In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity. It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states. Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.

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An elliptic equation arising from the study o fstatic solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field, is considered. We classify the solutions and establish the uniqueness of radially symmetric solutions. We also complete a classification of symmetric solutions of an elliptic equation on the sphere.