15 research outputs found
Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit
We study the small dispersion limit for the Korteweg-de Vries (KdV) equation
in a critical scaling regime where
approaches the trailing edge of the region where the KdV solution shows
oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an
asymptotic expansion for the KdV solution in a double scaling limit, which
shows that the oscillations degenerate to sharp pulses near the trailing edge.
Locally those pulses resemble soliton solutions of the KdV equation.Comment: 25 pages, 4 figure
Asymptotics for a special solution to the second member of the Painleve I hierarchy
We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page
Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory
We discuss universality in random matrix theory and in the study of
Hamiltonian partial differential equations. We focus on universality of
critical behavior and we compare results in unitary random matrix ensembles
with their counterparts for the Korteweg-de Vries equation, emphasizing the
similarities between both subjects.Comment: review paper, 19 pages, to appear in the proceedings of the MSRI
semester on `Random matrices, interacting particle systems and integrable
systems
Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions
We study numerically the small dispersion limit for the Korteweg-de Vries
(KdV) equation for and give a
quantitative comparison of the numerical solution with various asymptotic
formulae for small in the whole -plane. The matching of the
asymptotic solutions is studied numerically
Transformation of a shoaling undular bore
We consider the propagation of a shallow-water undular bore over a gentle monotonic
bottom slope connecting two regions of constant depth, in the framework of the variablecoefficient
Korteweg – de Vries equation. We show that, when the undular bore advances
in the direction of decreasing depth, its interaction with the slowly varying topography
results, apart from an adiabatic deformation of the bore itself, in the generation of a
sequence of isolated solitons — an expanding large-amplitude modulated solitary wavetrain
propagating ahead of the bore. Using nonlinear modulation theory we construct an
asymptotic solution describing the formation and evolution of this solitary wavetrain. Our
analytical solution is supported by direct numerical simulations. The presented analysis
can be extended to other systems describing the propagation of undular bores (dispersive
shock waves) in weakly non-uniform environments