Discretized fractional calculus
SIAM Journal on Mathematical Analysis
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Absorbing Boundary Conditions for One-dimensional Nonlinear Schrödinger Equations
Numerische Mathematik
Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations
Journal of Computational Physics
A perfectly matched layer approach to the nonlinear Schrödinger wave equations
Journal of Computational Physics
Journal of Computational Physics
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The paper addresses the problem of constructing non-reflecting boundary conditions for two types of one dimensional evolution equations, namely, the cubic nonlinear Schrodinger (NLS) equation, @?"tu+Lu-i@g|u|^2u=0 with L=-i@?"x^2, and the equation obtained by letting L=@?"x^3. The usual restriction of compact support of the initial data is relaxed by allowing it to have a constant amplitude along with a linear phase variation outside a compact domain. We adapt the pseudo-differential approach developed by Antoine et al. (2006) [5] for the NLS equation to the second type of evolution equation, and further, extend the scheme to the aforementioned class of initial data for both of the equations. In addition, we discuss efficient numerical implementation of our scheme and produce the results of several numerical experiments demonstrating its effectiveness.