On singular perturbation problems for the nonlinear Poisson equation (Q2730767)

The authors consider the following singular perturbation problem for the nonlinear Poisson equation NEWLINE\[NEWLINE\varepsilon^2{d^2\phi\over dx^2}= V'(\phi),\quad 0< x< 1,\quad \phi(0)= \phi_0,\quad \phi(1)= \phi_1,NEWLINE\]NEWLINE where \(V\) is a given smooth function, ususally referred to as the Sagdeev potential of the system, and \(\varepsilon\) is a small parameter. Depending on the properties of \(V\), this problem may not have a unique solution; a classification of its different solutions is provided, and their behavior as \(\varepsilon\) tends to \(0\) is studied.NEWLINENEWLINENEWLINEThis classification is based on their monotonicity properties, which mainly depend on the stationary points of the Sagdeev potential. For each class of solutions, necessary and sufficient conditions for the resolution of the problem are given and a boundary layer analysis is performed.NEWLINENEWLINENEWLINEThe periodic case is also investigated. A fluid model of electrostatic sheets is used to illustrate the results of the paper.