Rogue wave observation in a water wave tank

The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schr?dinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.     

Dynamics of magnetization in ferromagnet with spin-transfer torque

We review our recent works on dynamics of magnetization in ferromagnet with spin-transfer torque. Driven by constant spin-polarized current, the spin-transfer torque counteracts both the precession driven by the effective field and the Gilbert damping term different from the common understanding. When the spin current exceeds the critical value, the conjunctive action of Gilbert damping and spin-transfer torque leads naturally the novel screw-pitch effect characterized by the temporal oscillation of domain wall velocity and width. Driven by space- and time-dependent spin-polarized current and magnetic field, we expatiate the formation of domain wall velocity in ferromagnetic nanowire. We discuss the properties of dynamic magnetic soliton in uniaxial anisotropic ferromagnetic nanowire driven by spin-transfer torque, and analyze the modulation instability and dark soliton on the spin wave background, which shows the characteristic breather behavior of the soliton as it propagates along the ferromagnetic nanowire. With stronger breather character, we get the novel magnetic rogue wave and clarify its formation mechanism. The generation of magnetic rogue wave mainly arises from the accumulation of energy and magnons toward to its central part. We also observe that the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background, and the critical current condition is obtained analytically. At last, we have theoretically investigated the current-excited and frequency-adjusted ferromagnetic resonance in magnetic trilayers. A particular case of the perpendicular analyzer reveals that the ferromagnetic resonance curves, including the resonant location and the resonant linewidth, can be adjusted by changing the pinned magnetization direction and the direct current. Under the control of the current and external magnetic field, several magnetic states, such as quasi-parallel and quasi-antiparallel stable states, out-of-plane precession, and bistable states can be realized. Th     

自陡峭和自频移效应对有限背景解的影响

基于自聚焦的非线性薛定谔方程,研究了自陡峭效应和自频移效应对Peregrine怪波(PS)、Akhmediev呼吸子(AB)和Kuznetsov-Ma孤子(KMS)传输特性的影响。数值模拟结果表明:这两种效应使三种有限背景解分裂加快,相邻最大压缩脉冲间的距离减小,脉冲中心发生偏移,且参数越大,分裂得越早,脉冲中心偏移量越大。     

Peregrine solitons and gradient catastrophes in discrete nonlinear Schrödinger systems

In the present work, we examine the potential robustness of extreme wave events associated with large amplitude fluctuations of the Peregrine soliton type, upon departure from the integrable analogue of the discrete nonlinear Schrödinger (DNLS) equation, namely the Ablowitz–Ladik (AL) model. Our model of choice will be the so-called Salerno model, which interpolates between the AL and the DNLS models. We find that rogue wave events are drastically distorted even for very slight perturbations of the homotopic parameter connecting the two models off of the integrable limit. Our results suggest that the Peregrine soliton structure is a rather sensitive feature of the integrable limit, which may not persist under “generic” perturbations of the limiting integrable case.     

Hirota方程的怪波解及其传输特性研究

求出了高阶Hirota方程在可积条件下的一种精确呼吸子解,并基于此呼吸子解得到了Hirota方程的一种怪波解.在此怪波解的基础上研究了怪波的激发,发现对平面波进行周期性扰动可以激发怪波,对平面波进行高斯扰动可以更快地激发怪波,还可以直接在常数项上增加高斯扰动激发怪波.作为一个实例,采用分步傅里叶方法数值研究了在考虑自频移和拉曼增益时怪波的传输特性,自频移使怪波中心发生偏移,拉曼增益使得怪波分裂得更快,而且拉曼增益值越大怪波分裂得越快,但是拉曼增益对怪波的峰值强度没有明显影响.最后数值模拟了相邻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.     

Ultrashort light pulses generated from modulation instability: background removal and soliton content

Modulation instability can be used to convert a continuous light wave into a train of pulses on a constant background. It is a longstanding discussion whether these pulses can be converted into solitons. We clarify the situation by using a more general mathematical context, invoking the Akhmediev breather, Peregrine soliton and Kuznetsov-Ma soliton solutions of the wave equation, and suggest the use of a Mach–Zehnder interferometer to remove the background. Expressions for the pulse widths and peak powers thus obtained are presented, and their soliton content is determined. It turns out that more than 95 % of each pulse’s energy can be converted to a soliton.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrodinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.     

Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this paper, we obtained the exact breather-type kink soliton and breather-type periodic soliton solutions for the (3 + 1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation using the extended homoclinic test technique. Some new nonlinear phenomena, such as kink and periodic degeneracies, are investigated. Using the homoclinic breather limit method, some new rational breather solutions are found as well. Meanwhile, we also obtained the rational potential solution which is found to be just a rogue wave. These results enrich the variety of the dynamics of higher-dimensional nonlinear wave field.     

Observation of rogue wave triplets in water waves

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.     

Formation mechanism of asymmetric breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations

Based on the developed Darboux transformation, we investigate the exact asymmetric solutions of breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations. As an example, some types of exact breather solutions are given analytically by adjusting the parameters. Moreover, the interesting fundamental problem is to clarify the formation mechanism of asymmetry breather solutions and how the particle number and energy exchange between the background and soliton ultimately form the breather solutions. Our results also show that the formation mechanism from breather to rogue wave arises from the transformation from the periodic total exchange into the temporal local property.     

Magnetic rogue wave in a perpendicular anisotropic ferromagnetic nanowire with spin-transfer torque

We present the current controlled motion of a dynamic soliton embedded in spin wave background in ferromagnetic nanowire. With the stronger breather character we get the novel magnetic rogue wave and clarify its formation mechanism. The generation of magnetic rogue wave mainly arises from the accumulation of energy and magnons toward to its central part. We also observe that the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background, and the critical current condition is obtained analytically. Even more interesting is that the spin-transfer torque plays the completely opposite role for the cases below and above the critical value.     

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov (GI) equation is one of the most important integrable equations, which can be reduced from the third generic deformation of the derivative nonlinear Schrödinger equation. The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation. As applications, we obtain the bright-dark soliton, breather, rogue wave, kink, W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2n-fold Darboux transformation. These solutions show rich wave structures for selections of different parameters. In all these instances we practically show that these solutions have different properties than the ones for local case.     

General Higher-Order Breather and Hybrid Solutions of the Fokas System

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schr?dinger (NLS) equation (Eq.(2), Inverse Problems 10 (1994) L19-L22).By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of $n$-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits arange of interesting dynamics. For high-order breather, it has more rich dynamical behaviors.The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively.By taking a long waves limit of a part of exponential functions in $f$ and $g$ appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.     

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.     

Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa–Satsuma case

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.     

Soliton,breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

The nonlinear Schro¨dinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schro¨dinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schro¨dinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.     

Amplification of matter rogue waves and breathers in quasi-two-dimensional Bose-Einstein condensates

We construct rogue wave and breather solutions of a quasi-two-dimensionalGross-Pitaevskii equation with a time-dependent interatomic interaction and external trap.We show that the trapping potential and an arbitrary functional parameter that present inthe similarity transformation should satisfy a constraint for the considered equation tobe integrable and yield the desired solutions. We consider two different forms offunctional parameters and investigate how the density of the rogue wave and breatherprofiles vary with respect to these functional parameters. We also construct vectorlocalized solutions of a two coupled quasi-two-dimensional Bose-Einstein condensatesystem. We then investigate how the vector localized density profiles modify in theconstant density background with respect to the functional parameters. Our results mayhelp to manipulate matter rogue waves experimentally in the two-dimensional Bose-Einsteincondensate systems.     

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.     

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.