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Showing papers on "Rogue wave published in 2021"


Journal ArticleDOI
TL;DR: In this paper, the authors used the physics-informed neural network to solve a variety of femtosecond optical soliton solutions of the high-order nonlinear Schrodinger equation.
Abstract: We use the physics-informed neural network to solve a variety of femtosecond optical soliton solutions of the high-order nonlinear Schrodinger equation, including one-soliton solution, two-soliton solution, rogue wave solution, W-soliton solution and M-soliton solution. The prediction error for one-soliton, W-soliton and M-soliton is smaller. As the prediction distance increases, the prediction error will gradually increase. The unknown physical parameters of the high-order nonlinear Schrodinger equation are studied by using rogue wave solutions as data sets. The neural network is optimized from three aspects including the number of layers of the neural network, the number of neurons, and the sampling points. Compared with previous research, our error is greatly reduced. This is not a replacement for the traditional numerical method, but hopefully to open up new ideas.

104 citations


Journal ArticleDOI
TL;DR: In this article, the neural network model of test function for the (3+1)-dimensional Jimbo-Miwa equation is extended to the 4-2-3 model by giving some specific activation functions.
Abstract: It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.

95 citations


Journal ArticleDOI
TL;DR: The multi-layer PINN deep learning method is used to study the data-driven rogue wave solutions of the defocusing nonlinear Schr\"odinger (NLS) equation with the time-dependent potential by considering several initial conditions such as the rogue wave.

55 citations



Journal ArticleDOI
TL;DR: In this article, the authors investigated the Ivancevic option pricing model and constructed the rogue wave and dark wave solutions by means of symbolic computation, these analytical solutions are obtained with the Maple.
Abstract: Under investigation in this paper is the Ivancevic option pricing model. Based on trial function method, rogue wave and dark wave solutions are constructed. By means of symbolic computation, these analytical solutions are obtained with the Maple. Perturbation solutions are obtained through direct perturbation method. These results will enrich the existing literature of the Ivancevic option pricing model. Dynamical characteristics for rogue waves and dark waves are exhibited by using three-dimensional plots, curve plots, density plots and contour plots.

42 citations


Journal ArticleDOI
TL;DR: In this paper, an inhomogeneous Hirota equation with variable dispersion and nonlinearity is considered and a novel transformation is introduced to map this equation to a constant coefficient Hirota equations.
Abstract: We consider an inhomogeneous Hirota equation with variable dispersion and nonlinearity. We introduce a novel transformation which maps this equation to a constant coefficient Hirota equation. By employing this transformation we construct the rogue wave solution of the inhomogeneous Hirota equation. Furthermore, we demonstrate that one can control the rogue wave dynamics by suitably choosing the dispersion and the nonlinearity. These results suggest an efficient approach for controlling the basic features of the relevant rogue wave and may have practical implications for the management of the rogue waves in nonlinear optical systems.

42 citations


Journal ArticleDOI
TL;DR: Based on a direct variable transformation, the authors obtained multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order solutions.
Abstract: Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

41 citations


Journal ArticleDOI
TL;DR: In this paper, the generalized unstable space time fractional nonlinear Schrodinger equation may be converted into plane systems using various ansatz transformation to obtain these solutions, and the authors plot graphical representation of their results in distinct dimensions.
Abstract: This article possess lump, lump with one kink, lump with two kink, rogue wave and lump interactions with periodic and kink solitons for the generalized unstable space time fractional nonlinear Schrodinger equations. According to theory of dynamical systems, Schrodinger equation may be converted into plane systems. We use various ansatz transformation to obtain these solutions. At the end, we plot graphical representation of our results in distinct dimensions.

38 citations


Journal ArticleDOI
TL;DR: In this paper, the root structures of the Yablonskii-Vorob'ev polynomial hierarchy of the rogue wave are determined by the root structure of the polynomials, and their orientations are controlled by the phase of the large parameter.

37 citations


Journal ArticleDOI
TL;DR: In this article, a bilinear Backlund transformation based on which a Lax pair is constructed is presented. And the effects of the noise perturbations on the obtained solutions are investigated.
Abstract: In this paper, outcomes of the study on the Backlund 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 Backlund 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.

36 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like (CDGKS-like) equation.

Journal ArticleDOI
TL;DR: In this article, a family of eigenfunction solutions of the Lax pair of the Kadomtsev-Petviashvili 1 (KP1) equation is constructed in terms of the Taylor coefficients of a fundamental exponential function associated with the lax pair.

Journal ArticleDOI
TL;DR: The two-dimensional doubly localized rogue waves on a background of dark solitons for the Fokas system, which are described by semi-rational solutions, are investigated, which have great importance in nonlinear science and physical applications.

Journal ArticleDOI
TL;DR: In this paper, a two-component derivative nonlinear Schrodinger equation (cDNLS) is described by a theory of ordinary differential equation, and analytic solutions describing different waveforms of cDNLS are obtained by virtue of Darboux transformation.

Journal ArticleDOI
TL;DR: In this article, the authors investigated doubly localized two-dimensional rogue waves for the DSI equation in the background of dark solitons or a constant, by employing the Kadomtsev-Petviashvili hierarchy reduction method in conjunction with the Hirota's bilinear technique.
Abstract: Doubly localized two-dimensional rogue waves for the Davey–Stewartson I equation in the background of dark solitons or a constant, are investigated by employing the Kadomtsev–Petviashvili hierarchy reduction method in conjunction with the Hirota’s bilinear technique. These two-dimensional rogue waves, described by semi-rational type solutions, illustrate the resonant collisions between lumps or line rogue waves and dark solitons. Due to the resonant collisions, the line rogue waves and lumps in these semi-rational solutions become doubly localized in two-dimensional space and in time. Thus, they are called line segment rogue waves or lump-typed rogue waves. These waves arise from the background of dark solitons, then exist in the background of dark solitons for a very short period of time, and finally completely decay back to the background of dark solitons. In particular circumstances which are characterized by special parametric conditions, the dark solitons in the long wave component of the DSI equation can degenerate into the constant background. In this case, the rogue waves appear and disappear in a constant background.

Journal ArticleDOI
TL;DR: The Peregrine solution of the NLSE has wide range of applications in physics with spatial scales that vary from microns to kilometres and can be applied to water waves, optics, plasma and Bose-Einstein condensate as mentioned in this paper.
Abstract: The nonlinear Schroedinger equation (NLSE) has wide range of applications in physics with spatial scales that vary from microns to kilometres. Consequently, its solutions are also universal and can be applied to water waves, optics, plasma and Bose-Einstein condensate. The most remarkable solution presently known as the Peregrine solution describes waves that appear from nowhere. This solution describes unique events localised both in time and in space. Following the language of mariners they are called ‘rogue waves’. As thorough mathematical analysis shows, these waves have properties that differ them from any other nonlinear waves known before. Peregrine waves can serve as ‘elementary particles’ in more complex structures that are also exact solutions of the NLSE. These structures lead to specific patterns with various degrees of symmetry. Some of them resemble ‘atomic like structures’. The number of particles in these structures is not arbitrary but satisfies strict rules. Similar structures may be observed in systems described by other equations of mathematical physics: Hirota equation, Davey-Stewartson equations, Sasa-Satsuma equation, generalised Landau-Lifshitz equation, complex KdV equation and even the coupled Higgs field equations describing nucleons interacting with neutral scalar mesons. This means that the ideas of rogue waves enter nearly all areas of physics including the field of elementary particles.

Posted Content
TL;DR: In this paper, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the derivative nonlinear Schrodinger equation (DNLS) in complex space.
Abstract: The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural network (PINN) which has been put forward to uncover dynamical behaviors of nonlinear partial different equation from spatiotemporal data directly, an improved PINN method with neuron-wise locally adaptive activation function is presented to derive localized wave solutions of the DNLS in complex space. In order to compare the performance of above two methods, we reveal the dynamical behaviors and error analysis for localized wave solutions which include one-rational soliton solution, genuine rational soliton solutions and rogue wave solution of the DNLS by employing two methods, also exhibit vivid diagrams and detailed analysis. The numerical results demonstrate the improved method has faster convergence and better simulation effect. On the bases of the improved method, the effects for different numbers of initial points sampled, residual collocation points sampled, network layers, neurons per hidden layer on the second order genuine rational soliton solution dynamics of the DNLS are considered, and the relevant analysis when the locally adaptive activation function chooses different initial values of scalable parameters are also exhibited in the simulation of the two-order rogue wave solution.

Journal ArticleDOI
TL;DR: In this article, the authors report a kind of breather, rogue wave, and mixed interaction structures on a variational background height in the Gross-Pitaevskii equation in the Bose-Einstein condensate by the generalized Darboux transformation method.
Abstract: We report a kind of breather, rogue wave, and mixed interaction structures on a variational background height in the Gross–Pitaevskii equation in the Bose–Einstein condensate by the generalized Darboux transformation method, and the effects of related parameters on rogue wave structures are discussed. Numerical simulation can discuss the dynamics and stability of these solutions. We numerically confirm that these are correct and can be reproduced from a deterministic initial profile. Results show that rogue waves and mixed interaction solutions can evolve with a small amplitude perturbation under the initial profile conditions, but breathers cannot. Therefore, these can be used to anticipate the feasibility of their experimental observation.

Journal ArticleDOI
18 Feb 2021-Chaos
TL;DR: In this paper, a rogue wave solution on the periodic background for the fourth-order nonlinear Schrodinger (NLS) equation was constructed by combining the method of nonlinearization of spectral problem with the Darboux transformation method.
Abstract: In this paper, we construct rogue wave solutions on the periodic background for the fourth-order nonlinear Schrodinger (NLS) equation. First, we consider two types of the Jacobi elliptic function solutions, i.e., dn- and cn-function solutions. Both dn- and cn-periodic waves are modulationally unstable with respect to the long-wave perturbations. Second, on the background of both periodic waves, we derive rogue wave solutions by combining the method of nonlinearization of spectral problem with the Darboux transformation method. Furthermore, by the study of the dynamics of rogue waves, we find that they have the analogs in the standard NLS equation, and the higher-order effects do not have effect on the magnification factor of rogue waves. In addition, when the elliptic modulus approaches 1, rogue wave solutions can reduce to multi-pole soliton solutions in which the interacting solitons form weakly bound states.

Journal ArticleDOI
01 Jun 2021
TL;DR: In this paper, a polynomial conjecture associated with rational solutions including rogue wave solutions of the KdV equation is presented, which can be used to show that an arbitrary linear combination of two Wronskian polynomials with a difference two between the Wronkian orders will again be a solution.
Abstract: A polynomial conjecture, associated with rational solutions including rogue wave solutions of the KdV equation, is presented. The conjecture can be used to show that for the bilinear KdV equation, an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders will again be a solution.

Journal ArticleDOI
01 Apr 2021-Optik
TL;DR: New multiple visions for the soliton solution to the modified nonlinear Schrodinger equation (MNLSE) that describes successfully these phenomenon of rogue waves in ocean engineering will contribute effectively in marine safety and improvement the pulse devices which cooperate to protect human life effectively.

Journal ArticleDOI
TL;DR: The statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of extreme events, often called rogue waves, in Kerr cavities.
Abstract: We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D) localized structures consist of either an isolated light bullet (LB), bound together, or could occur in clusters forming well-defined 3D patterns. They can be seen as stationary states in the reference frame moving with the group velocity of light within the cavity. The number of LBs and their distribution in 3D settings are determined by the initial conditions, while their maximum peak power remains constant for a fixed value of the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation of homoclinic snaking for dissipative light bullets. However, when the strength of the injected beam is increased, LBs lose their stability and the cavity field exhibits giant, short-living 3D pulses. The statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of extreme events, often called rogue waves.

Journal ArticleDOI
TL;DR: In this paper, a bilinear solution of the (1+1)-dimensional three-wave resonant interaction system is derived by the bilinearly method. And the results of this solution are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction respectively.
Abstract: General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction respectively. It is shown that while the first family of solutions associated with a simple root exist for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue-wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained.

Journal ArticleDOI
TL;DR: Both second-order and third-order line rogue waves are explicitly derived and their complex dynamical behaviors are illustrated by three-dimensional plots.

Journal ArticleDOI
TL;DR: In this article, the cubic-quintic nonlinear Schrodinger system with variable coefficients for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide was investigated.
Abstract: Twin-core optical fibers are applied in the fiber optic sensing technique and optical communication. Non-Kerr media are seen in plasma physics, nonlinear quantum mechanics and nonlinear optics. Propagation of an optical beam and superradiance for an atom in the waveguide are reported. This paper investigates the cubic-quintic nonlinear Schrodinger system with variable coefficients for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide. For the two components of the electromagnetic fields, Darboux-dressing transformation, semi-rational solutions and breather solutions are obtained. We acquire the Akhmediev breathers (ABs) and Kuznetsov-Ma (KM) solitons. Interaction between the rogue waves and KM/bright-dark solitons is presented. When b(z) is a linear/quadratic/cosine function, the ABs, rogue waves, KM and bright-dark solitons appear parabolic, cubic and wavy, respectively, where b(z) presents the delayed nonlinear response effects. We conduct the modulation instability for the plane wave solutions for a non-Kerr medium, twin-core nonlinear optical fiber or waveguide via the linear stability analysis: If χ < 0, the solutions are modulationally stable; otherwise, modulationally unstable, where χ is the growth rate of the instability.

Journal ArticleDOI
TL;DR: Li et al. as discussed by the authors developed second-order theory for narrowbanded surface gravity wavepackets experiencing a sudden depth transition based on a Stokes and multiple-scales expansion, and they validated their theory through comparison with fully nonlinear numerical simulations.
Abstract: This paper develops second-order theory for narrow-banded surface gravity wavepackets experiencing a sudden depth transition based on a Stokes and multiple-scales expansion. As a wavepacket travels over a sudden depth transition, additional wavepackets are generated that propagate freely obeying the linear dispersion relation and arise at both first and second order in wave steepness in a Stokes expansion. In the region near the top of the depth transition, the resulting transient processes play a crucial role. At second order in wave steepness, free and bound waves coexist with different phases. Their different speeds of travel result in a local peak a certain distance after the depth transition. This distance depends on the water depth . We validate our theory through comparison with fully nonlinear numerical simulations. Experimental validation is provided in a companion paper (Li et al, J. Fluid Mech., 2021, 915, A72). We conjecture that the combination of the local transient peak at second order and the magnitude of the linear free waves provides the explanation for the rogue waves observed after a sudden depth transition reported in a significant number of papers and reviewed in Trulsen etal (J. Fluid Mech., vol. 882, 2020, R2).

Journal ArticleDOI
TL;DR: It is found that the width and angle of the beak-shaped rogue wave along the space axis enlarge with the increase of a real parameter in the higher-order coupled nonlinear Schrodinger system.

Journal ArticleDOI
TL;DR: In this article, it was shown that universal rogue wave patterns exist in integrable systems and that these patterns are determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy through a linear transformation.

Journal ArticleDOI
TL;DR: In this article, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients and their dynamic features are shown in some 3D and contour plots.
Abstract: In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a ( $$2+1$$ )-dimensional variable-coefficient Kadomtsev–Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamic features are shown in some 3D and contour plots.

Journal ArticleDOI
TL;DR: In this article, the improved physics-informed neural network (IPINN) approach with neuron-wise locally adaptive activation function is presented to derive the data-driven localized wave solutions, which contain rational solution, soliton solution, rogue wave, periodic wave and rogue periodic wave with initial and boundary conditions in complex space.