Using the connection between closed solution curves of the Vortex Filament Equation and the periodic problem for the nonlinear Schrr odinger equation (NLS), we investigate the possibility of relating the knot types of nite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply-covered circles.

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV eq...

We demonstrate "hidden solvability" of the nonlinear Schrödinger (NLS) equation whose nonlinearity coefficient is spatially modulated by Hermite-Gaussian functions of different orders and the external potential is appropriately chosen. By means of an explicit transformation, this equation is reduced to the stationary version of the classical NLS equation, which makes it possible to use the brig...

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation which was first derived by means of bi-Hamiltonian methods in [A. S. Fokas, Phys. D 87 (1995), 145–150]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as...

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offeri...

The scalar nonlinear Schrödinger (NLS) equation and a suitable discretization are well known integrable systems which exhibit the phenomena of “effective” chaos. Vector generalizations of both the continuous and discrete system are discussed. Some attention is directed upon the issue of the integrability of a discrete version of the vector NLS equation.

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offeri...

In this paper, the lattice Boltzmann method for convectiondiffusion equation with source term is applied directly to solve some important nonlinear complex equations, including nonlinear Schrödinger (NLS) equation, coupled NLS equations, Klein-Gordon equation and coupled Klein-Gordon-Schrödinger equations, by using complex-valued distribution function and relaxation time. Detailed simulations o...

We analyze self-focusing and singularity formation in the nonlinear Schrödinger equation (NLS) with high-order dispersion iψt±∆ψ+ |ψ|2σψ = 0, in the isotropic mixed-dispersion NLS iψt + ∆ψ + ǫ∆2ψ + |ψ|2σψ = 0, and in nonisotropic mixed-dispersion NLS equations which model propagation in fiber arrays.

Abstract The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz periodic traveling wave solutions by period-averaging conservation laws of NLS equation. resulting written vector form, which allows one to show that they preserve rotational invariance equation, as well with respect s...