Pseudo-spectral solution of nonlinear Schrödinger equations (Q582840)

This paper compares four discretization methods for solving the generalized nonlinear Schrödinger equation \(iu_ t+u_{xx}+q_ c| u|^ 2u+q_ q| u|^ 4u+iq_ m| u|^ 2_ xu+iq_ u| u|^ 2u_ x=0\) where \(q_ c\), \(q_ q\), \(q_ m\) and \(q_ u\) are real parameters. An initial value problem is considered so that \(u(x,0)=u_ 0(x)\) is specified. The solution may be represented in a Fourier series where the coefficients depend on time and the methods differ on their formalism connecting the time variable with the space function discretization at n collocation points. Numerical examples are given.