Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

     

Generalized Darboux transformation and localized waves in coupled Hirota equations

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an N$N$th-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) vector generalization of the first- and the second-order rogue wave solutions; (2) interactional solutions between a dark–bright soliton and a rogue wave, two dark–bright solitons and a second-order rogue wave; (3) interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.     

Dynamics of localized wave solutions for the coupled Higgs field equation

Nonlinear Dynamics - By virtue of Hirota bilinear form of the coupled Higgs field equation, some higher-order rogue wave, the Ma breather, the Akhmediev breather and the general breather solutions...     

On the characterization of breather and rogue wave solutions and modulation instability of a coupled generalized nonlinear Schrödinger equations

We construct Darboux transformation of a coupled generalized nonlinear Schrödinger (CGNLS) equations and obtain exact analytic expressions of breather and rogue wave solutions. We also formulate the conditions for isolating these solutions. We show that the rogue wave solution can be found only when the four wave mixing parameter becomes real. We also investigate the modulation instability of the steady state solution of CGNLS system and demonstrate that it can occur only when the four wave mixing parameter becomes real. Our results give an evidence for the connection between the occurrence of rogue wave solution and the modulation instability.     

Controllable rogue waves in coupled nonlinear Schrödinger equations with varying potentials and nonlinearities

Exact rogue wave solutions, including the first-order rogue wave solutions and the second-order ones, are constructed for the system of two coupled nonlinear Schrödinger (NLS) equations with varying potentials and nonlinearities. The method employed in this paper is the similarity transformation, which allows us to map the inhomogeneous coupled NLS equations with variable coefficients into the integrable Manakov system, whose explicit solutions have been well studied before. The result shows that the rogue wavelike solutions obtained by this transformation are controllable. Concretely, we illustrate how to control the trajectories of wave centers and the evolutions of wave peaks, and analyze the dynamic behaviors of the rogue wavelike solutions.     

Double degeneration on second-order breather solutions of Maxwell–Bloch equation

The breather solutions of the Maxwell–Bloch equations in a two-level resonant system associated with the self-induced transparency phenomenon are constructed by the Darboux transformation. After constructing the formulas of the second-order breather solutions, the double degeneration and hybrid solutions are studied by the analytical form as well as figures. Our results might be helpful in such application or prevention of the rogue waves from breather solution interactions and degeneration in the nonlinear optical systems associated with the Maxwell–Bloch equations.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

     

Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides

With the help of the similarity transformation connected the variable-coefficient nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions in two dimensional graded-index waveguides. Then, we investigate the nonlinear tunneling of controllable rogue waves when they pass through nonlinear barrier and nonlinear well. Our results indicate that the propagation behaviors of rogue waves, such as postpone, sustainment and restraint, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z _m and the effective propagation distance corresponding to maximum amplitude of rogue waves Z ₀. Postponed, sustained and restrained rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged and decreasing amplitudes by modulating the ratio of the amplitudes of rogue waves to barrier or well height.     

M-lump,rogue waves,breather waves,and interaction solutions among four nonlinear waves to new (3+1)-dimensional Hirota bilinear equation

In this research article, we study a new (3+1)-dimensional Hirota bilinear equation which can describe the dynamics of ion-acoustic wave and Alvin wave of small but finite amplitude in plasma physics and describe the propagation process of nonlinear waves in shallow water. First, we apply two methods to study the equation, namely the Hirota bilinear method and long-wave limit method M-lump solution, and line rogue waves are reported. Furthermore, we investigate the velocity, propagation trajectory, and interaction phenomenon of M-lump solution(M=2,3). Then, based on the multi-solitons, two cases of high-order breather solution are constructed by selecting some special parameters. Finally, four types interaction solutions are successfully obtained by employing long-wave limit method and selecting some special parameters. More importantly, we explore physical collision phenomenon of the interaction between nonlinear waves. In order to better illustrate the characteristics of the interaction solutions, the results are shown in three-dimensional plots and numerical simulation. To our knowledge, all of the obtained solutions in this article are novel. The results of this article may be provide an important theoretical basis for explaining some nonlinear phenomena in the field of fluid mechanics and shallow water.

     

Revisit of rogue wave solutions in the Yajima–Oikawa system

The general rogue wave solutions for the one-dimensional (1D) Yajima–Oikawa (YO) system are derived through Hirota’s bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method. Different from the previous work, we improve the construction of the differential operators to save the complicated recursiveness and obtain the rogue wave solutions in a purely algebraic expression. Based on this simple expression, the new shape of the third-order rogue waves’ arrangement is found. Moreover, three types of fundamental rogue waves and the rogue wave patterns from second to fifth order are graphically illustrated. In particular, there exist \(N-1\) (\(2\le N \)) polygonal configurations of Nth-order rogue waves for the 1D YO system, which is proven to be related to the Yablonskii–Vorob’ev polynomial hierarchy.

     

Prediction on dispersion in elastoplastic unsaturated granular media

This study focuses on the propagation of the plane wave in the elastoplastic unsaturated granular media, and the wave equations and dispersion equations are derived for the media under the framework of Cosserat theory. Due to symmetry, five different wave modes are considered and predicted for the elastoplastic unsaturated granular media based on the Cosserat theory, including two longitudinal waves, one rotational longitudinal wave and two coupled transverse–rotational transverse waves. The correspondence is discussed between these Cosserat wave modes and the classical wave modes. Based on the dispersion equations, the dispersion behaviors are obtained for the five Cosserat wave modes. The results indicated that the different stress-strain stages,including the elastic, hardening and softening stages, have obvious effect on the dispersion behaviors of the Cosserat wave modes.     

Extreme wave solutions: Parametric studies and wavelet analysis

Wave fields for near homoclinic, single mode rogue-wave solutions of the periodic nonlinear Schrödinger equation are presented. Parameters of candidate solutions are estimated and refined through an eigenvalue solution procedure. An overview of the estimation and refining procedure used by the authors is provided. Solutions are scaled to facilitate experimental implementation. The continuous wavelet transform is used to carry out time–frequency analyses and the results obtained are demonstrative of the dispersion relation as well as the time varying side band energy transfer associated with the Benjamin–Feir instability. The analysis framework and approach used are validated with the Peregrine solution. Other extreme wave solutions are analyzed as well. The framework presented here could serve as a basis for experimental investigations into single mode rogue waves as well as other localizations in wave fields.     

Discrete breathers and modulational instability in a discrete $$\varvec{\phi ^{4}}$$ nonlinear lattice with next-nearest-neighbor couplings

The properties of discrete breathers and modulational instability in a discrete \(\phi ^{4}\) nonlinear lattice which includes the next-nearest-neighbor coupling interaction are investigated analytically. By using the method of multiple scales combined with a quasi-discreteness approximation, we get a dark-type and a bright-type discrete breather solutions and analyze the existence conditions for such discrete breathers. It is found that the introduction of the next-nearest-neighbor coupling interactions will influence the existence condition for the bright discrete breather. Considering that the existence of bright discrete breather solutions is intimately linked to the modulational instability of plane waves, we will analytically study the regions of discrete modulational instability of plane carrier waves. It is shown that the shape of the region of modulational instability changes significantly when the strength of the next-nearest-neighbor coupling is sufficiently large. In addition, we calculate the instability growth rates of the \(q=\pi \) plane wave for different values of the strength of the next-nearest-neighbor coupling in order to better understand the appearance of the bright discrete breather.     

Characterizing JONSWAP rogue waves and their statistics via inverse spectral data

Rogue waves in random sea states modeled by the JONSWAP power spectrum are high amplitude waves arising over non-uniform backgrounds that cannot be viewed as small amplitude modulations of Stokes waves. In the context of Nonlinear Schrödinger (NLS) models for waves in deep water, this poses the challenge of identifying appropriate analytical solutions for JONSWAP rogue waves, investigating possible mechanisms for their formation, and examining the validity of the NLS models in these more realistic settings. In this work we investigate JONSWAP rogue waves using the inverse spectral theory of the periodic NLS equation for moderate values of the period. For typical JONSWAP initial data, numerical experiments show that the developing sea state is well approximated by the first few dominant modes of the nonlinear spectrum and can be described in terms of a 2- or 3-phase periodic NLS solution. As for the case of uniform backgrounds, proximity to instabilities of the underlying 2-phase solution appears to be the main predictor of rogue wave occurrence, suggesting that the modulational instability of 2-phase solutions of the NLS is a main mechanism for rogue wave formation and that heteroclinic orbits of unstable 2-phase solutions are plausible models of JONSWAP rogue waves. To support this claim, we correlate the maximum wave strength as well as the higher statistical moments with elements of the nonlinear spectrum. The result is a diagnostic tool widely applicable to both model or field data for predicting the likelihood of rogue waves. Finally, we examine the validity of NLS models for JONSWAP data, and show that NLS solutions with JONSWAP initial data are described by non-Gaussian statistics, in agreement with the TOPEX field studies of sea surface height variability.     

基于导波技术的螺柱轴力无损检测

根据弹性动力学理论,采用纵向导波与弯曲导波相结合的方法对螺柱所受轴向应力进行无损检测。计算了M22螺柱中纵向导波和弯曲导波的群速度频散曲线。根据频散曲线,确定了采用导波对螺柱轴向应力进行无损检测的最优检测信号频率范围(50~80 kHz),此频率范围的纵向导波与弯曲导波模态单一,并且频散性较低。分别计算了不同轴向应力σ作用下,多种频率的纵向导波和和弯曲导波在螺柱中传播的群速度值cgσrL和cgσrF。结果表明,随着轴向应力的增大,同频率纵向导波和弯曲导波的群速度皆呈线性递减趋势。利用纵向导波和弯曲导波群速度与轴向应力的线性关系及纵向导波和弯曲导波在轴向应力作用螺柱端面的反射时间tLσ和tσF,可以迅速确定螺柱所受轴向应力值。     

On different aspects of the optical rogue waves nature

Rogue waves are giant nonlinear waves that suddenly appear and disappear in oceans and optics. We discuss the facts and fictions related to their strange nature, dynamic generation, ingrained instability, and potential applications. We present rogue wave solutions to the standard cubic nonlinear Schrödinger equation that models many propagation phenomena in nonlinear optics. We propose the method of mode pruning for suppressing the modulation instability of rogue waves. We demonstrate how to produce stable Talbot carpets—recurrent images of light and plasma waves—by rogue waves, for possible use in nanolithography. We point to instances when rogue waves appear as numerical artefacts, due to an inadequate numerical treatment of modulation instability and homoclinic chaos of rogue waves. Finally, we display how statistical analysis based on different numerical procedures can lead to misleading conclusions on the nature of rogue waves.

     

Characteristics of rogue waves on a periodic background for the Hirota equation

We consider the rogue dn-periodic waves (the rogue wave solutions on the dn-periodic waves background) for the Hirota equation by using Darboux transformation. We take Jacobian elliptic function dn as a seed solution, which is modulationally unstable as regards long wave perturbations. Through nonlinearization of the Lax pair for Hirota equation, the corresponding periodic eigenfunctions are successfully obtained. Based on these periodic eigenfunctions, we further construct the solutions of the Lax pair equations with dn-periodic wave seed solutions. In addition, numerical simulations are presented to reveal the phenomena of these solutions under different parameters choices.     

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrödinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.