Analytic Solutions of the KPZ Equation Containing the Kummer’s Functions

It is well-known that with the Hopf-Cole (HC) transformation the Kardar-Parisi-Zhang (KPZ) non-linear surface growth equation can be transformed to the regular diffusion equation. The reverse operation is also possible. Along this logic we can transform our new-type of self-similar solutions of the regular diffusion equations - which contain the Kummer’s functions - to solutions which fulfill the KPZ equation. These solutions give us a glimpse into the true origin of the singularities of the KPZ equation. Due to the free self-similar exponent we discuss if our solutions could explain surface erosion additionally to surface growth. We investigate the KPZ equation without any noise term. At the second part of the manuscript we investigate an analytic source term in the diffusion equation and its transformed form in the KPZ equation.