A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation (Q1034644)

The paper deals with the Kuramoto-Tsuzuki equation \[ \frac{\partial w}{\partial t}=(1+ic_1)\frac{\partial^2 w}{\partial x^2}+w-(1+ic_2)|w|^2w, \; (x,t)\in (0,1) \times (0,T] \] with an initial condition and with the homogeneous Neumann boundary conditions. The authors propose an semi-explicit linearized difference scheme of Crank-Nicolson type with the truncation error \({\mathcal O}(h^2+\tau^2).\) It is proved that the scheme is stable and possesses the accuracy order \({\mathcal O}(h^2+\tau^2)\) in the \(L_{\infty}\)-norm. Numerical examples confirm the theoretical results.