The Modultional Instability for a Generalized KdV Equation

Abstract

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first index counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second index provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Haragus and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator - in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests geometric interpretations attached to the vanishing of the modulational instability index previously mentioned.

35 pages, 3 figures; fixed formula and derivation for modulational instability index

References in corpus (2)

• Orbital stability of periodic waves for the nonlinear Schroedinger equation
• Stability of periodic solutions of conservation laws with viscosity: Analysis of the Evans function

Cited by in corpus (13)

• Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids
• On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
• Full description of Benjamin-Feir instability of Stokes waves in deep water
• Modulational Instability in Equations of KdV Type
• Unstable Stokes waves
• Nonlinear Modulational Instability of Dispersive PDE Models
• Co-periodic stability of periodic waves in some Hamiltonian PDEs
• Stability of Traveling wave solutions of Nonlinear Dispersive equations of NLS type
• Global well-posedness for $H^{-1}(\mathbb{R})$ perturbations of KdV with exotic spatial asymptotics
• Full description of Benjamin-Feir instability for generalized Korteweg-de Vries equations
• Validity of the hyperbolic Whitham modulation equations in Sobolev spaces
• Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability
• On higher order isolas of unstable Stokes waves