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

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. (paper)