The Korteweg-de Vries Equation with Small Dispersion: Higher Order Lax-Levermore Theory

Abstract

We utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem:

$$\begin{gathered} {u_t} + 6u{u_x} + { \in ^2}{u_{xxx}} = 0, \hfill \\ u\left( {x,0} \right) = v\left( x \right), \hfill \\ \end{gathered} $$

when ∈ → 0. We refine the Lax-Levermore theory, which gives the weak limit of the solution as ∈ → 0, by introducing a quantum condition. This allows us to calculate the waveform of the local oscillations up to phase-shifts when ∈ is small.