The existence of contact transformations for evolution-type equations

We investigate the existence of a one-parameter group of contact transformations for evolution-type equations u_t = F(t, x, u, u_x, u_xx, . . . , u_(n)) (subscripts denote differentiation unless otherwise indicated), where u_(n) is the nth derivative of u with respect to x. We prove that contact transformations of evolution equations, where F is expandable as a power series in terms of all derivatives of order higher than one, are just extended Lie point transformations. This result is extended to the case with m independent space variables. As a consequence, we obtain an ansatz for determining Lie point transformations for nth-order evolution equations with m independent space variables. Examples are given to verify the results obtained as well as to show how Lie point transformations of these evolution-type partial differential equations can be calculated from this ansatz.