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

Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev–Petviashvili Equation

This is the published version, also available here: http://dx.doi.org/10.1137/090770758.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

A simple criterion for transverse linear instability of nonlinear waves

AbstractWe prove a simple criterion for transverse linear instability of nonlinear waves for partial differential equations in a spatial domain Ω×R⊂Rn×R. For stationary solutions depending upon x∈Ω only, the question of transverse (in)stability is concerned with their (in)stability with respect to perturbations depending upon (x,y)∈Ω×R. Starting with a formulation of the PDE as a dynamical system in the transverse direction y, we give sufficient conditions for transverse linear instability. We apply the general result to the Davey–Stewartson equations, which arise as modulation equations for three-dimensional water waves

On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.Comment: 19 pages

The phase shift of line solitons for the KP-II equation

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations