[en] Full text: We investigate modulational instability (Ml) for several discrete cubic-quintic complex Ginzburg-Landau (CQCGL) models and compare with Ml for the continuous CQCGL equation. The Ml of a discrete cubic-quintic complex Ginzburg-Landau model, which was recently proposed by us, depends on the wave number of a plane wave. We also study, analytically and numerically, bright solitons of the discrete CQCGL model which may be generated by Ml. Copyright (2005) Australian Institute of Physics

[en] We present an infinitely-extended KdV equation that contains an infinite number of arbitrary real coefficients controlling higher-order terms in the extended evolution equation. The higher-order terms are chosen in a way that maintains the integrability of the whole equation. Another significant step in this work is that this extended equation admits complex-valued solutions. This generalization allows us to consider both solitons and rogue waves in the form of rational solutions of this equation. Special choices of the arbitrary coefficients lead to particular cases—the basic KdV and its higher-order versions. Using the extended KdV, instead of the basic one, may improve the accuracy of the description of rogue waves in shallow water. (paper)

[en] We provide a simple technique for finding the correspondence between the solutions of Ablowitz–Ladik and nonlinear Schrödinger equations. Even though they belong to different classes, in that one is continuous and one is discrete, there are matching solutions. This fact allows us to discern common features and obtain solutions of the continuous equation from solutions of the discrete equation. We consider several examples. We provide tables, with selected solutions, which allow us to easily match the pairs of solutions. We show that our technique can be extended to the case of coupled Ablowitz–Ladik and nonlinear Schrödinger (i.e. Manakov) equations. We provide some new solutions. (paper)

[en] The spectra of the inverse scattering technique (IST) play a crucial role in the physics of nonlinear phenomena. They define the long term evolution of dynamical systems. We present the IST spectral portraits for the extensive three-parameter families of the first order doubly periodic solutions of the nonlinear Schrödinger equation that cover a wide range of physical phenomena such as modulation instability, rogue waves and many other problems with periodic boundary conditions. We relate these spectral portraits with the parameters of the family. We show that there are two qualitatively different types of spectral portraits. A-type spectra consist of two continuous bands: a band of purely imaginary eigenvalues within the interval $[-<!--\; ???\; -->i,i]$ and a finite band of complex eigenvalues. On the contrary, B-type spectra possess only continuous bands of imaginary eigenvalues all located within the interval $[-<!--\; ???\; -->i,i]$ and separated by a finite band gap. A physical interpretation of these results is given. (paper)

[en] We propose initial conditions that could facilitate the excitation of rogue waves. Understanding the initial conditions that foster rogue waves could be useful both in attempts to avoid them by seafarers and in generating highly energetic pulses in optical fibers.

[en] We demonstrate that a generalized nonlinear Schroedinger equation (NSE), which includes dispersion of the intensity-dependent group velocity, allows for exact solitary solutions. In the limit of a long pulse duration, these solutions naturally converge to a fundamental soliton of the standard NSE. In particular, the peak pulse intensity times squared pulse duration is constant. For short durations, this scaling gets violated and a cusp of the envelope may be formed. The limiting singular solution determines then the shortest possible pulse duration and the largest possible peak power. We obtain these parameters explicitly in terms of the parameters of the generalized NSE.

[en] We present the most general multi-parameter family of a soliton on background solutions to the Sasa–Satsuma equation. These solutions contain a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits nontrivial limiting cases, such as rogue waves and classical solitons, that are considered in detail. (paper)

[en] We suggest that crucial effect on Bose-Einstein condensation in systems with attractive potential is three-body interaction. We investigate stationary solutions of the Gross- Pitaevskii equation with negative scattering length and a higher-order stabilising term in presence of an external parabolic potential. Stability properties of the condensate are similar to those for thermodynamic systems in statistical physics which have first order phase transitions. We have shown that there are three possible type of stationary solutions corresponding to stable, metastable and unstable phases. Results are discussed in relation to recently observed ⁷Li condensate. (Copyright (1998) World Scientific Publishing Co. Pte. Ltd)

[en] Using the method of moments for dissipative optical solitons, we show that there are two disjoint sets of fixed points. These correspond to stationary solitons of the complex cubic-quintic Ginzburg-Landau equation with concave and convex phase profiles respectively. Numerical simulations confirm the predictions of the method of moments for the existence of two types of solutions which we call solitons and antisolitons. Their characteristics are distinctly different

[en] We present the lowest order rogue wave solution of the Sasa–Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of the Peregrine solution appears when the extension parameter of the SSE is reduced to zero. -- Highlights: ► For the first time a rogue wave solution of the Sasa–Satsuma equation is presented. ► It is given by fourth order polynomials rather than second order in the NLSE limit. ► SSE rogue waves are illustrated for various parameters, as deviation from the NLSE. ► A non-singular NLSE limit of a solution of the SSE has been demonstrated.