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

TL;DR

This paper proves that heteroclinic orbits are nondegenerate for a class of two-dimensional potentials expressed as the squared magnitude of holomorphic functions, providing explicit examples in the vector case.

Contribution

It establishes nondegeneracy of heteroclinic orbits for a specific class of potentials on the plane, which was previously lacking explicit examples.

Findings

Nondegeneracy holds for potentials W(z)=|f(z)|^2 with holomorphic f

Provides explicit examples of nondegenerate heteroclinic orbits in vector case

Extends known scalar results to a class of vector potentials

Abstract

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 $W:\mathbb{R}^m\to\mathbb{R}$, $m\geq 2$, 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 $W:\mathbb{R|^2\to [0,\infty)$ that can be written as $W(z)=|f(z)|^2$, with $f:\mathbb{C} \to \mathbb{C}$ a holomorphic function.