A numerical study of the semi-classical limit of the focusing nonlinear Schrödinger equation

Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes

“…In fact, the initial data are very similar to the pure soliton data studied by Miller and Kamvissis [30]: operator has pure imaginary eigenvalues and the reflection coefficients are exactly zero. For our data set I, the eigenvalues are almost pure imaginary and the reflection coefficients are exponentially small for small ε; for set II, the eigenvalues are symmetric about the pure imaginary axis and are located roughly on a convex "parabola" whose vertex is at the origin [9,7]. They illustrate the main numerical difficulties and qualitative phenomena for the focusing NLS.…”

“…In this case, due to the modulational instability (see [7]), the numerical solution is stable but qualitatively wrong for small ε with the accumulation of round-off errors. Therefore, we apply the Krasny filter [24] to the solution at each time step (see also [9] for similar applications). That is, we set to zero all the Fourier coefficients of the numerical solution whose magnitudes are below a certain filter threshold.…”

“…For benchmark comparisons, we also define another spectral method, the CrankNicolson-type spectral method (CNSP) which was proposed in [9] for the cubic focusing NLS (V (x) ≡ 0, τ = 0, and f (ρ) = −ρ in (3.1) ):…”

“…These sets of initial data were already used in [9] for numerical study of the semiclassical limits of the NLS with cubic focusing nonlinearity by using a different numerical method. Here we use them to study our numerical method SP2.…”

Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

“…In theory, the strong convergence of the integral as ε goes to zero would imply the weak convergence of ρ ε = |u ε (x, t)| 2 with respect to x. This is motivated by the works [22,31] on the zero dispersion limit of the KDV equation and used for the numerical study of focusing nonlinear Schrödinger equation [3,9,27]. Figure 1 also displays the indefinite integral at t = 2.0 for four different values of ε: 0.64, 0.32, 0.16 and 0.08 for the three different types data.…”