On the well-posedness of the Eckhaus equation

The Eckhaus equation, which is a nonlinear Schrödinger type equation that can be linearized to the free linear Schrödinger equation, is considered. A linearized analysis of the nonlinear problem indicates that the periodic boundary value problem is ill-posed. The exact solution also demonstrates the ill-posedness, even though the L² norm of the solution is a constant of the motion. The ill-posedness disappears on the infinite line with square integrable data, but the solitonic solutions, which are not square integrable, are seriously unstable.     

Solitary waves, steepening and initial collapse in the Maxwell–Lorentz system

We present a numerical study of Maxwell’s equations in nonlinear dispersive optical media describing propagation of pulses in one Cartesian space dimension. Dispersion and nonlinearity are accounted for by a linear Lorentz model and an instantaneous Kerr nonlinearity, respectively. The dispersion relation reveals various asymptotic regimes such as Schrödinger and KdV branches. Existence of soliton-type solutions in the Schrödinger regime and light bullets containing few optical cycles together with dark solitons are illustrated numerically. Envelope collapse regimes of the Schrödinger equation are compared to the full system and an arrest mechanism is clearly identified when the spectral width of the initial pulse broadens beyond the applicability of the asymptotic behavior. We show that beyond a certain threshold the carrier wave steepens into an infinite gradient similarly to the canonical Majda–Rosales weakly dispersive system. The weak dispersion in general cannot prevent the wave breaking with instantaneous or delayed nonlinearities.     

Influence of high-order dispersion on self-focusing. II. Numerical investigation

The self-focusing described by the nonlinear Schrödinger equation with a higher-order derivative term is investigated numerically. We demonstrate that, depending on the sign before this term, it may lead in the final stage either to defocusing or to a steady homogeneous wave beam. In both cases the transition to the final state is shown to be oscillatory.     

On the non-integrability of the spherically-symmetric nonlinear Schrödinger equation in the Langmuir collapse problem

In the present paper we give some analytic considerations of the spherically-symmetric nonlinear Schrödinger equation arising in the Langmuir collapse problem. We will make a systematic exploration of the various group symmetries of thie equation and show that the latter possesses only a three-parameter symmetry group. We will then give a variational formulation of this equation and use the three-parameter symmetry group to show that the equation in question possesses apparently only two polynomial conservation laws. Finally, we will make a study of the singularity structure of the present equation and show that it does not seem to possess the Painlevé property. The conclusion is that the spherically-symmetric nonlinear Schrödinger equation in question is apparently not integrable.     

Influence of high-order dispersion on self-focusing. I. Qualitative investigation

The self-focusing, described by the nonlinear Schrödinger equation with a higher-order derivative term, appearing from dispersive corrections, is considered. A qualitative investigation shows that this term, even if it is small, may play an important role in the final state of the self-focusing. Depending on the sign of a coefficient before this term, it may lead either to a tunneling of the self-trapped radiation, which finally results in defocusing, or to a steady homogeneous wave beam.     

Localized solutions of (N+1)-dimensional evolution equations

We give a method of constructing nonlinear Schrödinger equations which are similar to the Davey-Stewartson equation, but in N+1 dimensions, and posses, by construction, solutions describing interacting soliton-like objects.     

Hydrodynamic symmetries for the Whitham equations for the sine-Gordon equation

The effective structure of the hydrodynamic symmetries for the Whitham equations, derived for simplest one-phase solutions of the sine-Gordon equation, is presented. This structure is analogous to the one, that was obtained for the Whitham equations for the KdV equation and the nonlinear Schrödinger equation (NSE). The hydrodynamic symmetries for the gas dynamic equations are described.     

Quantum inverse scattering method and the Heisenberg ferromagnet

The quantum version of the inverse scattering method is formulated for the Heisenberg ferromagnet. The elements of the transition matrix for the corresponding auxiliary linear problem are the quantum analogue of the action-angle variables. In a continuous limit the model is shown to yield the quantum nonlinear Schrödinger equation.     

Spatiotemporal solitons in inhomogeneous nonlinear media

We investigate the possibility of forming spatiotemporal solitons (optical bullets) in inhomogeneous, dispersive nonlinear media using a graded-index Kerr medium as an example. We use a variational approach to solve the multidimensional, inhomogeneous, nonlinear Schrödinger equation and show that spatiotemporal solitons can be stabilized under certain conditions. We verify their existence by means of a full numerical analysis and show that such solitons should be observable experimentally.     

Apodized chirped fibre Bragg gratings for dispersion compensation in a 10 Gbit/s IM-DD semiconductor laser system

Different fibre Bragg grating dispersion compensation schemes are studied for a directly modulated 1550 nm single-mode semiconductor laser signal through a standard nonlinear fibre link. The laser diode is simulated by its stochastic rate equations, while the nonlinear Schrödinger equation is used to simulate the propagation. The optimum length for dispersion compensation after transmission through 100 km SSM fibre is studied. Pulses with a FWHM of the order of 65 ps with any linewidth-enhancement factor are reconstructed using pre-compensation or post-compensation with an apodized 5.75 cm chirped fibre Bragg grating.     

Kink-like solution in U(1, 1) symmetry bose-condensate

A peculiar single-soliton solution of the vector nonlinear Schrödinger equation with noncompact U(1, 1) isogroup under nontrivial boundary conditions is found via the inverse scattering method. The existence of phase transitions within the model of gas mixtures through the power constant is discussed as well.     

On the fluid approximation to a nonlinear Schrödinger equation

We present a heuristic proof that the nonlinear Schrödinger equation (NLS) - in 2 + 1 dimensions has a family of solutions which can be well approximated by a collection of point vortices for a planar incompressible fluid. The novelty of our approach is that we begin with a representation of the NLS as a compressible perturbation of Euler's equations for hydrodynamics.     

Optimized Rayleigh–Schrödinger expansion of the effective potential

An optimized Rayleigh–Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ⁴ field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.     

The one-dimensional soliton state of Bose-Einstein condensate

Under certain conditions, the Bose-Einstein condensate confined in a two-dimensional harmonic trap is associated with a one-dimensional nonlinear Schrödinger equation and a soliton solution. The influence of the initial condition on the solution is discussed. The critical initial number of condensed atoms to maintain a soliton is evaluated. In the Li case, this number is of the order of thousand. The energy per particle of the solitons is also calculated.     

Stability of fundamental solitons of coupled nonlinear Schrödinger equations

We present analytical stability criteria for the fundamental solitons of two coupled nonlinear Schrödinger equations. The derived stability criteria are applied to the coupled fundamental soliton states in isotropic nonlinear media, in birefringent fibres and in nonlinear couplers. The predictions from the analytical stability criteria are consistent with numerical results.     

Ultrabroad bandwidth solitons in fiber amplifier

The amplification of an ultrabroad bandwidth pulse in an Erbium-doped fiber amplifier in the case where the spectral bandwidth of pulse is comparable to the gain bandwidth of the amplifier is investigated. Based on the numerical simulation of a modified nonlinear Schrödinger equation, it is shown that the amplifier limits the efficiency of amplification. However, the limitation can be minimized by choosing the corresponding optimal fiber length for different gain bandwidths. In addition, the effect of gain bandwidth on the amplification of a soliton with initial chirp depends on the sign of the chirp.     

A note on the mathematical structure of the optical cubic-quintic Schrödinger equation

The mathematical structure of the optical cubic-quintic Schrödinger equation is investigated in a special way by considering a potential depending upon the modulus of the wave-functions involved. In this context, an associated operator is defined.     

On the time dependence of a degenerate two-level system interacting with a pulse of radiation

The solution of the time-dependent Schrödinger equation is presented for a degenerate two-level system interacting with a pulse of radiation. The sinusoidal limit of the solution is used to illustrate the importance of averaging transition probabilities over the phase of an applied sinusoidal field.     

Rapid numerical difference recurrent formula of nonlinear Schrödinger equation and its application

In this paper, a rapid numerical difference recurrent formula, in which it has been taken that the chromatic dispersion and the nonlinearity act together along each fiber segment, is established in the time domain by applying a Maclaurin expansion to the differential form of the nonlinear Schrödinger equation (NLSE) in the frequency domain. The calculated results by using the established formula are contrasted with the known analytical results and the results of the split-step Fourier method (SSFM) and indicated that the rapid numerical difference recurrent formula is very accurate and more reasonable because it abandons an assumption that the dispersive and nonlinear effects can be assumed to act independently as the optical field propagates over each fiber segment. It has been concluded that the established formula in this paper is a scientific, reasonable and effective numerical method for the study of light pulse propagation in a nonlinear optical medium.     

Wave functions and low-order density matrices for a class of two-electron ‘artificial atoms’ embracing Hookean and Moshinsky models

A class of model two-electron ‘artificial atoms’ is proposed which embraces both Hookean and Moshinsky models. Particle densities and spinless first-order density matrices are obtained for this class of models. These quantities and the interacting system kinetic energy can be calculated using the ground-state solution of an explicit single-particle radial Schrödinger equation.