6533b821fe1ef96bd127b87c

RESEARCH PRODUCT

Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation

Tamara GravaChristian KleinChristian Klein

subject

PhysicsLeading edgeSmall dispersion limitComputer Science::Information RetrievalGeneral MathematicsMathematical analysisGeneral EngineeringMathematics::Analysis of PDEsGeneral Physics and AstronomyNonlinear equationsDispersive partial differential equationShock wavesAmplitudeNonlinear Sciences::Exactly Solvable and Integrable SystemsInitial value problemWavenumberDispersive shockDispersion (water waves)Constant (mathematics)Korteweg–de Vries equationDevries equationAsymptoticsSettore MAT/07 - Fisica MatematicaNonlinear Sciences::Pattern Formation and SolitonsMathematical Physics

description

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations. These oscillations are approximately described by the elliptic solution of KdV where the amplitude, wave-number and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as $\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a more accurate description near the leading edge of the oscillatory zone we present a multiscale expansion of the solution of KdV in terms of the Hastings-McLeod solution of the Painlev\'e-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order $\epsilon^{2/3}$.

yearjournalcountryeditionlanguage
2007-08-04
10.1098/rspa.2007.0249http://arxiv.org/abs/0708.0638