Financial Rogue Waves

We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black-Scholes model. These rogue wave solutions may be used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.     

Analysis on Lump,Lumpoff and Rogue Waves with Predictability to a Generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt Equation *

In this paper, we investigate a (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Ku-pershmidt equation. The lump waves, lumpoff waves, and rogue waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the moving path as well as the appearance time and place of the lump waves are given. Moreover, the special rogue waves are considered when lump solution is swallowed by double solitons. Finally, the corresponding characteristics of the dynamical behavior are displayed.     

Ion-Acoustic Rogue Waves in Multi-ion Plasmas *

The basic properties of nonlinear ion-acoustic (IA) waves (IAWs), particularly finite amplitude IA rogue waves (IARWs) in a plasma medium (containing pair ions, iso-thermal positrons, and non-thermal electrons) are theoretically investigated by deriving the nonlinear Schr?dinger equation (NLSE). The criteria for the modulational instability of IAWs, and the basic features of finite amplitude IARWs are identified. The modulationally stable and unstable regions are determined by the sign of the ratio of the dispersive coefficient to the nonlinear rcoefficient of NLSE. The latter is analyzed to obtain the region for the existence of the IARWs, which corresponds to the unstable region. The shape of the profile of the rogue waves depends on the non-thermal parameter$\alpha$ and the ratio of electron temperature to positron temperature. It is found that the increase in the value of the non-thermal parameter enhances both the amplitude and width of IARWs, and that the enhancement of electron (positron) temperature reduces (enhances) the amplitude and width of IARWs. It is worth to mention that our present investigation may be useful for understanding the salient features of IARWs in space (viz., upper region of Titan's atmosphere, cometary comae, and Earth's ionosphere, etc.) and laboratory (viz., plasma process ingreactor and neutral beam sources, etc.) plasmas.      

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 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.     

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.     

暗怪波的相互作用

以耦合非线性薛定谔方程为理论模型,数值研究了两个一阶暗怪波在正常色散单模光纤中的相互作用.基于一阶暗怪波精确解,采用分步傅里叶数值模拟法,从间距、相位差和振幅系数比方面讨论相邻两个一阶暗怪波之间的相互作用.基于二阶暗怪波精确解,讨论了两个一阶暗怪波的非线性相互作用.研究结果表明:同相位情况下,间距参数T1为0、5、20时,相邻两个一阶暗怪波相互作用激发产生“扭结型”暗怪波.相比较于单个暗怪波发生能量的弥散,“扭结型”暗怪波分裂形成多个次暗怪波.反相位情况下,间距参数T1为2、7、12时,相邻两个一阶暗怪波相互作用也可以激发产生“扭结型”暗怪波.并且“扭结型”暗怪波初始激发的空间位置偏离原始单个暗怪波的位置5.振幅系数比越大,该空间位置越接近5.二阶暗怪波可以看作是两个一阶暗怪波的非线性叠加,复合型和三组分型二阶暗怪波与相邻两个一阶暗怪波的相互作用略有相似.     

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.     

Dynamics of Controllable Optical Rogue Waves in Presence of Quintic Nonlinearity and Nonlinear Dispersion Effects

We analytically study optical rogue waves in the presence of quintic nonlinearity and nonlinear dispersion effects. Dynamics of the rogue waves are investigated through the precise expressions of their peak, valley, trajectory, and width. Based on this, the properties of a few specific rogue waves are demonstrated in detail, and the dynamical evolution of rogue waves can be well controlled under different nonlinearity management. It shows that the peak reaches its maximum and the valley becomes minimized when the width evolves to the minimum value. Moreover, we find that the higher-order effects here achieve balance due to the integrability, and they only influence the rogue waves' trajectory.     

Breathers and Rogue Waves Derived from an Extended Multi-dimensional N-Coupled Higher-Order Nonlinear Schrödinger Equation in Optical Communication Systems

In this paper, an extended multi-dimensional N-coupled higher-order nonlinear Schrödinger equation (NCHNLSE), which can describe the propagation of the ultrashort pulses in wavelength division multiplexing (WDM) systems, is investigated. By the bilinear method, we construct the breather solutions for the extended (1+1), (2+1) and (3+1)-dimensional N-CHNLSE. The rogue waves are derived as a limiting form of breathers with the aid of symbolic computation. The effect of group velocity dispersion (GVD), third-order dispersion (TOD) and nonlinearity on breathers and rogue waves solutions are discussed in the optical communication systems.     

Twenty Parameters Families of Solutions to the NLS Equation and the Eleventh Peregrine Breather

The Peregrine breather of order eleven(P_(11) breather) solution to the focusing one-dimensional nonlinear Schrdinger equation(NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given: when all parameters are equal to 0 we recover the famous P_(11) breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the(x; t) plane, in function of the different parameters.     

Spatiotemporal Rogue Waves for the Variable-Coefficient (3+1)-Dimensional Nonlinear Schrödinger Equation

With the help of the similarity transformation connected the variable-coefficient (3+1)-dimensional nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions. Then, we investigate the controllable behaviors of these rogue waves in the hyperbolic dispersion decreasing profile. Our results indicate that the integral relation between the accumulated time T and the real time t is the basis to realize the control and manipulation of propagation behaviors of rogue waves, such as sustainment and restraint. We can modulate the value T₀ to achieve the sustained and restrained spatiotemporal rogue waves. Moreover, the controllability for position of sustainment and restraint for spatiotemporal rogue waves can also be realized by setting different values of X₀.     

Rogue Wave with a Controllable Center of Nonlinear Schrödinger Equation

The rogue waves with a controllable center are reported for the nonlinear Schrödinger equation in terms of rational-like functions by using a direct method. The position of these solutions can be controlled by choosing different center parameters and this may describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, Bose-Einstein condensates respectively.     

Rogue waves in multi‐pair plasma medium

The non‐linear propagation of ion acoustic (IA) waves, which is governed by the non‐linear Schrödinger equation, in multi‐pair plasmas (MPPs) containing adiabatic positive and negative ion fluids as well as non‐extensive (q‐distributed) electrons and positrons is theoretically investigated. It is observed that the MPP under consideration supports two types of modes, namely fast and slow IA modes, and the modulationally stable and unstable parametric regimes for the fast and slow IA modes are determined by the sign of the ratio of the dispersive coefficient to the non‐linear one. It is also found that the modulationally unstable regime generates highly energetic IA rogue waves (IARWs), and the amplitude as well as the width of the IARWs decreases with increase in the value of q (for both q > 0 and q < 0 limits). These new striking features of the IARWs are found to be applicable in the space (i.e., D‐region [], and F‐region [H⁺, H^?] of the Earth's ionosphere) and laboratory MPPs (i.e., fullerene [C⁺, C^?]).     

Improved Speed and Shape of Ion-Acoustic Waves in a Warm Plasma

The basic set of fluid equations can be reduced to the nonlinear Kortewege-de Vries (KdV) and nonlinear Schrödinger (NLS) equations. The rational solutions for the two equations has been obtained. The exact amplitude of the nonlinear ion-acoustic solitary wave can be obtained directly without resorting to any successive approximation techniques by a direct analysis of the given field equations. The Sagdeev's potential is obtained in terms of ion acoustic velocity by simply solving an algebraic equation. The soliton and double layer solutions are obtained as a small amplitude approximation. A comparison between the exact soliton solution and that obtained from the reductive perturbation theory are also discussed.     

Soliton and Rogue Wave Solution of the New Nonautonomous Nonlinear Schrödinger Equation

In this paper, a new type (or the second type) of transformation which is used to map the variable coefficient nonlinear Schrödinger (VCNLS) equation to the usual nonlinear Schrödinger (NLS) equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is
introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically.
The main differences between the two types of transformation mentioned above are listed by three items.     

高阶效应对怪波传输的影响

求出高阶Hirota方程在可积条件下的一种精确呼吸子解,并在解的基础上得到Hirota方程的一种怪波解。从怪波解的形式和图形中深刻认识到怪波的两个特征——时空局域性和高能量特点,认识到怪波产生的物理机制——平面波和其他波的叠加。利用分布傅立叶方法数值研究了怪波在考虑自频移和拉曼增益时的传输特性,自频移使怪波中心发生偏移,拉曼增益使怪波分裂得更快;数值模拟了怪波之间的相互作用特点——随着怪波之间距离的减小,怪波将合二为一,成为一束怪波,之后再分裂,并分析了拉曼增益和自频移对怪波相互作用的影响。