Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional 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 develop a new robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithm reported here is developed in the exterior algebra representation, which leads to a fast algorithm with some novel properties. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg–de Vries equation, and the long-wave–short-wave resonance equations are presented. Part 1 considers the case of a four-dimensional phase space, and Part 2 considers the case of a 2n2n-dimensional phase space with n>2n>2.