Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

Abstract

We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

The wave breaking for Whitham-type equations revisited

“…On the other hand those equations have a (weak) dispersive part, with the possibility of existence of global strong small solutions. This is suggested by some numerical simulations in [11] and has been proven for the cubic fKdV equation with −1 < α < 0 in the work [17] of the Authors.…”

Breather Solutions to the Cubic Whitham Equation

“…Indeed while it has been shown for example that the Peregrine breather is unstable to virtually all perturbations [24,48], breathers are nevertheless observable in the laboratory [49]. Moreover asymptotic model equations such as the KdV and Whitham equations are generally valid physical models only on a time scale inversely proportional to the amplitude [31,[50][51][52].…”

Global dynamics of the generalized fifth-order KdV equation with critical nonlinearity

“…Nevertheless, we prefer to present an alternative approach to understand the asymptotics of the large time solutions of (1.1) since the argument we are using is also flexible in fractional dispersive models (e.g. [18]). Different with [15], our proof is fully carried out in Fourier space whose main idea is to identify the ODE of ∂ t f (t, ξ) precisely.…”