Second-order Rogue Waves Research Articles

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.

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.

Elusive exotic structures and their collisional dynamics in (2+1)-dimensional Boiti-Leon-Pempinelli equation

In this paper, we investigate the (2+1) dimensional Boiti-Leon-Pempinelli (BLP) equation employing truncated Painlevé expansion approach and extract a plethora of localized nonlinear waves, including multi-dromions, multi-lumps, multi-rogue waves, generalized-breathers etc. The dromions are characterized as bright, dark and mixed (bright-dark) based on their intensity. The collisional dynamics of dromions shows that they change their shape or form upon interaction in addition to undergoing a phase change. The lump solutions of orders one and two are also extracted through appropriate test functions and observed to be non-interacting in nature. Also, the first-order and second-order rogue waves are also obtained through rational polynomials and shown to be unstable. The generalized breathers are obtained by utilizing the three-wave test function. The highlights of our investigation is that one encounters a strange coherent structure called ‘dromion filter’ which contains a dynamic and a stationary dromion. In addition, we are also able to unearth a ‘coexistent dromion-line soliton’.

Two-dimensional rogue wave clusters in self-focusing Kerr-media

In this paper we consider two-dimensional (2D) rogue waves that can be obtained from the approximate solution of the (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation with Kerr nonlinearity. The approximate method proposed in this paper not only reduces complex operations needed for the solution of high-dimensional nonlinear partial differential equations, but also furnishes an effective mechanism for constructing stable high-dimensional rogue wave clusters, and also provides for more abundant local rogue wave excitations. Such an investigation is important in view of the known tendency for instability of localized solutions to the 2D and 3D NLS equations with focusing Kerr nonlinearity. We adopt the (2 + 1)-dimensional NLS equation in cylindrical coordinates as the basic model, and reduce the complexity of its solution by looking for the radially-symmetric ring-like solutions with the specific feature that the radius of the ring L is much greater than the ring thickness w. In this manner, we obtain an approximate analytical solution with the separated radial and angular functions which depend on the two integer parameters, the azimuthal mode number m and the rogue wave solution's order number n. With the two parameters L and w also included in the approximate solution, we show the profiles of the first-order and the second-order rogue waves, and discuss their characteristics. The method for finding approximate rogue wave solutions of the (2 + 1)-dimensional NLS equation proposed in this paper can be used as an effective way to obtain higher-order rogue wave excitations, and can be extended to other (2 + 1)-dimensional nonlinear systems.

Vector dark-bright second-order rogue wave and triplets for a (3+1)-dimensional CNLSE with the partially nonlocal nonlinearity

Vector dark-bright second-order rogue wave and triplets for a (3+1)-dimensional CNLSE with the partially nonlocal nonlinearity

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.

Studies on the rogue waves of a (3 + 1)-dimensional Gross–Pitaevskii equation in the Bose–Einstein condensation

On the basis of the application in the Bose–Einstein condensation, we investigate a (3 + 1)-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients. With the aid of the Kadomtsev–Petviashvili hierarchy reduction method, we construct the [Formula: see text]th-order rogue-wave solutions in terms of the Gram determinant by introducing appropriate constraints. Using different coefficients for the diffraction [Formula: see text] and gain/loss [Formula: see text], we demonstrate the behaviors of the first- and second-order rogue waves by analytical and graphical means. We find that only if [Formula: see text], the rogue waves appear on the constant backgrounds; otherwise, the heights of the backgrounds change as time goes on. With the different choices of [Formula: see text] and [Formula: see text], the long-live, rapid-reducing and periodic rogue waves are discussed. The separated and aggregated second-order rogue waves are also shown on the constant and periodical backgrounds.

The splitting mechanism of the second-order rogue wave—Interaction between two component first-order Akhmediev breathers

Based on the nonlinear Schrödinger equation, it's illustrated that the peak intensity of Akhmediev breather is negatively correlated with the modulation frequency firstly. Then, two splitting modes of the second-order rogue wave have been studied in detail through the second-order rogue wave train formed by the nonlinear superposition of two first-order Akhmediev breathers whose modulation frequency ratio is 1:2. It is revealed that the process of the mode B splitting can be divided into three stages: the restructuring stage where the intensity of the latter sub-peaks of the main peak decreases first and then increases, the competition stage where the fine structure forms and the relaxation stage. The process of the mode A splitting can be divided into two stages: the restructuring stage and the relaxation stage. And both are due to the interaction between the two first-order component Akhmediev breathers. The results can be used to explain the different characteristics of the two splitting modes of the second-order rogue wave. In addition, the inevitability of the fine structure appearance in the process of the mode B splitting of the second-order rogue wave, which causes the transformation point on the peak intensity curve of the second order rogue wave, is also revealed. We anticipate that it will be a meaningful and novel way to study the characteristics of the high-order rogue wave through interaction between breathers with certain modulation frequency ratio.

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.

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.

Higher-order optical rogue waves in spatially inhomogeneous multimode fiber

The objective is to explore the mathematical analysis to understand and manipulate the dynamics of higher-order rogue waves in an inhomogeneous multi-mode optical media. For this purpose, we consider a theoretical model represented by the coupled nonlinear Schrödinger equations consisting of the spatially varying dispersion and nonlinearities arising from both coherent and incoherent nonlinear effects, which drives the spatially modulated linear and nonlinear refractive index of the media. We construct controllable spatiotemporally (doubly) localized rogue wave solutions through Darboux transformation and by designing a similarity transformation. Explicitly, we investigate the evolution of first- and second-order vector rogue waves possessing different characteristics such as degenerate and non-degenerate type bright, gray, and dark localized profiles that can be controlled by the inhomogeneity arising in the medium and polarization parameters available in the solutions. To highlight the consequences of the spatially inhomogeneous nature of the medium, we demonstrate the transition mechanism of rogue waves by adopting four different spatial modulations, namely periodic, hyperbolic, and localized behavior, by incorporating Jacobi elliptic functions. Significantly, these modulations lead to the transition of rogue waves to spatially-periodic Akhmediev breathers of single- and multi-peak symmetric and asymmetric behavior arising on top of snoidal, cnoidal, and dnoidal backgrounds. Further, the rogue waves experience amplification, suppression, splitting, and trapping due to localized soliton type background modulations. The observed results will facilitate improvement in understanding other types of localized nonlinear waves in different inhomogeneous media.

Dissipative electrostatic wave modulation in warm multi-ion dusty plasmas with superthermal electrons

Abstract A theoretical investigation has been carried out to explore the modulational instability (MI) of electrostatic waves in a warm multi-ion dusty plasma system containing positive ions, negative ions and positively or negatively charged dust in presence of superthermal electrons. With the help of the standard perturbation technique, it is found that the dynamics of the modulated wave is governed by a damped nonlinear Schrödinger equation (NLSE). Regions of MI of the electrostatic wave are precisely determined and the analytical solutions predict the formation of dissipative bright and dark solitons as well as dissipative first- and second-order rogue wave solutions. It is found that the striking features (viz., instability criteria, amplitude and width of rogue waves, etc.) are significantly modified by the effects of relevant plasma parameters such as degree of the electron superthermality, dust density, etc. The time dependent numerical simulations of the damped NLSE reveal that modulated electrostatic waves exhibit breather like structures. Moreover, phase plane analysis has been performed to study the dynamical behaviors of NLSE by using the theory of dynamical system. It is remarked that outcome of present study may provide physical insight into understanding the generation of several types of nonlinear structures in dusty plasma environments, where superthermal electrons, positive and negative ions are accountable (e.g. Saturn’s magnetosphere, auroral zone, etc.).

Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations

The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a 3 × 3 matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The n -fold Darboux transformation as well as the multiparametric n th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.

Three-component coupled nonlinear Schrödinger system in a multimode optical fiber: Darboux transformation induced via a rank-two projection matrix

In this paper, the investigation is on a three-component coupled nonlinear Schrödinger (NLS) system which governs the wave evolution in a multimode optical fiber. We construct a Darboux transformation (DT) induced via a rank-two projection matrix, and then derive an (N,m)-generalized DT and the Nth-order solution, where the positive integers N and m denote the numbers of iterative times and distinct spectral parameters, respectively. Focusing on the Nth-order solution on the nonzero–zero–zero background, we derive two kinds of waves which could not be derived by the DT induced via a rank-one projection matrix, that is, the so-called fundamental nonlinear wave (N=1) and degenerate fundamental nonlinear wave (N=2 and m=1). Via the asymptotic analysis, we find that the fundamental nonlinear wave is the nonlinear superposition of two dark–bright–bright solitons and a breather/Kuznetsov–Ma​ breather/rogue wave; and the degenerate fundamental nonlinear wave is the nonlinear superposition of four dark–bright–bright solitons and two breathers/two Kuznetsov–Ma​ breathers/a second-order rogue wave. Since such phenomena are not admitted for the one-component NLS equation and two-component coupled NLS system, they are more useful to understand the three-component coupled NLS system than what the latter two models admit. For other three-component coupled systems, more phenomena may be expected when the rank of projection matrix used to construct a DT is two rather than one, because our study presents an example.

Phase engineering of chirped rogue waves in Bose–Einstein condensates with a variable scattering length in an expulsive potential

We consider a cubic Gross-Pitaevskii (GP) equation governing the dynamics of Bose-Einstein condensates (BECs) with time-dependent coefficients in front of the cubic term and inverted parabolic potential. Under a special condition imposed on the coefficients, a combination of phase-imprint and modified lens-type transformations converts the GP equation into the integrable Kundu-Eckhaus (KE) one with constant coefficients, which contains the quintic nonlinearity and the Raman-like term producing the self-frequency shift. The condition for the baseband modulational instability of CW states is derived, providing the possibility of generation of chirped rogue waves (RWs) in the underlying matter-wave (BEC) model. Using known RW solutions of the KE equation, we present explicit first- and second-order chirped RW states. The chirp of the first- and second-order RWs is independent of the phase imprint. Detailed solutions are presented for the following configurations: (i) the nonlinearity exponentially varying in time; (ii) time-periodic modulation of the nonlinearity; (iii) a stepwise time modulation of the strength of the expulsive potential. Singularities of the local chirp coincide with valleys of the corresponding RWs. The results demonstrate that the temporal modulation of the s-wave scattering length and strength of the inverted parabolic potential can be used to manipulate the evolution of rogue matter waves in BEC.

Three-Dimensional Rogue Waves in Earth’s Ionosphere

The modulational instability of ion-acoustic waves (IAWs) in a four-component magneto-plasma system consisting of positive–negative ions fluids and non-Maxwellian (r,q) distributed electrons and positrons, is investigated. The basic system of fluid equations is reduced to a three-dimensional (3D) nonlinear Schrödinger Equation (NLS). The domains of the IAWs stability are determined and are found to be strongly affected by electrons and positrons spectral parameters r and q and temperature ratio Tp/Te (Tp and Te are positrons and electrons temperatures, respectively). The existence domains, where we can observe the ion-acoustic rogue waves (IARWs) are determined. The basic features of IARWs are analyzed numerically against the distribution parameters and the other system physical parameters as Tp/Te and the external magnetic field strength. Moreover, a comparison between the first- and second-order rogue waves solution is presented. Our results show that the nonlinearity of the system increases by increasing the values of the non-Maxwellian parameters and the physical parameters of the system. This means that the system gains more energy by increasing r, q, Tp, and the external magnetic field through the cyclotron frequency ωci. Finally, our theoretical model displays the effect of the non-Maxwellian particles on the MI of the IAWs and RWs and its importance in D–F regions of Earth’s ionosphere through (H+,O2−) and (H+,H−) electronegative plasmas.

Modulational instability of dust-ion-acoustic waves and associated first and second-order rogue waves in a super-thermal plasma

A proper theoretical research has been carried out to explore the modulational instability (MI) conditions of dust-ion-acoustic (DIA) waves (DIAWs) in a three-component dusty plasma system containing inertialess κ-distributed electrons, and inertial warm positive ions and negative dust grains. The nonlinear Schrödinger equation (NLSE) has been derived by employing the reductive perturbation method. The numerical analysis under consideration demonstrates two types of modes, namely, fast and slow DIA modes. The MI conditions of DIAWs and the configuration of the energetic rogue waves (RWs) associated with DIAWs in the modulationally unstable parametric regime have been rigorously changed by the variation of various plasma parameters, namely, charge, mass, temperature, and number density of the plasma species. The findings of our present investigation will be useful in understanding the criteria for the formation of electrostatic RWs in both astrophysical environments (viz., Jupiter’s magnetosphere, cometary tails, pulsar magneto-sphere, interstellar medium, Earth’s mesosphere, Saturn’s rings, etc.) and laboratory experiments.