A new time dependent approach for solving electrochemical interfaces Part I: theoretical considerations using lie group analysis

In this paper we introduce a nonlinear partial differential equation (nPDE) of the third order to the first time. This new model equation allows the extension of the Debye-Hückel Theory (DHT) considering time dependence explicitly. This also leads to a new formulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE) and therefore we call it the modified Poisson-Boltzmann Equation (mPBE). In the present first part of this extensive study we derive the equation from the electromagnetics from a quasistatic perspective, or more precisely the electroquasistatic approximation (EQS). Our main focus will be the analysis via the Lie group formalism and since that up to now no symmetry calculation is available we believe that it seems indispensable to apply this method yielding a deeper insight into the behaviour of the solution manifold of this new equation following electrochemical considerations. We determine the classical Lie point symmetries including algebraic properties. Similarity solutions in a most general form and suitable nonlinear transformations are obtained. In addition, a note relating to potential and generalized symmetries is drawn. Moreover we show how the equation leads to approximate symmetries and we apply the method to the first time. The second part appearing shortly after will deal with algebraic solution methods and we shall show that closed-form solutions can be calculated without any numerical methods. Finally the third part will consider appropriate electrochemical experiments proving the model under consideration.