Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel’nikov equation and the multicomponent Schrödinger–Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel’nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrödinger–Boussinesq system are generated.     

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrödinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.     

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, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. The cut off distance δ_cutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.     

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.     

Rogue Waves in the (2+1)-Dimensional Nonlinear Schrodinger Equation with a Parity-Time-Symmetric Potential

The(2+1)-dimension nonlocal nonlinear Schrodinger(NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110(2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the(x,y) plane.     

Characteristics of Rogue Waves on a Soliton Background in the General Coupled Nonlinear Schrödinger Equation

Under investigation in this work is the general coupled nonlinear Schrödinger (gCNLS) equation, which can be used to describe a wide variety of physical processes. By using Darboux transformation, the new higher-order rogue wave solutions of the equation are well constructed. These solutions exhibit rogue waves on a multi-soliton background. Moreover, the dynamics of these solutions is graphically discussed. Our results would be of much importance in enriching and predicting rogue wave phenomena arising in nonlinear and complex systems.     

Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations

Under investigation in this paper are the inhomogeneous nonlinear Schrödinger Maxwell–Bloch (INLS-MB) equations which model the propagation of optical waves in an inhomogeneous nonlinear light guide doped with two-level resonant atoms. Higher-order nonautonomous breather as well as rogue wave solutions in terms of the determinants for the INLS-MB equations are presented via the n$n$-fold variable-coefficient modified Darboux transformation. The interactions among two nonautonomous breathers are graphically discussed, including the fundamental breather, bound breather, two-breather compression and two-breather evolution, etc. Moreover, several patterns of the higher-order rogue waves are also exhibited, such as the square rogue wave, two- and three-order periodic rogue waves, periodic fission and fusion, two-order stationary rogue waves, and recurrence of the two-order rogue waves. The character of the trajectory of the two-order periodic rogue wave is analyzed. Additionally, a novel type of interaction, namely, the collision between the breather and long-lived rogue waves, is found to be elastic. Our results could be useful for controlling the nonautonomous optical breathers and rogue waves in the inhomogeneous erbium doped fiber.     

Controllable rogue waves in the nonautonomous nonlinear system with a linear potential

Based on the similarity transformation connected the nonautonomous nonlinear Schrödinger equation with the autonomous nonlinear Schrödinger equation, we firstly derive self-similar rogue wave solutions (rational solutions) for the nonautonomous nonlinear system with a linear potential. Then, we investigate the controllable behaviors of one-rogue wave, two-rogue wave and rogue wave triplets in a soliton control system. Our results demonstrate that the propagation behaviors of rogue waves, including postpone, sustainment, recurrence and annihilation, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z _m and the parameter Z ₀. Moreover, the excitation time of controllable rogue waves is decided by the parameter T ₀.     

Mixed lump-stripe,bright rogue wave-stripe,dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics. We derive the mixed lump-stripe waves, bright mixed rogue wave-stripe, dark mixed rogue wave-stripe and dark rogue waves solutions by virtue of the symbolic computation. We observe the fission and fusion phenomena between the lump and one-stripe wave through the mixed-stripe wave solutions. Then, we observe that the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the mixed lump-stripe waves, where l₁ and l₂ represent the dispersion and nonlinear effects, l₃, l₆, l₇ and l₈ are the perturbed effects, while l₄ and l₅ stand for the disturbed wave velocities along the transverse spatial coordinates y and z, respectively. We graphically present the interaction between a rogue wave and a pair of stripe waves through the mixed rogue wave-stripe solutions. We derive a dark mixed rogue wave-stripe when l₁ < 0. We study the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the rogue wave and a pair of stripe waves. We present the dark rogue wave with certain parameters and observe that two stripe waves merge into one stripe wave.     

Rogue Waves in the Three-Dimensional Kadomtsev-Petviashvili Equation

Breathers and rogue waves as exact solutions of the three-dimensional Kadomtsev-Petviashvili equation are obtained via the bilinear transformation method.The breathers in three dimensions possess different dynamics in different planes,such as growing and decaying periodic line waves in the(x,y),(x,z) and(y,t) planes.Rogue waves are localized in time,and are obtained theoretically as a long wave limit of breathers with indeGnitely larger periods.It is shown that the rogue waves possess growing and decaying line profiles in the(x,y) or(x,z)plane,which arise from a constant background and then retreat back to the same background again.     

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.     

Vector financial rogue waves

The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black-Scholes option pricing model. In this Letter, we analytically propose vector financial rogue waves of the coupled nonlinear volatility and option pricing model without an embedded w-learning. Moreover, we exhibit their dynamical behaviors for chosen different parameters. The vector financial rogue wave (rogon) solutions may be used to describe the possible physical mechanisms for the rogue wave phenomena and to further excite the possibility of relative researches and potential applications of vector rogue waves in the financial markets and other related fields.     

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.     

Dark and composite rogue waves in the coupled Hirota equations

The intriguing dark and composite rogue wave dynamics in a coupled Hirota system are unveiled, based on the exact explicit rational solutions obtained under the assumption of equal background height. It is found that a dark rogue wave state would occur as a result of the strong coupling between two field components with large wavenumber difference, and there would appear plenty of composite structures that are attributed to the specific wavenumber difference and the free choice of three independent structural parameters. The coexistence of different fundamental rogue waves in such a coupled system is also demonstrated.     

Waves that appear from nowhere and disappear without a trace

The title (WANDT) can be applied to two objects: rogue waves in the ocean and rational solutions of the nonlinear Schrödinger equation (NLSE). There is a hierarchy of rational solutions of ‘focussing’ NLSE with increasing order and with progressively increasing amplitude. As the equation can be applied to waves in the deep ocean, the solutions can describe “rogue waves” with virtually infinite amplitude. They can appear from smooth initial conditions that are only slightly perturbed in a special way, and are given by our exact solutions. Thus, a slight perturbation on the ocean surface can dramatically increase the amplitude of the singular wave event that appears as a result.     

Rogue wave solutions for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

A generalized Darboux transformation for the coupled cubic–quintic nonlinear Schrödinger equation is constructed by the Darboux matrix method. As applications, the Nth-order rogue wave solutions of the coupled cubic–quintic nonlinear Schrödinger equation have been obtained. In particular, the dynamics of the general first- and second-order rogue waves are discussed and illustrated through some figures.     

Dynamics of breathers and rogue waves in scalar and multicomponent nonlinear systems

In this paper, we propose a new method, the variable separation technique, for obtaining a breather and rogue wave solution to the nonlinear evolution equation. Integrable systems of the derivative nonlinear Schrödinger type are used as three examples to illustrate the effectiveness of the presented method. We then obtain a family of rational solutions. This family of solutions includes the Akhmediev breather, the Kuznetsov-Ma breather, versatile rogue waves, and various interactions of localized waves. Moreover, the main characteristics of these solutions are discussed and some graphics are presented. More importantly, our results show that more abundant and novel localized waves may exist in the multicomponent coupled equations than in the uncoupled ones.     

Multiple-order rogue wave solutions to a (2+1)-dimensional Boussinesq type equation

In this paper, based on the Hirota bilinear method and symbolic computation approach, multiple-order rogue waves of (2+1)-dimensional Boussinesq type equation are constructed. The reduced bilinear form of the equation is deduced by the transformation of variables. Three kinds of rogue wave solutions are derived by means of bilinear equation. The maximum and minimum values of the first-order rogue wave solution are given at a specific moment. Furthermore, the second-order and third-order rogue waves are explicitly derived. The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.