Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

Rogue waves in Lugiato-Lefever equation with variable coefficients

In this paper, we theoretically investigate the generation of optical rogue waves from a Lugiato-Lefever equation with variable coefficients by using the nonlinear Schrödinger equation-based constructive method. Exact explicit rogue-wave solutions of the Lugiato-Lefever equation with constant dispersion, detuning and dissipation are derived and presented. The bright rogue wave, intermediate rogue wave and the dark rogue wave are obtained by changing the value of one parameter in the exact explicit solutions corresponding to the external pump power of a continuous-wave laser.     

Exciting rogue waves,breathers, and solitons in coherent atomic media

We propose a scheme that excites rogue waves via electromagnetically induced transparency(EIT), which can also excite breathers and solitons. The system is a resonant Λ-type atomic ensemble. Under EIT conditions, the envelope equation of the probe field can be reduced to several different models, such as the saturable nonlinear Schr?dinger equation(SNLSE), and SNLSE with the trapping potential provided by a far-detuned laser field or a magnetic field. In this scheme, rogue waves can be generated by different initial pulses, such as the Gaussian wave with(or without) the uniform background. The scheme can be used to obtain rogue waves,breathers and solitons. We show the existence regions of rogue waves, breathers, and solitons as the function of the amplitude and width of the initial pulse. The novelty of our paper is that, we not only show rogue waves in the integrable system numerically, but also present the method to generate and control rogue waves in the non-integrable system.     

Rogue-wave pair and dark-bright-rogue wave solutions of the coupled Hirota equations

Novel explicit rogue wave solutions of the coupled Hirota equations are obtained by using the Darboux transformation.In contrast to the fundamental Peregrine solitons and dark rogue waves, we present an interesting rogue-wave pair that involves four zero-amplitude holes for the coupled Hirota equations. It is significant that the corresponding expressions of the rogue-wave pair solutions contain polynomials of the fourth order rather than the second order. Moreover, dark-brightrogue wave solutions of the coupled Hirota equations are given, and interactions between Peregrine solitons and dark-bright solitons are analyzed. The results further reveal the dynamical properties of rogue waves for the coupled Hirota equations.     

Rogue waves in exciton–polariton condensates under homogeneous pumping

We predict the emergence of rogue wave solutions in one-dimensional exciton–polariton condensates under homogeneous pumping. We model the condensate dynamics in a microwire using the dissipative Gross–Pitaevskii equation for the polariton field, with considers attractive nonlinearity, coupled to the rate equation of the excitonic reservoir density. With the help of the direct ansatz method and similarity transformation, deformed first order rogue wave solutions are constructed and its dynamics analyzed. We show that the deformed rogue wave has a curved background controlled by the pump power and the strength of the nonlinear interaction of polaritons. Moreover, the maximal population of the polaritons appears where high energy of rogue wave is concentrated.     

Optical Rogue Wave Excitation and Modulation on a Bright Soliton Background

We study rogue waves in an inhomogeneous nonlinear optical fiber with variable coefficients.An exact rogue wave solution that describes rogue wave excitation and modulation on a bright soliton pulse is obtained.Special properties of rogue waves on the bright soliton,such as the trajectory and spectrum,are analyzed in detail.In particular,our analytical results suggest a way of sustaining the peak shape of rogue waves on the soliton background by choosing an appropriate dispersion parameter.     

signal of NMSSM at the LHC

For the Benjamin Ono equation, the Hirota bilinear method and long wave limit method are applied to obtain the breathers and the rogue wave solutions. Bright and dark rogue waves exist in the Benjamin Ono equation, and their typical dynamics are analysed and illustrated. The semirational solutions possessing rogue waves and solitons are also obtained, and demonstrated by the three-dimensional figures. Furthermore, the hybrid of rogue wave and breather solutions are also found in the Benjamin Ono equation.     

Nonautonomous Dark Solitons and Rogue Waves in a Graded-Index Grating Waveguide

We analytically present a family of nonautonomous dark solitons and rogue waves in a planar graded-index grating waveguide with an additional long-period grating.The dark solitons whose dynamics described by the explicit expressions such as the valley,background and wave central position are investigated.We find that dark soliton's depth and the long-period grating have effects on soliton's wave central position;the gain or loss term affects directly both the background and valley of the soliton.For rogue waves,it is reported that one can modulate the distribution of the light intensity by adjusting the parameters of the long-period grating.Additionally,more rogue waves with different evolution behaviors in this special waveguide are demonstrated clearly.     

Rogue waves of a (3+1)-dimensional BKP equation

We investigate certain rogue waves of a (3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method. We obtain semi-rational solutions in the determinant form, which contain two special interactions: (i) one lump develops from a kink soliton and then fuses into the other kink one; (ii) a line rogue wave arises from the segment between two kink solitons and then disappears quickly. We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time, which performs like a rogue wave. Furthermore, the higher-order semi-rational solutions describing the interaction between two lumps (one line rogue wave) and three kink solitons are presented.     

Generalized Darboux Transformations,Rogue Waves,and Modulation Instability for the Coherently Coupled Nonlinear Schrödinger Equations in Nonlinear Optics

Nonlinear optics plays a central role in the advancement of optical science and laser‐based technologies. The second‐order rogue‐wave solutions and modulation instability for the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling in nonlinear optics are reported in this paper. Generalized Darboux transformations for such coupled equations are derived, with which the second‐order rational solutions for the purpose of modelling the rogue waves are obtained. With respect to the slowly‐varying complex amplitudes of two interacting optical modes, it is observed that 1) number of valleys of the second‐order rogue waves increases and peak value of the second‐order rogue wave decreases first and then increases; 2) single‐hump second‐order rogue wave turns into the double‐hump second‐order rogue wave; 3) single‐hump bright second‐order rogue wave turns into the dark second‐order rogue wave and finally becomes the three‐hump bright second‐order rogue wave. Meanwhile, baseband modulation instability through the linear stability analysis is seen.     

Interactions of Localized Wave Structures on Periodic Backgrounds for the Coupled Lakshmanan–Porsezian–Daniel Equations in Birefringent Optical Fibers

Nonlinear waves on periodic backgrounds play an important role in physical systems. In this study, nonlinear waves that include solitons, breathers, rogue waves, and semi-rational solutions on periodic backgrounds for the coupled Lakshmanan-Porsezian-Daniel equations are investigated. Moreover, the interactions between different types of nonlinear waves are examined and their dynamic behaviors are studied. In particular, it is observed that bright-dark rogue waves interact with bright-dark breathers or solitons on periodic backgrounds, four-petaled breathers interact with two eye-shaped breathers on periodic backgrounds, and a four-petal rogue wave interplays with a rogue wave on periodic backgrounds. Furthermore, it is found that the value of the parameter γ₃ affects the weak and strong interactions of these nonlinear waves. These results may be useful in the study of nonlinear wave dynamics in coupled nonlinear wave models.     

Deep learning neural networks for the third-order nonlinear Schrödinger equation: bright solitons,breathers, and rogue waves

The dimensionless third-order nonlinear Schrödinger equation (alias the Hirota equation) is investigated via deep leaning neural networks. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g. bright soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and perturbated (a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of bright solitons.     

Bright‐Dark Rogue Waves

During the last two decades, revealing mechanisms of origin waves with anomalous amplitude (rogue waves) have been in the focus of researchers from different fields ranging from oceanography to laser physics. Mode‐locked lasers, as a test bed system, provide a unique opportunity to collect more data on rogue waves in the form of random pulses (soliton rain) and to clarify the mechanisms of rogue‐wave emergence caused by soliton–soliton and soliton–dispersive wave interactions. Here, for the first time, for an Er‐doped mode‐locked laser, a new type of vector rogue waves is demonstrated experimentally and theoretically, which is driven by desynchronization of the orthogonal linear states of polarization, so leading to output power oscillations in the form of anomalous spikes‐dips (bright‐dark rogue waves). The results can pave the way to unlocking the universal nature of the origin of rogue waves and thus can be of interest to the broad scientific community.     

Rogue waves in a multistable system

Clear evidence of rogue waves in a multistable system is revealed by experiments with an erbium-doped fiber laser driven by harmonic pump modulation. The mechanism for the rogue wave formation lies in the interplay of stochastic processes with multistable deterministic dynamics. Low-frequency noise applied to a diode pump current induces rare jumps to coexisting subharmonic states with high-amplitude pulses perceived as rogue waves. The probability of these events depends on the noise filtered frequency and grows up when the noise amplitude increases. The probability distribution of spike amplitudes confirms the rogue wave character of the observed phenomenon. The results of numerical simulations are in good agreement with experiments.     

On the extension of solutions of the real to complex KdV equation and a mechanism for the construction of rogue waves

Rogue waves are more precisely defined as waves whose height is more than twice the significant wave height. This remarkable height was measured (by Draupner in 1995). Thus, the need for constructing a mechanism for the rogue waves is of great utility. This motivated us to suggest a mechanism, in this work, that rogue waves may be constructed via nonlinear interactions of solitons and periodic waves. This suggestion is consolidated here, in an example, by studying the behavior of solutions of the complex (KdV). This is done here by the extending the solutions of its real version.     

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients

The research of rogue waves is an advanced field which has important practical and theoretical significances in mathematics, physics, biological fluid mechanics, oceanography, etc. Using the reductive perturbation theory and long wave approximation, the equations governing the movement of blood vessel walls and the flow of blood are transformed into high-order nonlinear Schrödinger (NLS) equations with variable coefficients. The third-order nonlinear Schrödinger equation is degenerated into a completely integrable Sasa-Satsuma equation (SSE) whose solutions can be used to approximately simulate the real rogue waves in the vessels. For the first time, we discuss the conditions for generating rogue waves in the blood vessels and effects of some physiological parameters on the rogue waves. Based on the traveling wave solutions of the fourth-order nonlinear Schrödinger equation, we analyze the effects of the higher order terms and the initial deformations of the blood vessel on the wave propagation and the displacement of the tube wall. Our results reveal that the amplitude of the rogue waves are proportional to the initial stretching ratio of the tube. The high-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave steepness.     

Darboux Transformations,Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.     

Lump solitons,rogue wave solutions and lump-stripe interaction phenomena to an extended (3+1)-dimensional KP equation

By using a direct method, we construct the Hirota bilinear form for an extended (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Based on this bilinearization, the lump solitons and rogue wave solutions are investigated. Constraint conditions for the wave propagation and velocity for lump solitons are found and verified by figures. Also the lump-stripe interaction was investigated to show that the lump solitons will be swallowed by the stripe soliton. Finally, the dynamic behaviour for the obtained lump solution, rogue wave and lump-stripe soliton interaction by suitable special parameters is shown graphically.     

Analysis on lump,lumpoff and rogue waves with predictability to the (2 + 1)-dimensional B-type Kadomtsev–Petviashvili equation

In this work, we investigate the ($2+1$)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation, which can be used to describe weakly dispersive waves propagating in the quasi media and fluid mechanics. We construct the more general lump solutions, localized in all directions in space, with more arbitrary autocephalous parameters. By considering a stripe soliton generated completely by lump solution, a lumpoff solution is presented. Its lump part is cut by soliton part before or after a specific time, with a specific divergence relationship. Furthermore, combining a pair of stripe solitons, we obtain the special rogue waves when lump solution is cut by double solitons. Our results show that the emerging time and place of the rogue waves can be caught through tracking the moving path of lump solution, and confirming when and where it happens a collision with the visible soliton. Finally, some graphic analysis are discussed to understand the propagation phenomena of these solutions.