Diversity of Rogue Wave Solutions to the (1+1)-Dimensional Boussinesq Equation

A periodically homoclinic solution and some rogue wave solutions of (1+1)-dimensional Boussinesq equation are obtained via the limit behavior of parameters and different polynomial functions. Besides, the mathematics reasons for different spatiotemporal structures of rogue waves are analyzed using the extreme value theory of the two-variables function. The diversity of spatiotemporal structures not only depends on the disturbance parameter u0 but also has a relationship with the other parameters c0, α, β.

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 thepossible physical mechanisms for rogue wave phenomenon in financial markets and related fields.

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

The nonlinear Schrodinger 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 Schrodinger 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 Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.

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

In this paper, a new type （or the second type） of transformation which is used to map the variable coefficient nonlinear Schr6dinger （VCNLS） equation to the usual nonlinear Schrodinger （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.

Analytical solutions and rogue waves in （3＋1）-dimensional nonlinear SchrSdinger equation

Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.

A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation

In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.

Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation

The nonlinear Schrodinger （NLS） equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright-bright, bright-dark, and dark-dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright-bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright-bright or bright-dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.

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

The (2+1)-dimension nonlocal nonlinear Schrödinger (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 plane.

Soliton, Breather and Rogue Wave Solutions for the Nonlinear Schrdinger Equation Coupled to a Multiple Self-Induced Transparency System

We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown.

Rogue Waves in Nonintegrable KdV-Type Systems

It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.

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 of a(3+1)-dimensional BKP equation

We investigate certain rogue waves of a(3+1)-dimensional BKP equation via the Kadomtsev-Petviashili hierarchy reduction method.We obtain semi-rational solutions in the determinant form,which contain two special interactions:(i)one lump develops from a kink soliton and then fuses into the other kink one;(ii)a line rogue wave arises from the segment between two kink solitons and then disappears quickly.We find that such a lump or line rogue wave only survives in a short time and localizes in both space and time,which performs like a rogue wave.Furthermore,the higher-order semi-rational solutions describing the interaction between two lumps(one line rogue wave)and three kink solitons are presented.

Data-driven parity-time-symmetric vector rogue wave solutions of multi-component nonlinear Schrödinger equation

Rogue waves are a class of nonlinear waves with extreme amplitudes,which usually appear suddenly and disappear without any trace.Recently,the parity-time(PT)-symmetric vector rogue waves(RWs)of multi-component nonlinear Schrödinger equation(n-NLSE)are usually derived by the methods of integrable systems.In this paper,we utilize the multi-stage physics-informed neural networks(MS-PINNs)algorithm to derive the data-driven symmetric vector RWs solution of coupled NLS system in elliptic and X-shapes domains with nonzero boundary condition.The results of the experiment show that the multi-stage physics-informed neural networks are quite feasible and effective for multi-component nonlinear physical systems in the above domains and boundary conditions.

Collisions Between Lumps/Rogue Waves and Solitons for A(3+1)-Dimensional Generalized Variable-Coefficient Shallow Water Wave Equation

In this paper,we investigate a(3+1)-dimensional generalized variable-coefficient shallow water wave equation,which can be used to describe the flow below a pressure surface in oceanography and atmospheric science.Employing the Kadomtsev−Petviashvili hierarchy reduction,we obtain the semi-rational solutions which describe the lumps and rogue waves interacting with the kink solitons.We find that the lump appears from one kink soliton and fuses into the other on the x−y and x−t planes.However,on the x−z plane,the localized waves in the middle of the parallel kink solitons are in two forms:lumps and line rogue waves.The effects of the variable coefficients on the two forms are discussed.The dispersion coefficient influences the speed of solitons,while the background coefficient influences the background’s height.

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 Schrodinger(NLS)equations with variable coefficients.The third-order nonlinear Schrodinger 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 Schrodinger 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.

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.

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg–de Vries equation

Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.

Nonplanar dust acoustic solitary and rogue waves in an ion beam plasma with superthermal electrons and ions

The propagation characteristics of dust acoustic solitary and rogue waves are investigated in an unmagnetized ion beam plasma with electrons and ions following kappa-type distribution in nonplanar geometry. The reductive perturbation method （RPM） is employed to derive the cylindrical/spherical Korteweg-de Vries （KdV） equation, which is further transformed into standard KdV equation by neglecting the geometrical effects. Using new stretching coordinates, nonlinear Schrrdinger equation （NLSE） has been derived from the standard KdV equation to study the different order rational solutions of dust acoustic rogue waves （DARWs）. The impact of various physical parameters on the characteristics of dust acoustic solitary waves （DASWs） is elaborated specifically in nonplanar geometry. Further, the effects of ion beam and superthermality of electrons/ions on the characteristics of DARWs are studied. The results obtained in the present investigation may be useful in comprehending a variety of phenomena in Earth＇s magnetosphere polar cap region where the presence of positive ion beam has been detected and also in other regions of space/astrophysical environments where dust along with superthermal electrons and ions exists.

Dynamical interactions between higher-order rogue waves and various forms ofn-soliton solutions(n→∞)of the(2+1)-dimensional ANNV equation

We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.

Rogue Wave with a Controllable Center of Nonlinear Schrdinger Equation

The rogue waves with a controllable center are reported for the nonlinear Schroedinger 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.