Numerical stability analysis of nonlinear Schrödinger equation via congruence transformation Original Research Article

In this paper, we present a congruence transformation for studying the numerical stability condition of the nonlinear Shrödinger equation View the MathML sourceiut+uxx+pu+q|u|2u=0,i2=−1, subject to homogeneous Neumann boundary conditions, where pp and qq are real parameters. The congruence transformation takes advantage of a special structure of the matrices that result from Galerkin’s discretization with piecewise linear functions and product approximation for the nonlinear term by decoupling the system of perturbation into independent subsystems. This transformation can also be applied to solving any linear system Tx=bTx=b if TT is of the same special structure.