Computing the Maslov index of solitary waves. Part 2:Phase space with dimension greater than four

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

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 CMLA - Centre de Mathématiques et de Leurs Applications
3 School of Mathematical Sciences [Dublin]
4 UNIS - University of Surrey

This paper extends the theory of the Maslov index of solitary waves in Part 1 to the case where the phase space is of dimension greater than four. The starting point is Hamiltonian PDEs, in one space dimension and time, whose steady part is a Hamiltonian ODE with a phase space of dimension six or greater. This steady Hamiltonian ODE is the main focus of the paper. Homoclinic orbits of the steady ODE represent solitary waves of the PDE, and one of the properties of the homoclinic orbits is the Maslov index. We develop formulae for the Maslov index, the Maslov angle and its subangles, in an exterior algebra framework, and develop numerical algorithms to compute them. In addition, a new numerical approach based on a discrete QR algorithm is proposed. The Maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time-dependent problem. The theory is applied to a class of reaction-diffusion equations, the longwave-shortwave resonance equations and the seventh-order KdV equation.

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie symplectique [math.SG]; Systèmes dynamiques [math.DS]; Analyse numérique [math.NA]

Frédéric Chardard, Frédéric Dias, Thomas Bridges. Computing the Maslov index of solitary waves. Part 2:Phase space with dimension greater than four. Physica D: Nonlinear Phenomena, 2011, 240 (17), p.1334-1344. ⟨10.1016/j.physd.2011.05.014⟩. ⟨hal-00793165⟩