Showing papers by "Kenneth D.T-R McLaughlin published in 2018"

Rigorous asymptotics of a KdV soliton gas.

Abstract: We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

A Riemann--Hilbert Problem Approach to Periodic Infinite Gap Hill's Operators and the Korteweg--de Vries Equation

Abstract: We formulate the inverse spectral theory of infinite gap Hill's operators with bounded periodic potential as a Riemann--Hilbert problem on a typically infinite collection of spectral bands and gaps. We establish a uniqueness theorem for this Riemann--Hilbert problem, which provides a new route to establishing unique determination of periodic potentials from spectral data. As the potential evolves according to the KdV equation, we use integrability to derive an associated Riemann--Hilbert problem with explicit time dependence. Basic principles from the theory of Riemann--Hilbert problems yield a new characterization of spectra for periodic potentials in terms of the existence of a solution to a scalar Riemann--Hilbert problem, and we derive a similar condition on the spectrum for the temporal periodicity for an evolution under the KdV equation.