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Fence of saddle solutions of the Allen–Cahn equation in the plane

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  22 November 2024

Yong Liu  and
Yitian Zhang
Yong Liu
Affiliation:
Department of Mathematics, University of Science and Technology of China, 96 Jinzhai Road, Baohe District, Hefei, Anhui Province 230026, China ([email protected]) (corresponding author)
Yitian Zhang
Affiliation:
Department of Mathematics, University of Science and Technology of China, 96 Jinzhai Road, Baohe District, Hefei, Anhui Province 230026, China ([email protected])
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Abstract

In a two-dimensional plane, entire solutions of the Allen–Cahn type equation with a finite Morse index necessarily have finite ends. In the case that the nonlinearity is a sine function, all the finite-end solutions have been classified. However, for the classical Allen–Cahn nonlinearity, the structure of the moduli space of these solutions remains unknown. We construct in this paper new finite-end solutions to the Allen–Cahn equation, which will be called fence of saddle solutions, by gluing saddle solutions together. Our construction can be generalized to the case of gluing multiple four-end solutions, with some of their ends being almost parallel.

Keywords

Allen–Cahn equationmultiple-end solutionsaddle solutionsreduction methodToda systemCauchy-data matching

MSC classification

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 42
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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