No abstract

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.

No abstract

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.

Using the Darboux transformation technique and numerical simulations, we study the hierarchy of rational solutions of the nonlinear Schrödinger equation that can be considered as higher order rogue waves in this model. This analysis reveals the existence of rogue wave clusters with a high level of symmetry in the (x,t) plane. These structures arise naturally when the shifts in the Darboux scheme are taken to be eigenvalue dependent. We have found single-shell structures where a central higher order rogue wave is surrounded by a ring of first order peaks on the (x,t) plane.

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...

No abstract