Sasa-Satsuma equation: soliton on a background and its limiting cases

Dynamics of perturbations at the critical points between modulation instability and stability regimes

We study numerically the evolutions of perturbations at critical points between modulational instability and stability regimes. It is demonstrated that W-shaped solitons and rogue waves can be both excited from weak resonant perturbations at the critical points. The rogue wave excitation at the critical points indicates that rogue wave comes from modulation instability with resonant perturbations, even when the baseband modulational instability is absent. The perturbation differences for generating W-shaped solitons and rogue waves are discussed in detail. These results can be used to generate W-shaped solitons and rogue waves controllably from weak perturbations.

Excitation conditions of several fundamental nonlinear waves on continuous-wave background

We study the excitation conditions of antidark solitons and nonrational W-shaped solitons in a nonlinear fiber with both third-order and fourth-order effects. We show that the relative phase can be used to distinguish antidark solitons and nonrational W-shaped solitons. The excitation conditions of these well-known fundamental nonlinear waves (on a continuous-wave background) can be clarified clearly by the relative phase and three previously reported parameters (background frequency, perturbation frequency, and perturbation energy). Moreover, the numerical simulations from the nonideal initial states also support these theoretical results. These results provide an important complement for the studies on relationship between modulation instability and nonlinear wave excitations, and are helpful for controllable nonlinear excitations in experiments.

Dark-bright solitons and semirational rogue waves for the coupled Sasa-Satsuma equations

In this paper, we investigate the coupled Sasa-Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Darboux-dressing transformation is applied to obtain the dark-bright soliton and semirational rogue-wave solutions. Dark-bright one solitons with the single-hump, double-hump, and even breather-like structures are presented. Interactions between the double-peak breather and different kinds of dark-bright solitons are studied. We show that the double-peak (or single-peak) rogue wave can coexist and interact with different kinds of dark-bright solitons. Coexistence of the solitons with different velocities and rogue waves is also found. Numerical stabilities of the dark-bright solitons and semirational rogue waves are exhibited. It is expected that those localized wave phenomena can be experimentally observed and have potential applications.

Weak-light vector rogue waves, breathers, and their Stern-Gerlach deflection via electromagnetically induced transparency

We propose a scheme for generating and manipulating vector (or two-component) optical rogue waves using Akhmediev and Kuznetsov-Ma breathers in a coherent atomic system with an M-type five-level configuration via electromagnetically induced transparency (EIT). We show that the propagation velocity of these nonlinear excitations can be reduced to 10^-4c and their generation power can be lowered to microwatts. We also show that the motion trajectories of the two polarization components in these excitations can be deflected significantly by using a transversal gradient magnetic field, similar to the Stern-Gerlach effect of an atomic beam. We find that the deflection angle can reach to 10^-4 radian within the propagation distance of only several centimeters; at variance with the atomic Stern-Gerlach effect, the deflection angle can be made different for different polarization components and may be actively adjusted in a controllable way. The results obtained may have promising applications, including the precise measurement of gradient magnetic fields.

Semi-rational solutions of the third-type Davey-Stewartson equation

General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers, and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers, and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.

Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects

We study the correspondence between modulational instability and types of fundamental nonlinear excitation in a nonlinear fiber with both third-order and fourth-order effects. Some soliton excitations are obtained in the modulational instability regime which have not been found in nonlinear fibers with second-order effects and third-order effects. Explicit analysis suggests that the existence of solitons is related to the modulation stability circle in the modulation instability regime, and they just exist in the modulational instability regime outside of the modulational stability circle. It should be emphasized that the solitons exist only with two special profiles on a continuous-wave background at a certain frequency. The evolution stability of the solitons is tested numerically by adding some noise to initial states, which indicates that they are robust against perturbations even in the modulation instability regime. Further analysis indicates that solitons in the modulational instability regime are caused by fourth-order effects.

Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov's theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime

We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.

W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation

We study rational solutions of continuous wave backgrounds with the critical frequencies of the Sasa-Satsuma equation, which can be used to describe the evolution of the optical field in a nonlinear fiber with some high-order effects. We find a striking dynamical process that two W-shaped solitons are generated from a weak modulation signal on the continuous wave backgrounds. This provides a possible way to obtain stable high-intensity pulses from a low-intensity continuous wave background. The process involves both modulational instability and modulational stability regimes, in contrast to the rogue waves and W-shaped solitons reported before which involve modulational instability and stability, respectively. Furthermore, we present a phase diagram on a modulational instability spectrum plane for the fundamental nonlinear localized waves obtained already in the Sasa-Satsuma equation. The interactions between different types of nonlinear localized waves are discussed based on the phase diagram.

State transition induced by higher-order effects and background frequency

The state transition between the Peregrine rogue wave and W-shaped traveling wave induced by higher-order effects and background frequency is studied. We find that this intriguing transition, described by an exact explicit rational solution, is consistent with the modulation instability (MI) analysis that involves a MI region and a stability region in a low perturbation frequency region. In particular, the link between the MI growth rate and the transition characteristic analytically demonstrates that the localization characteristic of transition is positively associated with the reciprocal of the zero-frequency growth rate. Furthermore, we investigate the case for nonlinear interplay of multilocalized waves. It is interesting that the interaction of second-order waves in the stability region features a line structure rather than an elastic interaction between two W-shaped traveling waves.

Coexisting rogue waves within the (2+1)-component long-wave-short-wave resonance

The coexistence of two different types of fundamental rogue waves is unveiled, based on the coupled equations describing the (2+1)-component long-wave-short-wave resonance. For a wide range of asymptotic background fields, each family of three rogue wave components can be triggered by using a slight deterministic alteration to the otherwise identical background field. The ability to trigger markedly different rogue wave profiles from similar initial conditions is confirmed by numerical simulations. This remarkable feature, which is absent in the scalar nonlinear Schrödinger equation, is attributed to the specific three-wave interaction process and may be universal for a variety of multicomponent wave dynamics spanning from oceanography to nonlinear optics.

Rogue-wave pattern transition induced by relative frequency

We revisit a rogue wave in a two-mode nonlinear fiber whose dynamics is described by two-component coupled nonlinear Schrödinger equations. The relative frequency between two modes can induce different rogue wave patterns transition. In particular, we find a four-petaled flower structure rogue wave can exist in the two-mode coupled system, which possesses an asymmetric spectrum distribution. Furthermore, spectrum analysis is performed on these different type rogue waves, and the spectrum relations between them are discussed. We demonstrate qualitatively that different modulation instability gain distribution can induce different rogue wave excitation patterns. These results would deepen our understanding of rogue wave dynamics in complex systems.

Vector rogue waves and baseband modulation instability in the defocusing regime

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.

Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology.

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

We investigate the solution in rational form for the Sasa-Satsuma equation on a continuous background which describes a nonlinear fiber system with higher-order effects including the third-order dispersion, Kerr dispersion, and stimulated inelastic scattering. The W-shaped soliton in the system is obtained analytically. It is found that the height of hump for the soliton increases with decreasing the background frequency in certain parameter regime. The maximum height of the soliton can be three times the background's height and the corresponding profile is identical with the one for the well-known eye-shaped rogue wave with maximum peak. The numerical simulations indicate that the W-shaped soliton is stable with small perturbations. Particularly, we show that the W-shaped soliton corresponds to a stable supercontinuum pulse by performing exact spectrum analysis.

Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance

Exact explicit rogue-wave solutions of intricate structures are presented for the long-wave-short-wave resonance equation. These vector parametric solutions feature coupled dark- and bright-field counterparts of the Peregrine soliton. Numerical simulations show the robustness of dark and bright rogue waves in spite of the onset of modulational instability. Dark fields originate from the complex interplay between anomalous dispersion and the nonlinearity driven by the coupled long wave. This unusual mechanism, not available in scalar nonlinear wave equation models, can provide a route to the experimental realization of dark rogue waves in, for instance, negative index media or with capillary-gravity waves.

Twisted rogue-wave pairs in the Sasa-Satsuma equation

Exact explicit rogue wave solutions of the Sasa-Satsuma equation are obtained by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the rogue wave can exhibit an intriguing twisted rogue-wave pair that involves four well-defined zero-amplitude points. This exotic structure may enrich our understanding on the nature of rogue waves.

Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation

In this paper, using the Darboux transformation, we demonstrate the generation of first-order breather and higher-order rogue waves from a generalized nonlinear Schrödinger equation with several higher-order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fiber. The same nonlinear evolution equation can also describe the soliton-type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ(1), denoting the strength of the higher-order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ(1) are discussed in detail.