We present a method for finding the hierarchy of rational solutions of the self-focusing nonlinear Schrödinger equation and present explicit forms for these solutions from first to fourth order. We also explain their relation to the highest amplitude part of a field that starts with a plane wave perturbed by random small amplitude radiation waves. Our work can elucidate the appearance of rogue waves in the deep ocean and can be applied to the observation of rogue light pulse waves in optical fibers.

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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.

The Hirota equation is a modified nonlinear Schrödinger equation (NLSE) that takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity. In describing wave propagation in the ocean and optical fibers, it can be viewed as an approximation which is more accurate than the NLSE. We have modified the Darboux transformation technique to show how to construct the hierarchy of rational solutions of the Hirota equation. We present explicit forms for the two lower-order solutions. Each one is a regular (nonsingular) rational solution with a single maximum that can describe a rogue wave in this model. Numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution.

Rare events of extremely high optical intensity are experimentally recorded at the output of a mode-locked fiber laser that operates in a strongly dissipative regime of chaotic multiple-pulse generation. The probability distribution of these intensity fluctuations, which highly depend on the cavity parameters, features a long-tailed distribution. Recorded intensity fluctuations result from the ceaseless relative motion and nonlinear interaction of pulses within a temporally localized multisoliton phase.

The complex Ginzburg-Landau equation (CGLE) is a standard model for pulse generation in mode-locked lasers with fast saturable absorbers. We have found complicated pulsating behavior of solitons of the CGLE and regions of their existence in the five-dimensional parameter space. We have found zero-velocity, moving and exploding pulsating localized structures, period doubling (PD) of pulsations and the sequence of PD bifurcations. We have also found chaotic pulsating solitons. We have plotted regions of parameters of the CGLE where pulsating solutions exist. We also demonstrate the coexistence (bi- and multistability) of different types of pulsating solutions in certain regions of the parameter space of the CGLE.

We present three novel pulsating solutions of the cubic-quintic complex Ginzburg-Landau equation. They describe some complicated pulsating behavior of solitons in dissipative systems. We study their main features and the regions of parameter space where they exist. PACS numbers: 42.65.Tg, 05.45.Yv, 05.70.Ln, 47.20.Ky A soliton is a self-localized solution of a nonlinear partial differential equation describing the evolution of a nonlinear dynamical system with an infinite number of degrees of freedom. Solitons are usually attributed to integrable systems. In this instance, solitons remain unchanged during interactions, apart from a phase shift. They can be viewed as "modes" of the system, and, along with radiation modes, they can be used to solve initial value problems, using a nonlinear superposition of modes [1]. The two main features of solitons in integrable systems which are of interest to us are, first, that they are one-(or few-)parameter families and, second, that the superposition of solitons with zero velocity produces a pulsating solution [2], which is sometimes called a "breather."Reductions to integrable systems are extreme simplifications of the complex systems existing in nature. They can be considered as a subclass of the more general Hamiltonian systems [3]. Indeed, such a simplification allows us to analyze the systems quantitatively and to completely understand the behavior of the solitons. Solitons in Hamiltonian (but nonintegrable) systems can also be regarded as nonlinear modes, but in the sense that they allow us to describe the behavior of systems with an infinite number of degrees of freedom in terms of a few variables, thus allowing us to effectively reduce the number of degrees of freedom. Solitons in these systems collide inelastically and interact with radiation waves, thus showing that they are qualitatively different from those in integrable systems. However, as in the integrable case, the solitons are still a one-(or few-)parameter family of solutions. Another interesting property of Hamiltonian systems is that there are no pulsating solutions. If the system is near integrable, then the two-soliton solutions of the nonlinear Schrödinger equation (NLSE) which are initially excited will gradually split into two solitons or transform into a single soliton solution, depending on the type of the perturbation [4]. If the system is far from integrable, pulsations may exist if a single soliton solution is excited with a perturbation; however, they die out, so that the pulse gradually converges to a stationary soliton.Dissipative systems are more complicated than Hamiltonian ones in the sense that, in addition to nonlinearity and dispersion, they include energy exchange with external sources. The generic equation which describes dissipative systems above the point of bifurcation is the complex Ginzburg-Landau equation (CGLE) [5,6]. A review of experiments described by the CGLE is given in [7]. In optics, it describes laser systems [8][9][10][11], soliton transmission lines [12], nonline...

We present novel stable solutions which are soliton pairs and trains of the 1D complex GinzburgLandau equation (CGLE), and analyze them. We propose that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations. We present a two-dimensional phase plane ("interaction plane") for analyzing the stability properties and general dynamics of two-soliton solutions of the CGLE. [S0031-9007(97) [3], and spatially extended nonequilibrium systems [4,5]. In optics, it is useful in analyzing optical transmission lines [6,7], passively mode-locked fiber lasers [8,9], and spatial optical solitons [10]. In each case, the problem of the interaction of two, individually stable, juxtaposed elementary coherent structures (i.e., solitons) is crucial for understanding the general behavior of the system [11,12].Stable pulse-like solutions of the quintic CGLE have been found by Thual and Fauve [13]. Minimal requirements for their stability have been obtained in [14]. In the conservative limit, these solutions can be considered as perturbations of the nonlinear Schrödinger equation (NLSE) solitons [15]. The continuous transition of these solutions from the conservative limit to the gradient limit in the parameter space of the CGLE has been studied in Ref. [16]. Although the dynamical properties of these pulselike solutions, their collisions and interactions are different from those of solitons of integrable systems, they have been called "solitons" in a number of works. We follow this tradition, and also call them "solitons" or "soliton solutions."For the nonlinear Schrödinger equation, two solitons have zero binding energy. Hence, any nonlinear superposition of two solitons is neutrally stable, and can be made unstable with a very small perturbation. On the other hand, for the NLSE, there is no stationary solution in the form of two solitons with equal amplitudes and velocities and with a fixed separation. Frequently, real systems are not described by integrable equations (e.g., the NLSE), but by Hamiltonian generalizations of the NLSE. For these systems, the interaction between the pulses becomes inelastic, so that two-soliton solutions of the perturbed NLSE (when they exist) are unstable due to the energy exchange between the pulses [17]. The situation changes completely for nonconservative systems. Each soliton then has its own internal balance of energy which maintains its constant amplitude. Fixing the amplitudes effectively reduces the number of degrees of freedom in the system of two solitons and can make it stable. Bound states of two solitons in these systems were first analyzed by Malomed [18]. Using standard perturbation analysis for soliton interaction, he showed that stationary solutions in the form of bound states of two solitons, which are in-phase or out-of-phase, may exist. We also confirm that they do exist. However, careful numerical 0031-9007͞97͞79(21)͞4047(5)$10.00