On convergence and stability of difference schemes for derivative nonlinear evolution equations (Q1366369)

The authors consider the initial-boundary value problem for a class of nonlinear evolution equations \[ {\partial u\over\partial t}- A {\partial^2u\over\partial x^2}- g(u) {\partial u\over\partial x}= f(u), \] where \(A\) is a complex diagonal matrix, \(f\), \(g\) are complex vector functions, and its numerical solution by means of a Crank-Nicolson type difference scheme (in space and time), combined with an iterative method. Convergence and stability are proven without a restriction to the discretization parameters. The class of equations considered by the authors includes nonlinear Schrödinger-type equations (e.g. Kuramoto-Tsuzuki equation) as well as reaction-diffusion equations.