Analytic representation of the order parameter profiles and susceptibility of a Ginzburg-Landau type model with strongly adsorbing competing walls

In this work, we study a critical thermodynamic system (say, a binary liquid mixture) of plane film geometry whose stable states, at given temperature and external ordering field, are determined by the minimizers of the one-dimensional counterpart of the standard ϕ4 Ginzburg-Landau Hamiltonian in terms of the order parameter. We focus on the case in which both bounding walls are strongly adsorbing but competing, e.g., prefer different components of the mixture, that is the order parameter tends to infinity at one of the boundaries and to minus infinity at the other one. Assuming that the boundaries of the system are positioned at a finite distance from one another, we solve the corresponding (+, −) boundary-value problem in terms of Weierstrass and Jacobi elliptic functions and give analytic representation of the order parameter profiles and local and total susceptibilities depending on the temperature and ordering field.