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math.AP2025
The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach
Iryna Karpenko, Dmitry Shepelsky
We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation , $\t…
math.AP2025
Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions
I. Karpenko, D. Shepelsky, G. Teschl
We consider the Cauchy problem for the modified Camassa-Holm equation \[ u_t+\left((u^2-u_x^2)m\right)_x=0,\quad m\coloneqq u-u_{xx}, \quad t>0,\ \ -\infty<x<+\infty \] subject to…
math.AP2025
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 ini…