Nondegeneracy of heteroclinic orbits for a class of potentials on the plane

In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential W:...

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Bibliographic Details
Published in:Applied mathematics letters 2022-02, Vol.124, p.107681, Article 107681
Main Authors: Jendrej, Jacek, Smyrnelis, Panayotis
Format: Article
Language:English
Subjects:
Analysis of PDEs
Hamiltonian systems
Heteroclinic orbit
Mathematics
Minimizer
Nondegenerate
Phase transition
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Summary:In the scalar case, the nondegeneracy of heteroclinic orbits is a well-known property, commonly used in problems involving nonlinear elliptic, parabolic or hyperbolic P.D.E. On the other hand, Schatzman proved that in the vector case this assumption is generic, in the sense that for any potential W:Rm→R, m≥2, there exists an arbitrary small perturbation of W, such that for the new potential minimal heteroclinic orbits are nondegenerate. However, to the best of our knowledge, nontrivial explicit examples of such potentials are not available. In this paper, we prove the nondegeneracy of heteroclinic orbits for potentials W:R2→[0,∞) that can be written as W(z)=|f(z)|2, with f:ℂ→ℂ a holomorphic function.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2021.107681