227 citations · 685 across the 24 of their papers we have counts for
16 papers · 1 filter
The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach
Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky +1
This work addresses the development of the Riemann-Hilbert problem (RHP) formalism (the Fokas method) for the Camassa-Holm equation under periodic boundary conditions. Particularly…
Global conservative solutions of the nonlocal NLS equation beyond blow-up
Yan Rybalko, Dmitry Shepelsky
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with…
A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
Iryna Karpenko, Dmitry Shepelsky, Gerald Teschl
The paper aims at developing the Riemann-Hilbert (RH) approach for the modified Camassa-Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solu…
Focusing nonlocal nonlinear Schrödinger equation with asymmetric boundary conditions: large-time behavior
Anne Boutet de Monvel, Yan Rybalko, Dmitry Shepelsky
We consider the focusing integrable nonlocal nonlinear Schrödinger equation \[\mathrm{i}q_{t}(x,t)+q_{xx}(x,t)+2q^{2}(x,t)\bar{q}(-x,t)=0\] with asymmetric nonzero boundary conditi…
Asymptotic stage of modulation instability for the nonlocal nonlinear Schrödinger equation
Yan Rybalko, Dmitry Shepelsky
We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 \] with symmetric bounda…
The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky
This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\in…