Differential invariants of the one-dimensional quasi-linear second-order evolution equation

We consider evolution equations of the form ut = f(x, u, ux)uxx + g(x, u, ux) and ut = uxx + g(x, u, ux). In the spirit of the recent work of Ibragimov [Ibragimov NH. Laplace type invariants for parabolic equations. Nonlinear Dynam 2002;28:125–33] who adopted the infinitesimal method for calculating invariants of families of differential equations using the equivalence groups, we apply the method to these equations. We show that the first class admits one differential invariant of order two, while the second class admits three functional independent differential invariants of order three. We use these invariants to determine equations that can be transformed into the linear diffusion equation.