In this paper we generalize the classification of self-adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

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In this paper we consider a (2 + 1) dimensional generalized Burgers equation. After having written this equation as a system in two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law for this system.

We consider a class of second-order partial differential equations which arises in diffusion phenomena and, following a new approach, we look for a Lie invariance classification via equivalence transformations. A class of exact invariant solutions containing an arbitrary function is obtained.

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