SPECTRAL-ANALYSIS OF LONG-WAVELENGTH PERIODIC-WAVES AND APPLICATIONS

This paper presents a spectral analysis of large wavelength periodic t ravelling wave solutions of nonlinear evolutionary p.d.e.'s in one spa ce variable. The framework is general enough to include waves occurrin g in a variety of different equations such as the generalized KdV and other dispersive equations, and also, parabolic systems. It is assumed that the equations admit a family of large wavelength periodic waves which, as a parameter alpha tends to zero, tend to a limiting homoclin ic (or solitary) wave. In regions of the spectral plane in which the h omoclinic wave has isolated eigenvalues, the main result, Theorem 1.2, is that the periodic waves have continua of eigenvalues in a neighbor hood of each isolated eigenvalue of the homoclinic wave for sufficient ly small alpha. Generically, these continua will form loops in the spe ctral plane. The main result is applied to the spectral analysis of lo ng wavelength periodic waves arising in several applications, includin g periodic viscous profiles of solutions of degenerate 2x2 conservatio n laws, the generalized KdV, BBM, and Boussinesq equations, the FitzHu gh-Nagumo system, and the Gray-Scott model.