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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Zhiyou Chen · Jannlong Chern. Topological multivortex solutions for the Chern–Simons system with two Higgs particles. 2016.