Rigorous justification of the short-pulse equation

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.Comment: 15 pages, no figure

Modulation equations near the Eckhaus boundary: the KdV equation

We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*} \partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi |\Psi|^2, \end{align*} near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters $\alpha$, $\beta$ a number of modulation equations can be derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the KdV approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping we work in spaces of analytic functions.Comment: 44 pages, 8 figure

Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation

It is the purpose of this paper to give a simple proof of the fact that solutions of the KdV equation can be approximated via solutions of the NLS equation. The proof is based on an elimination of the quadratic terms of the KdV equation via the Miura transformation

The KdV approximation for a system with unstable resonances

The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. We expect that the approach will allow to handle more complicated systems without a detailed discussion of the resonances and without finding a suitable energy