Abstract: In this talk I will describe a construction of new solutions to some classical autonomous semilinear elliptic equations in the plane. These solutions constitute a "gluing" of one-dimensional profiles with a single transition, located very far apart one to each other. In the case of the Allen-Cahn equation, solutions with a finite number of nearly parallel transition layers are built, while for the stationary nonlinear Schrodinger equation multiple bump-line patterns are found. The Toda system is shown to rule the asymptotic shape of these transition lines.
This is joint work with Michal Kowalczyk, Frank Pacard and Juncheng Wei.