3,465 research outputs found
Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation
In this paper, we study the orbital stability for a four-parameter family of
periodic stationary traveling wave solutions to the generalized Korteweg-de
Vries equation. In particular, we derive sufficient conditions for such a
solution to be orbitally stable in terms of the Hessian of the classical action
of the corresponding traveling wave ordinary differential equation restricted
to the manifold of periodic traveling wave solution. We show this condition is
equivalent to the solution being spectrally stable with respect to
perturbations of the same period in the case of the Korteweg-de Vries equation,
and in neighborhoods of the homoclinic and equilibrium solutions in the case of
a power-law nonlinearity.Comment: 24 page
Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case
Extending previous results of Oh--Zumbrun and Johnson--Zumbrun, we show that
spectral stability implies linearized and nonlinear stability of spatially
periodic traveling-wave solutions of viscous systems of conservation laws for
systems of generic type, removing a restrictive assumption that wave speed be
constant to first order along the manifold of nearby periodic solutions.Comment: Fixed minor typo
Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
In this paper, we investigate the spectral instability of periodic traveling
wave solutions of the generalized Korteweg-de Vries equation to long wavelength
transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By
analyzing high and low frequency limits of the appropriate periodic Evans
function, we derive an orientation index which yields sufficient conditions for
such an instability to occur. This index is geometric in nature and applies to
arbitrary periodic traveling waves with minor smoothness and convexity
assumptions on the nonlinearity. Using the integrable structure of the ordinary
differential equation governing the traveling wave profiles, we are then able
to calculate the resulting orientation index for the elliptic function
solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem
corrected. Exposition updated and references added
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