scholarly journals Riemann–Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the steplike initial data

2010 ◽  
Vol 51 (9) ◽  
pp. 093506 ◽  
Author(s):  
Vladimir Kotlyarov ◽  
Alexander Minakov
Keyword(s):  
Initial Data ◽  
Vries Equation ◽  
Hilbert Problem ◽  
Time Dynamics ◽  
De Vries ◽  
Long Time ◽  
Riemann Hilbert Problem ◽  
Long Time Dynamics

scholarly journals Long-time asymptotics for the Korteweg–de Vries equation with step-like initial data

Nonlinearity ◽  
2013 ◽  
Vol 26 (7) ◽  
pp. 1839-1864 ◽  
Author(s):  
Iryna Egorova ◽  
Zoya Gladka ◽  
Volodymyr Kotlyarov ◽  
Gerald Teschl
Keyword(s):  
Initial Data ◽  
Vries Equation ◽  
Time Asymptotics ◽  
De Vries ◽  
Long Time ◽  
Long Time Asymptotics

scholarly journals A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

2019 ◽  
Author(s):  
Kenneth T-R McLaughlin ◽  
Patrik V Nabelek
Keyword(s):  
Vries Equation ◽  
Hilbert Problem ◽  
Kdv Equation ◽  
Unique Determination ◽  
Existence Of A Solution ◽  
Spectral Bands ◽  
Periodic Potentials ◽  
De Vries ◽  
Explicit Time Dependence ◽  
Riemann Hilbert Problem

Abstract We formulate the inverse spectral theory of infinite gap Hill’s operators with bounded periodic potentials 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 potentials evolve according to the Korteweg–de Vries Equation (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.

Sign in / Sign up

Export Citation Format

Share Document