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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
##### Keywords:
moduli space of solutions; multiple-end solutions
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##### References:
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