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We derive the new infinite Sasa–Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 × 3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa–Satsuma hierarchy functionals. All these operators are parts of a generalised Sasa–Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multiparameter dynamical system. [ABSTRACT FROM AUTHOR]

We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly. [ABSTRACT FROM AUTHOR]

We present the first four exact rational solutions of the set of rational solutions of the modified Korteweg–de Vries equation. These solutions can be considered as rogue waves of the corresponding equation. Comparison with rogue wave solutions of the nonlinear Schrödinger equation shows a strong analogy between their characteristics, especially for amplitude-to-background ratio. The new solutions may be useful in the theory of rogue waves in shallow water and for light propagation in cubic nonlinear media involving only a few optical cycles. [ABSTRACT FROM AUTHOR]

We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems. [ABSTRACT FROM AUTHOR]

Abstract: Doubly-localised breather solutions of the nonlinear Schrödinger equation (NLS) are considered to be appropriate models to describe rogue waves in water waves as well as in other nonlinear dispersive media such as fibre optics. Within the hierarchy of this type of formations, the Peregrine breather (PB) is the lowest-order rational solution. Higher-order solutions of this kind may be understood as a nonlinear superposition of fundamental Peregrine solutions. These superpositions are nontrivial and admit only a fixed well prescribed number of elementary breathers in each higher-order solution. Here, we report first observation of second-order solution which in reality is a triplet of rogue waves. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

The full text of this article is available in the PDF provided. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Following the first experimental observation of a new mechanism leading to optical rogue wave (RW) formation briefly reported in Lecaplain et al (2012 Phys. Rev. Lett.108 233901), we provide an extensive study of the experimental conditions under which these RWs can be detected. RWs originate from the nonlinear interactions of bunched chaotic pulses that propagate in a fiber laser cavity, and manifest as rare events of high optical intensity. The crucial influence of the electrical detection bandwidth is illustrated. We also clarify the observation of RWs with respect to other pulsating regimes, such as Q-switching instability, that also lead to L-shaped probability distribution functions. [ABSTRACT FROM AUTHOR]

Abstract: We consider a combined model of dissipative solitons that are generated due to the balance between gain and loss of energy as well as to the balance between input and output of matter. The system is governed by the generic complex Ginzburg–Landau equation, which is coupled to a common reaction–diffusion (RD) system. Such a composite dynamical system may describe nerve pulses with a significant part of electromagnetic energy involved. We present examples of such composite dissipative solitons and analyse their internal balances between energy and matter generation and dissipation. [Copyright &y& Elsevier]

We present a multiparameter family of a soliton on a background solution to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the nonlinear Schrödinger equation limit and the limit of rogue wave solutions. [ABSTRACT FROM AUTHOR]

We present experimental results on the study of spatiotemporally localized surface wave events on deep water that can be modeled using the Peregrine breather solution of the nonlinear Schrödinger equation. These are often considered as prototypes of oceanic rogue waves that can focus wave energy into a single wave packet. For small steepness values of the carrier gravity waves the Peregrine breathers are relatively wide, thus providing an excellent agreement between the theory and experimental results. For larger steepnesses the focusing leads to temporally and spatially shorter events. Nevertheless, agreement between measurements and the Peregrine breather theory remains reasonably good, with discrepancies of modulation gradients and spatiotemporal symmetries being tolerable. Lifetimes and travel distances of the spatiotemporally localized wave events determined from the experiment are in good agreement with the theory. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: We have numerically calculated chaotic waves of the focusing nonlinear Schrr̈odinger equation (NLSE), starting with a plane wave modulated by relatively weak random waves. We show that the peaks with highest amplitude of the resulting wave composition (rogue waves) can be described in terms of exact solutions of the NLSE in the form of the collision of Akhmediev breathers. [Copyright &y& Elsevier]