group, which is then used to give a complete symmetry classification of this class. The infinite-dimensional symmetry is used to reduc
e such equations to (1+1)-dimensional PDEs. Special attention is paid to group-theoretical properties of a class of generalized dispersionless KP (gdKP) or Zabolotskaya--Khokhlov equations as a subclass of gKZK equations. The conditions are determined under which a gdKP equation is invariant under a Lie algebra containing the Virasoro algebra as a subalgebra. This occurs if and only if this equation is completely integrable. A similar connection is shown to hold for generalized KP equations.
We give a Lie-algebraic classification of third order quasilinear equations which admit non-trivial Lie point symmetries.
The conditions for a generalized Burgers equation which a priori involves nine arbitrary functions of one, or two variables to allow an infinite dimensional symmetry algebra are determined. Though this algebra can involve up to two arbitrary function
s of time, it does not allow a Virasoro algebra. This result confirms that variable coefficient generalizations of a non-integrable equation should be expected to remain as such.
