Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data

Numerical methods for the inverse nonlinear fourier transform

“…4 can be used to determine, for each method, what is the resolution δ α (and, correspondingly, the number of points N F = 2T /δ α ) required for the computation of the integrals in order to find the solution u(t) with a given accuracy. This, in turn, will affect the complexity of the method according to (4), (5), or (7). Another relevant parameter that affects the computational complexity of the NCG and IC methods (but not that of the NT and IC1 methods) is the resolution δ t = cδ α /2 with which the solution u(t) is needed (or, equivalently, the number of points N u = N F /c in which the solution must be computed).…”

“…Fig. 5 compares the computational complexity required by the various methods, computed according to (4), (5), and (7), to obtain the solution u(t) with a desired RMSE of 2 × 10 −3 (about 2 −9 in Fig. 4), as a function of the desired resolution δ t .…”

“…The solution is obtained after iterated convolutions with the GLME kernel, which are efficiently computed trough the FFT algorithm. The accuracy and computational complexity of the proposed method have been investigated and compared to those of two other methods available in the literature, both exploiting the Nyström method in combination with either the conjugate gradient algorithm [5] or the Trench algorithm [4]. The obtained results show that, for a desired accuracy, the proposed method requires the lowest number of operations.…”

“…The computational complexity of the methods presented in[4] and[5] is not indicated by the authors. The values reported here refer to the most efficient implementations that we were able to devise.…”

On the Integrability of the Poisson Driven Stochastic Nonlinear Schrödinger Equations

“…It can be proven then that certain associated scattering functions K ( x , z ) satisfy and hence the potential Θ( x ) = K ( x , x ) is also bounded. In [31, 32, 33] a complete discussion and comparison between the former and other alternative conditions which guarantee solvability of the direct problem is given.…”