First-order Rogue Wave Research Articles

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Voltage manipulation of magnetic rogue waves for controllable electromagnetic interference

In this paper, we investigated the excitation of magnetic rogue waves and their voltage control in ferromagnetic nanowires. Starting from the Maxwell equations coupled with the Landau–Lifshitz–Gilbert equation, we derived the generalized derivative nonlinear Schrödinger equation under the long-wave approximation, which describes the nonlinear wave evolution in anisotropic ferromagnetic nanowires. By constructing the corresponding Lax pair, we obtained the exact solutions of first-order and second-order magnetic rogue waves via a generalized Darboux transformation. After spectral analysis and modulation instability analysis, we present the excitation regions of magnetic rogue waves. Furthermore, we tested and achieved precise control over the occurrence positions of first-order and second-order magnetic rogue waves by applying bias voltage. We further discussed the physical implications of these results, highlighting their potential applications in controlled electromagnetic interference for data destruction.

A Kundu-nonlinear Schrödinger equation: Rogue waves, breathers, and mixed interaction solutions.

A Kundu-nonlinear Schrödinger equation that can be utilized to simulate the pulse propagation in optical fibers is researched in this paper. First, the Lax integrability of the above equation is proved and its modulational instability (i.e., the main mechanism for producing the rogue wave solutions and the breather solutions) is calculated. Subsequently, using the generalized perturbation (n,N-n)-fold Darboux transformation, the rogue waves, breathers, and mixed interaction solutions are acquired, as well as the impact of various parameters on the solutions is examined. In particular, when we assume that the coefficient of the equation is θ=mx+dt, some new wave structures are found based on parameter variations, such as the rotational separation of first-order rogue waves, scale-like structures generated by second-order breathers, etc., which offer novel ideas for producing different signals via optical fibers. Ultimately, the classification numbers of mixed solutions of rogue waves and breathers are provided, which can better observe how the two types of the local waves are combined, in accordance with the distributions of the increasing numbers of algebraic equations.

Baseband modulational instability and interacting localized mixed waves in coherently coupled optical media

We consider a coherently coupled nonlinear Schrödinger equations (model equations) that describe interacting localized waves in coherently coupled optical media. The baseband modulational instability in such regimes is investigated, leading the result that the modulational instability does not necessary imply the baseband modulational instability. By means of a set of linear transformations, the model equations are converted into two independent scalar nonlinear Schrödinger equations. Based on the generalized perturbation (n,N−n)–fold Darboux transformation, we report and discuss analytical mixed localized wave solutions of these scalar equations. A first-order rogue wave solution of the scalar nonlinear Schrödinger equations is also presented. We then show that linear superpositions of different found analytical mixed localized wave solutions of the derived scalar equations results into several kinds of coherent nonlinear mixed structures such as, coexisting bright/dark soliton-breather, rogue wave-breather-soliton, interacting rogue waves, bright/dark breathers, and bright/dark solitons.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

Rogue waves and their patterns for the coupled Fokas–Lenells equations

In this work, we explore the rogue wave patterns in the coupled Fokas–Lenells equation by using the Darboux transformation. We demonstrate that when one of the internal parameters is large enough, the general high-order rogue wave solutions generated at a branch point of multiplicity three can be decomposed into some first-order outer rogue waves and a lower-order inner rogue wave. Remarkably, the positions and the orders of these outer and inner rogue waves are intimately related to Okamoto polynomial hierarchies.

Annular rogue waves in whispering gallery mode optical resonators

Based on the variable-coefficient Lugiato-Lefever equation, nonlinear dynamics of rogue waves in whispering gallery mode optical resonators are studied. By means of the analytical solution in the absence of pump, dynamical behaviors of the rogue waves in the periodic system are discussed. Firstly, the first-order rogue wave under the condition of no pumping has obvious periodic nature, and the periodicity of width and amplitude of rogue wave can be consistent by controlling the wavefront curvature, and the energy of rogue wave remains stable under the small gain parameter, and the inverse diffraction effect makes the center position of rogue wave produce a small offset. In addition, the amplitudes of second-order rogue waves have more complicated periodicity and longer period lengths both with the good stability. At last, the dynamical behaviors of rogue wave after adding external excitation are investigated. Under the periodical external excitation with the lower intensity, rogue wave can achieve a longer stable transmission time, while the smaller wavefront curvature can effectively prolong the stable transmission time of rogue wave. Moreover, we investigate the influence of different forms of external excitation on rogue wave, and the periodical external excitation makes rogue waves more stable than the constant external excitation. These results extend the application of higher-order rogue waves in (2 + 1)-dimensional nonlinear systems, and have implications for the development of high-energy optical pulses.

Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE

Under investigation in this paper is the cubic–quintic nonlinear Schrödinger equation, which describes the propagation of optical on resonant-frequency fields in the inhomogeneous fiber. According to abundant previous researches on the model, exact soliton solutions and rogue wave solutions have been derived through Darboux transformation. The modulation instability phenomenon has been analyzed to evaluate the ability of an initially perturbed plane wave to split into localized energy packets when propagating in a dispersive and nonlinear medium.Numerical solutions with high accuracy are needed in fields of production and engineering. Nonetheless, the data acquisition costs of the optical pulse transmission system is high, which will limit the accuracy and the efficiency of typical numerical and data-driven methods. With the physical knowledge embedded into neural networks in the form of loss function, the problem of big data dependence has been solved. For dynamic analysis on optical pulses with small amount of known information, we strive to obtain high accuracy numerical solutions. Considering the case that the cubic–quintic nonlinear Schrödinger equation is converted to the Kundu–Eckhaus equation with simplified coefficient constraints through variable transformation, we construct modified physics-informed neural networks, where conversions on the input and output are attached to deep neural networks. Training networks with the given initial and boundary data, we effectively derive the expected soliton and rogue wave solutions, where the approximated one-soliton, two-soliton, first-order and second-order rogue waves are included. In general, the modified network reaches low prediction errors with small data available. With the coefficients of equations, the weights and the bias of networks combined as parameters to be trained, we deduce the corresponding value of condition settings for different systems. Moreover, we simulate diverse localized waves in the context of nonlinear electrical transmissions with different environment settings and compare the evolution process to reach conclusions on the parameter discovery.

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

In the text, rational solutions to a (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The first-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing different dispersion coefficient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two first-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is different from that in branches of line rogue waves. Besides, propagation of the first-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the plane.

Physics-informed Neural Network method for the Modified Nonlinear Schrödinger equation

In this paper, we apply the Physics-informed Neural Network (PINN) method to investigate some nonlinear wave solutions of the Modified Nonlinear Schrödinger (MNLS) equation, which characterizes the ultrashort pulse propagation well in optical fibers. We generate the training dataset by the Latin Hypercube Sampling (LHS) method under Dirichlet boundary conditions. We use the Xavier initialization method to generate the parameters of the neural network, and then use the Adam algorithm to optimize them. Finally, we derive the data-driven solutions of single soliton solution, double solitons solution, single breather solution, double breather solution, first-order rogue wave and second-order rogue wave solutions for the MNLS equation. Through loss and L2 error of the experimental results, the PINN method has been verified to have prominent stability and effectiveness.

Higher-order rogue waves with controllable fission and asymmetry localized in a (3 + 1)-dimensional generalized Boussinesq equation

The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations. Such a nonlinear model considered in this paper as the concrete example is the (3 + 1)-dimensional generalized Boussinesq (gB) equation, and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters. It is indicated by the (3 + 1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave, but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes. The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.

Mixed localized waves and their dynamics for a matrix Lakshmanan–Porsezian–Daniel equation

Interactions between different localized waves are of great significance to physical systems. In this paper, we study the mixed localized waves and their dynamics based on the matrix Lakshmanan–Porsezian–Daniel equation. First, we construct the Nth-order mixed localized solutions describing the interactions between the (N−1) th-order rogue waves and breathers. Using these solutions, we discuss the second- and third-order mixed localized waves, as well as their dynamics. Furthermore, we describe five types of interactions between rogue waves and breathers: between the first-order rogue waves and temporal period breathers, the first-order rogue waves and spatial period breathers, the first-order rogue waves and spatial-temporal period breathers, the second-order rogue waves and temporal period breathers, and the second-order rogue waves and spatial period breathers. These results may be useful for the study of nonlinear wave interactions in physical systems.

Rogue waves of the coupled modified nonlinear Schrödinger equations

In this paper, we employ the generalized Darboux transformation to obtain rogue wave solutions of the coupled modified nonlinear Schrödinger equations. By considering both double-root and triple-root situations of the spectral characteristic equation, we derive the first-order and second-order rogue wave solutions. We find that this coupled system admits abundant spatiotemporal patterns of rogue waves including the double, triple, quadruple and sextuple structures. In addition, we also demonstrate the coexistence dynamics of rogue waves, which depend on the choice of structural parameters.

Controllable optical rogue waves in inhomogeneous media

This paper investigates new rogue wave solutions of the nonlinear Schrödinger equation with variable coefficients, utilizing the self-similar transformation method. A new rogue wave family is introduced, which displays different wave structures. When one chooses appropriate variable coefficients, a series of first-order, second-order, and third-order rogue waves are obtained. We explore the generation and interaction of rational polynomial rogue waves in detail, and discuss some characteristics of their structures. The paper provides theoretical basis for studying the dynamic behavior of optical rogue wave in inhomogeneous media, and presents potential application value for control of rogue waves in fibers, plasmas and other fields of physics.

Multiple-order rogue wave solutions to a (2+1)-dimensional Boussinesq type equation

In this paper, based on the Hirota bilinear method and symbolic computation approach, multiple-order rogue waves of (2+1)-dimensional Boussinesq type equation are constructed. The reduced bilinear form of the equation is deduced by the transformation of variables. Three kinds of rogue wave solutions are derived by means of bilinear equation. The maximum and minimum values of the first-order rogue wave solution are given at a specific moment. Furthermore, the second-order and third-order rogue waves are explicitly derived. The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.

Continuous limit and position adjustable rogue wave solutions for the semi-discrete complex coupled system associated with 4 × 4 Lax pair

In this letter, we propose a new semi-discrete complex coupled system associated with 4 × 4 Lax pair. Firstly, the continuous limit technique is used to map this semi-discrete system to two new continuous complex equations. Then, the discrete generalized (m,N−m)-fold Darboux transformation for this system is first proposed. Finally, through applying the resulting Darboux transformation, some novel rogue wave and periodic wave structures with position adjustable parameters are investigated and discussed graphically. In particular, it is found that in the same semi-discrete system, the first-order rogue wave can have one or two fundamental rogue waves, and the second-order rogue wave can have three or six fundamental rogue waves.

Hybrid structures of rogue waves and breathers for the coupled Hirota system with negative coherent coupling

Hybrid waves play the significant parts in the nonlinear physical systems. In this work, we study the hybrid waves for the coupled Hirota system with negative coherent coupling. Firstly, based on the generalized Darboux transformation method, we derive the Nth-order hybrid wave solutions with two spectral parameters, which can describe the mth-order rogue waves and (N−M)th-order breathers. According to such solutions, we graphically illustrate the properties of the second-, third- and fourth-order hybrid waves. Moreover, we exhibit five kinds of hybrid structures consisting of the rogue waves and breathers: (i) consisting of the first-order rogue waves and first-order breathers; (ii) consisting of the second-order degenerate breathers; (iii) consisting of the first-order rogue waves and second-order degenerate breathers; (iv) consisting of the second-order rogue waves and first-order breathers; (v) consisting of the second-order rogue waves and second-order degenerate breathers. In addition, we show that the angle between the two-hump bright breathers and the negative direction of the -axis increases with the value of the real parameter in the coupled Hirota system with negative coherent coupling.

Engineering dissipative chirped solitons in the cardiac tissue under electromagnetic induction

This work deals with the engineering dissipative chirped solitary waves in the cardiac tissue under electromagnetic induction. Employing the a two-dimensional Ginzburg-Landau (GL) equation describing the spatiotemporal evolution of transmembrane potential in cardiac cells under magnetic flow effect, we use the phase imprint technique to reduce the model equation into a derivative GL equation (alias chirping model equation) that allow us to investigate the dynamics of dissipative chirp solitary wave propagating through a cardiac tissue. The baseband modulational instability (MI) of the chirping model is investigated and the analytical expression of the MI growth rate is derived. In the baseband MI regime, we use the approximative solitonlike solution of the chirping model to investigate the dynamics of transmembrane potential in cardiac cells. We show that the frequency chirp associated to the solitary wave is directly proportional to the wave amplitude and the nonlinear chirping parameter can be used to modulate the chirping amplitude. Also, our results show that the approximate solution of the chirping model can be used to generate bright (dissipative) solitons, dark solitons, breather solitons and first-order rogue wave in the cardiac cell under electromagnetic induction. Our theoretical results are confirmed by numerical simulations. The results of this work can motivate the investigations of qualitative new phenomena in the presence of large dissipation.

Higher-order rogue wave solutions of the Sasa–Satsuma equation

Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton.

Solitons, rogue waves and interaction behaviors of a third-order nonlinear Schrödinger equation

In this paper, we investigate a third-order nonlinear Schrödinger equation (NLSE) which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The higher-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detailed calculations.