Long time asymptotics for the nonlocal mKdV equation with finite density initial data

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schrödinger Equation with Finite Density Initial Data. I. Solitonless Sector

The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as t→±∞ (x/t∼O(1)) of solutions to the Cauchy problem for the defocusing non-linear Schrödinger equation (DfNLSE), i∂tu+∂ 2 xu−2(|u|−1)u=0, with finite density initial data u(x, 0)=x→±∞ exp( i(1∓1)θ 2 )(1+o(1)), wher...

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert problem approach is used to derive the leading-order asymptotics as |t| → ∞ (x/t ∼ O(1)) of solutions (u = u(x, t)) to the Cauchy problem for the defocusing...

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

The mKdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This Riemann-Hilbert problem has explicit (x, t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of t...

Long–time Asymptotics for Solutions of the Nls Equation with Initial Data in a Weighted Sobolev Space

Here Γ is the gamma function and the function r is the so-called reflection coefficient for the potential q0(x) = q(x, t = 0), as described below. The error term O( log t t ) is uniform for all x ∈ R. The above asymptotic form was first obtained in [ZaMa], but without the error estimate. Based on the nonlinear steepest descent method introduced in [DZ1], the error estimate in (1.1) was derived ...