65 citations · 204 across the 13 of their papers we have counts for
13 papers
Riemann-Hilbert approach for the nonlocal modified Korteweg-de Vries equation with a step-like oscillating background
Yan Rybalko
This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation with the oscillating step-like b…
The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: long-time asymptotics
Yan Rybalko, Dmitry Shepelsky, Shou-Fu Tian
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger equation \[ \I q_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0, \] subject to the step-like init…
Hölder continuity of functions in the fractional Sobolev spaces: 1-dimensional case
Yan Rybalko
This paper deals with the embedding of the Sobolev spaces of fractional order into the space of Hölder continuous functions. More precisely, we show that the function $f\in H^s(\ma…
On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system
Kenneth Karlsen, Yan Rybalko
We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic non…
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…
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…
Defocusing nonlocal nonlinear Schrödinger equation with step-like boundary conditions: long-time behavior for shifted initial data
Yan Rybalko, Dmitry Shepelsky
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schrödinger equation $ iq_{t}(x,t)+q_{x…
Curved wedges in the long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
Yan Rybalko, Dmitry Shepelsky
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation wit…
Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of step-like initial data
Yan Rybalko, Dmitry Shepelsky
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schrödinger (NNLS) equation with the step-like in…
Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation with step-like initial data
Yan Rybalko, Dmitry Shepelsky
We study the Cauchy 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 a step-like initial dat…
Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation
Yan Rybalko, Dmitry Shepelsky
We study the initial value problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation \[ iq_{t}(x,t)+q_{xx}(x,t)+2σq^{2}(x,t)\bar{q}(-x,t)=0 \] with decaying (as $x\…
Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the uniform transform method
Yan Rybalko
We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as…