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Moduli space theory for the Allen-Cahn equation in the plane. (English) Zbl 1286.35018
The paper is concerned with the space of solutions of the semilinear elliptic equation defined in the plane \(\mathbb R^2\) (i.e. entire solutions) \(\Delta u-F'(u)=0\), where \(F\) is an even and bistable function. The authors describe the moduli space of solutions with special structure at infinity, called multiple-end solutions, which have their zero set asymptotic to \(2k\) oriented affine half-lines at infinity \((k\geq 2)\). They prove that if a multiple-end solution \(u\) is nondegenerate, then the set of solutions locally near \(u\) is a smooth manifold of dimension \(2k\).

MSC:
35B08 Entire solutions to PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35J61 Semilinear elliptic equations
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