The Maslov index of solitary waves

 Frédéric Chardard  (École Normale Supérieure de Lyon, France)
 Frédéric Dias  (ENS Cachan & University College Dublin, Ireland)
 Thomas J. Bridges  (University of Surrey, England, UK)

Transversality, the Maslov index, and the 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.
  F. Chardard & T.J. Bridges. Transversality of homoclinic orbits, the Maslov index, and the symplectic Evans function Nonlinearity 28 77-102 (2015)
Nonlinearity website

Hamiltonian systems on a 4D phase space
  When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We are interested in developing a robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithms reported here are developed in the exterior algebra representation, which leads to a robust and fast algorithm with some novel properties. We use two different representations for the Maslov index, one based on an intersection index and one based on approximating the homoclinic orbit by a sequence of periodic orbits. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg-De Vries equation, and the longwave-shortwave resonance equations are presented. Part 1 considers the case of four-dimensional phase space, and Part 2 (below) considers the case of 2n-dimensional phase space with n>2.
  F. Chardard, F. Dias & T.J. Bridges. Computing the Maslov index of solitary waves. Part 1: Hamiltonian systems on a 4-dimensional phase space. Physica D 238 1841-1867 (2010)
doi:10.1016/j.physd.2009.05.008    Final form preprint
  F. Chardard, F. Dias & T.J. Bridges. Computational aspects of the Maslov index of solitary waves, HAL hal-00383888 (2009)  

Maslov index of multi-pulse homoclinic orbits
  Multi-pulse homoclinic orbits of Hamiltonian systems on on four-dimensional phase space have been classified by Buffoni, Champneys & Toland (1996) by assigning a sequence of integers to each orbit. In this paper we find the surprising result that this string of integers encodes the value of the Maslov index of the homoclinic orbit. Our results include a computable formulation of the Maslov index for homoclinic orbits and a robust numerical method for the evaluation of the Maslov index.
  F. Chardard, F. Dias & T.J. Bridges. On the Maslov index of multi-pulse homoclinic orbits, Proc. Royal Soc. London A 465 2897-2910 (2009)
doi:10.1098/rspa.2009.0155    Final form preprint

Hamiltonian systems on a 2nD (n>2) phase space
  The theory of the Maslov index of solitary waves in Part I (above) is extended to the case where the phase space has 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. It is this steady Hamiltonian ODE that 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.
  F. Chardard, F. Dias & T.J. Bridges. Computing the Maslov index of solitary waves. Part 2: Phase space with dimension greater than four, Physica D 240 1334-1344 (2011)   Final form preprint.pdf
Physica D website for the paper

The Maslov index of hyperbolic periodic orbits
  The Maslov index is a topological property of periodic orbits of finite-dimensional Hamiltonian systems that is widely used in semiclassical quantization, quantum chaology, stability of waves and classical mechanics. In this paper a numerical scheme is devised to compute the Maslov index for hyperbolic periodic orbits in systems of low phase space dimension. The idea is to compute on the exterior algebra of the ambient vector space, where the Lagrangian subspace representing the unstable subspace is reduced to a line. The idea is illustrated by application to Hamiltonian systems on a phase space of dimension four. The theory is used to compute the Maslov index for periodic solutions of the fifth-order KdV equation.
  F. Chardard, F. Dias & T.J. Bridges. Fast computation of the Maslov index for hyperbolic periodic orbits, J. Phys. A: Math. Gen. 39 14545-14557 (2006)
doi:10.1088/0305-4470/39/47/002,    Final form preprint .pdf
  F. Chardard. Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves, C. R. Acad. Sci. Paris, Ser. I 345 689-694 (2007)

Matlab codes
  The goal of these MATLAB routines is to compute numerically the Maslov index of solitary waves by using a framework based on exterior algebra. The method is described in the above publications. Two examples are available in the following files: kdv5_setup.m and kdv7_setup.m which can be downloaded below.


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  Homepages:    F. Chardard    F. Dias    T.J. Bridges