Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363-376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.