| The stability analysis of travelling waves in Hamiltonian systems is developed from a geometrical point of view, in which the eigenvalue problem is linked to Lagrangian intersection theory. If the linearized differential operator has a certain variational structure, the Maslov index, a higher dimensional analog of a winding number, can be used to gain information about the distribution of the spectrum. Moreover, in certain cases, the monotonicity of the Maslov index, with respect to the eigenvalue parameter lambda, is related to the exact count of the number of positive real eigenvalues for the linearized differential operator. In this work, properties of the Maslov index are developed and the instability of viscous shock waves for a class of conservation law PDE is analyzed as an application of the Maslov index theory developed. For the pattern case, a symplectic interpretation of the Morse index for the pattern (steady state) solution is given in terms of the Maslov index, expressed via the geometry of the solution set for the linearized partial differential operator. |
