We consider the nonlinear diffusive Burgers' equation as a model equation for signals propagation on the nonlinear electrical transmission line with intersite nonlinearities. By applying the extend sine-cosine method and using an appropriate modification of the Double-Exp function method, we successfully derived on one hand the exact analytical solutions of two types of solitary waves with strictly finite extension or compact support: kinks and pulses, and on the other hand the exact solution for two interacting pulse solitary waves with compact support. These analytical results indicate that the speed of the pulse compactons doesn't depends explicitly on the pulse amplitude as has been expected for long, but rather on the dc-component associated to this trigonometric solution. More interesting, the interactions between the two pulse compactons induce only a phase shift even though they are close together. These analytical solutions are checked by means of numerical simulations. (c) 2018 Elsevier B.V. All rights reserved.