406 research outputs found
A numerical implementation of the unified Fokas transform for evolution problems on a finite interval
We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. To our knowledge, the numerical evaluation of the solution so far could only be obtained by a time-stepping scheme, that must also take into account the issue, generically non-trivial, of the imposition of the boundary conditions. Instead of computing the evolution numerically, we evaluate the novel solution representation formula obtained by the unified transform, also known as Fokas transform. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results
White noise for KdV and mKdV on the circle
We survey different approaches to study the invariance of the white noise for
the periodic KdV. We mainly discuss the following two methods. First, we
discuss the PDE method, following Bourgain \cite{BO4}, in a general framework.
Then, we show how it can be applied to the low regularity setting of the white
noise for KdV by introducing the Besov-type space \hat{b}^s_{p, \infty}, sp<
-1. Secondly, we describe the probabilistic method by Quastel, Valk\'o, and the
author \cite{OQV}. We also use this probabilistic approach to study the white
noise for mKdV.Comment: 26 pages. To appear in RIMS Kokyuroku Bessats
Dispersive dam-break flow of a photon fluid
We investigate the temporal photonic analogue of the dam-break phenomenon for
shallow water by exploiting a fiber optics setup. We clearly observe the decay
of the step-like input (photonic dam) into a pair of oppositely propagating
rarefaction wave and dispersive shock wave. Our results show evidence for a
critical transition of the dispersive shock into a self-cavitating state. The
detailed observation of the cavitating state dynamics allows for a fully
quantitative test of the Whitham modulation theory applied to the universal
defocusing nonlinear Schroedinger equation
KdV equation under periodic boundary conditions and its perturbations
In this paper we discuss properties of the KdV equation under periodic
boundary conditions, especially those which are important to study
perturbations of the equation. Next we review what is known now about long-time
behaviour of solutions for perturbed KdV equations
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