D. J. Kaup

An exact solution for a derivative nonlinear Schrödinger equation

A method of solution for the ’’derivative nonlinear Schrodinger equation’’ iqt=−qxx±i (q*q2)x is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

Conditions for stationary pulse propagation in the strong dispersion management regime

Abstract Using the variational method, we obtain analytical conditions for stationary propagation of a Gaussian pulse in a fibre with strong dispersion management. We consider both the lossless fibre and the one with losses and periodic amplification. The analytical predictions have been checked against direct numerical simulations, and a good agreement between the two has been demonstrated. In particular, we find that in a certain region of parameters, the average dispersion necessary to support the stationary propagation is negative (normal). We also show that under a certain assumption, the variance of the Gordon-Haus timing jitter for the pulse in a strongly dispersion-managed system approximately equals that for the conventional soliton, reduced by an energy enhancement factor. Using our analytical conditions, we obtain an estimate for this factor. In particular, we show that in the presence of losses and periodic amplification, this jitter suppression factor can be made to be as large as that for the lossless case, by properly choosing the segment lengths in the dispersion map.

Embedded solitons: solitary waves in resonance with the linear spectrum

It is commonly held that a necessary condition for the existence of solitons in nonlinear-wave systems is that the soliton’s frequency (spatial or temporal) must not fall into the continuous spectrum of radiation modes. However, this is not always true. We present a new class of codimension-one solitons (i.e., those existing at isolated frequency values) that are embedded into the continuous spectrum. This is possible if the spectrum of the linearized system has (at least) two branches, one corresponding to exponentially localized solutions, and the other to radiation modes. An embedded soliton (ES) is obtained when the latter component exactly vanishes in the solitary-wave’s tail. The paper contains both a survey of recent results obtained by the authors and some new results, the aim being to draw together several different mechanism underlying the existence of ESs. We also consider the distinctive properties of semi-stability of ESs, and moving ESs. Results are presented for four different physical models, including an extended fifth-order KdV equation describing surface waves in inviscid fluids, and three models from nonlinear optics. One of them pertains to a resonant Bragg grating in an optical fiber with a cubic nonlinearity, while two others describe second-harmonic generation (SHG) in the temporal or spatial domain (i.e., respectively, propagating pulses in nonlinear-optical fibers, or stationary patterns in nonlinear planar waveguides). Special attention is paid to the SHG model in the temporal domain for a case of competing quadratic and cubic nonlinearities. In particular, a new result is that when both harmonics have anomalous dispersion, an ES can exist which is, virtually, completely stable.

The lump solutions and the Bäcklund transformation for the three‐dimensional three‐wave resonant interaction

A Backlund transformation is found for the three‐dimensional three‐wave resonant interaction, and from it, N‐lump exact solutions may be constructed. The one‐lump solution is analyzed in detail, and it is shown that it describes such effects as pulse decay, upconversion, and explosive instabilities, all in three dimensions.

The inverse scattering solution for the full three dimensional three- wave resonant interaction

Abstract The direct and inverse scattering solution for the full three dimensional three-wave resonant interaction is solved, and an infinite set of conservation laws are found.

Stability and evolution of solitary waves in perturbed generalized nonlinear Schrödinger equations

In this paper, we study the stability and evolution of solitary waves in perturbed generalized nonlinear Schrodinger (NLS) equations. Our method is based on the completeness of the bounded eigenstates of the associated linear operator in L2 space and a standard multiple-scale perturbation technique. Unlike the adiabatic perturbation method, our method details all instability mechanisms caused by perturbations of such equations and explicitly specifies when such instabilities will occur. In particular, our method uncovers the instability caused by bifurcation of nonzero discrete eigenvalues of the linearization operator. As an example, we consider the perturbed cubic-quintic NLS equation in detail and determine the stability regions of its solitary waves. In the instability region, we also specify where the solitary waves decay, collapse, develop movingfronts, or evolve into a stable spatially localized and temporally periodic state. The generalization of this method to other perturbed nonlinear wave systems is also discussed.

Evolution equations, singular dispersion relations, and moving eigenvalues

Abstract A new approach for finding the class of integrable evolution equations associated with a given eigenvalue problem is developed. The key point to note is that the squares of the eigenfunctions form a natural basis in which to expand the solutions of the evolution equation. Once this step is taken, the class of integrable equations may usually be read off by inspection. Of particular interest are those equations for which the bound state eigenvalues are not invariant but move in a way prescribed by the coefficients of the evolution equation. The corresponding solitons have the property that they retain their identity on collision with other solution components, but this identity is no longer a constant one. The Hamiltonian structure and the causality properties of these systems are also explored.

Interchannel pulse collision in a wavelength-division-multiplexed system with strong dispersion management

We develop a perturbation theory to analytically calculate the effects of complete and incomplete interchannel collisions of Gaussian pulses in a wavelength-division-multiplexed system with strong dispersion management. We show that, for complete collisions, the collision-induced frequency shift of a Gaussian pulse is negligible, whereas its position shift is significant and can be found in a simple analytical form. For strong dispersion management we find that incomplete collisions can be neglected, whereas for dispersion management of moderate strength the contribution of the incomplete collisions can be significant. The analytical predictions are in satisfactory agreement with numerical results. We also give an estimate of the limit imposed on the transmission distance by such collisions.

The method of solution for stimulated Raman scattering and two-photon propagation

Abstract A uniquely different inverse scattering transform is presented for solving stimulated Raman Scattering and two-photon propagation. This transform is based on the generalized Zakharov-Shabat problem, and in certain cases one will not have r = ±q ∗ , bound-state eigenvalues may move, and the space dependence of the scattering data may itself depend on a second inverse scattering transform.

Influence of the Raman effect on dispersion-managed solitons and their interchannel collisions

We calculate the self-frequency shift experienced by a soliton in a dispersion-managed fiber that is due to the Raman effect, as well as the energy and frequency shifts that result from a collision of such solitons with different wavelengths. We find that dispersion management suppresses both types of frequency shift but does not significantly affect the energy shift that is accumulated over a large propagation distance. The latter shift may represent a potential problem for wavelength-division-multiplexed systems with several gigabits per second in a single channel.