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.     

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.     

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.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

Rogue waves in the nonlocal $${\mathcal {PT}}$$ PT -symmetric nonlinear Schrödinger equation

Letters in Mathematical Physics - Rogue waves in the nonlocal $${\mathcal {PT}}$$ -symmetric nonlinear Schrödinger (NLS) equation are studied by Darboux transformation. Three types of rogue...     

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.     

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.     

Rogue Wave Solutions for Nonlinear Schrödinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrödinger (NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.     

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.     

Rogue waves and solitons on a cnoidal background

Solutions of the nonlinear Schrödinger equation, appearing as rogue waves on a spatially-periodic background envelope, are obtained using the Darboux transformation scheme. Several particular examples are illustrated numerically. These include soliton and breather solutions on a periodic background as well as higher-order structures. The results enrich our knowledge of possible analytic solutions that describe the appearance of rogue waves in a variety of situations. This work is prepared on the occasion of Prof. Helmut Brand's 60th birthday. He has made significant contributions to the science of solitons and his ideas have inspired our research into localised formations in various physical contexts.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

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.     

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.     

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.     

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.     

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.     

Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves

We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped nonlinear Schrödinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |Γt|?1$|\Gamma t|?1$, with Γ the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.     

Invariant interaction solutions for a supersymmetric mKdV equation

The bosonic supersymmetric modified KdV (BSmKdV) system is obtained by the bosonization approach. The nonlocal symmetry for the BSmKdV equation is obtained by the truncated Painlevé method. By introducing multiple new fields, the finite symmetry transformation for the BSmKdV equation is derived by applying Lie’s first principle to the prolonged systems. The similarity reductions related to the nonlocal symmetry are studied. The interaction solutions among the solitons and other complicated waves, including Painlevé II waves and periodic cnoidal waves, are presented through the reduction theorems. The concrete soliton-cnoidal interaction solutions are illustrated in detail by using the mapping and deformation method.     

Spectral stability analysis for periodic traveling wave solutions of NLS and CGL perturbations

We consider cnoidal traveling wave solutions to the focusing nonlinear Schrödinger equation (NLS) that have been shown to persist when the NLS is perturbed to the complex Ginzburg-Landau equation (CGL). We show that while these periodic traveling waves are spectrally stable solutions of NLS with respect to periodic perturbations, they are unstable with respect to bounded perturbations. Furthermore, we use an argument based on the Fredholm alternative to find an instability criterion for the persisting solutions to CGL.     

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions,for 1 and 2 unstable modes

The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the x-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of $N=1,2$ unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearance, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.