Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function

Frédéric Chardard (1, 2, 3) , Thomas Bridges (4)

1 MMCS - Modélisation mathématique, calcul scientifique
2 ICJ - Institut Camille Jordan
3 UJM - Université Jean Monnet - Saint-Étienne
4 UNIS - University of Surrey

Partial differential equations in one space dimension and time, which are gradient-like in time with Hamiltonian steady part, are considered. The interest is in the case where the steady equation has a homoclinic orbit, representing a solitary wave. Such homoclinic orbits have two important geometric invariants: a Maslov index and a Lazutkin–Treschev invariant. A new relation between the two has been discovered and is moreover linked to transversal construction of homoclinic orbits: the sign of the Lazutkin–Treschev invariant determines the parity of the Maslov index. A key tool is the geometry of Lagrangian planes. All this geometry feeds into linearization about the homoclinic orbit in the time-dependent system, which is studied using the Evans function. A new formula for the symplectification of the Evans function is presented, and it is proven that the derivative of the Evans function is proportional to the Lazutkin–Treschev invariant. A corollary is that the Evans function has a simple zero if, and only if, the homoclinic orbit of the steady problem is transversely constructed. Examples from the theory of gradient reaction–diffusion equations and pattern formation are presented.

Mots clés: Evans function; Lazutkin–Treschev invariant; transversality; homoclinc orbits; Maslov index; solitary waves; symplectic

Domaines: Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Frédéric Chardard, Thomas Bridges. Transversality of homoclinic orbits, the Maslov index and the symplectic Evans function. Nonlinearity, 2015, 28 (1), pp.77-102. ⟨10.1088/0951-7715/28/1/77⟩. ⟨hal-01092979v1⟩