We present a new system of modulation equations that approximate the focusing of the nonlinear Schrr odinger equation in the presence of small normal time dispersion (TDNLS). Since the modulation equations are much easier for analysis and for numerical simulations, they can be used to get a general picture of the TDNLS focusing. Analytical and numerical agreement between the modulation equation...

The possibility of physically relevant singular solutions of the nonlinear Schrodinger equation (NLSE) with sustained dissipation into the singularity is considered through numerical study of a dissipative regularisation and its small dissipation limit. A new form of such dissipative solutions is conjectured for certain parameter ranges where this behaviour was previously not expected, involvi...

The cubic nonlinear Schrr odinger equation with repulsive nonlinearity and elliptic function potential in two-dimensions models a repulsive dilute gas Bose{Einstein condensate in a lattice potential. A family of exact stationary solutions is presented and its stability is examined using analytical and numerical methods. All stable trivial-phase solutions are oo-set from the zero level. Our resu...

1. A general criterion for pointwise Fourier inversion 2. Pointwise Fourier inversion on R n (n = 3) 3. Fourier inversion on R 2 4. Fourier inversion on R n (general n) 5. Fourier inversion on spheres 6. Fourier inversion on complex projective space, and variants 7. Fourier inversion on hyperbolic space, and variants 8. Fourier inversion on strongly scattering manifolds 9. Hermite expansions an...

We report the rst application of complex symmetric wavelets to the numerical simulation of a nonlinear signal propagation model. This model is the so-called nonlinear Schrodinger equation that describes, for instance, the evolution of the electric eld amplitude in nonlinear optical bers. We propose and study a new way to implement a global space-time adaptive grid, based on interpolation prope...

Let u(x; t) be the solution to the free time-dependent Schrr odinger equation at the point (x; t) in space-time R n+1 with initial data f. We characterize the size of u in terms L p (L q)-estimates with power weights. Our bounds are given by norms of f in homogeneous Sobolev spaces _ H s (R n). Our methods include use of spherical harmonics, uniformity properties of Bessel functions and interpo...

The known correspondence between the Kronig{Penney model and certain Jacobi matrices is extended to a wide class of Schrr odinger operators on graphs. Examples include rectangular lattices with and without a magnetic eld, or comb{shaped graphs leading to a Maryland{type model. Schrr odinger operators on L 2 (?) , where ? is a graph, were introduced into quantum mechanics long time ago 1]. In re...

In this paper we consider the nonlinear Schrr odinger equation with an oscillatory, mean-zero dispersion, which has recently been proposed as an alternative method of dispersion compensation for pulse transmission in optical bers. Under the assumption that the time scale on which the dispersion changes is short in comparison with the dispersion and nonlinearity time scales, we are able to facto...

An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schr odinger equation that is valid from the early stages of self-focusing until the blowup point. The adiabatic law leads to an analytical formula for the location of the blowup point and can be used to estimate the effects of various small perturbations on self-focusing. The results of the analysis are confir...

Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-d nonlinear Schrr odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering oo an interface separating two regions of constant potential. We nd that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitar...