Rogue Waves Research Articles

Modulation instability spectrum and rogue waves of the repulsive lattices

Modulation instability spectrum and rogue waves of the repulsive lattices

Dispersive Wave Focusing on a Shear Current: Part 1—Linear Approximations

AbstractWe consider the evolution and kinematics during dispersive focussing, for a group of waves propagating atop currents varying with depth. Our analysis assumes long-crested linear waves propagating at arbitrary angles relative to the current. Although low steepness is assumed, the linear model is often a reasonable approximation for understanding rogue waves. A number of analytical approximate relations are derived assuming different sub-surface current profiles, including linearly varying current, exponentially varying current, and currents of arbitrary depth profile which are weakly sheared following the approximation of Stewart and Joy (Deep Sea Res. Abs. 21, 1974). The orbital velocities are likewise studied. While shear currents have modest influence on the motion of the envelope of the wave group, they significantly change wave kinematics. Horizontal orbital velocities are either amplified or suppressed depending on whether the shear is opposing or following, respectively. To illustrate these phenomena we consider a real-world example using velocity profiles and wave spectra measured in the Columbia River estuary. Near the surface at the point where focusing occurs, horizontal orbital velocities are, respectively, increased and decreased by factors of 1.4 and 0.7 for focusing groups propagating on following and opposing shear (respectively, upstream and downstream in the earth-fixed reference system). The implications for the forces a focusing wave group can exert on vessels and installations are profound, emphasising the importance of considering current profiles in maritime operations.

Clinical and electroencephalogram characteristics of methylmalonic acidemia with MMACHC and MUT gene mutations

ObjectiveThis study investigated the clinical, imaging, and electroencephalogram (EEG) characteristics of methylmalonic acidemia (MMA) with nervous system damage as the primary manifestation.MethodsFrom January 2017 to November 2022, patients with nervous system injury as the main clinical manifestation, diagnosed with methylmalonic acidemia by metabolic and genetic testing, were enrolled and analyzed. Their clinical, imaging, and electroencephalogram data were analyzed.ResultsA total of 18 patients were enrolled, including 15 males and 3 females. The clinical symptoms were convulsions, poor feeding, growth retardation, disorder of consciousness, developmental delay, hypotonia, and blood system changes. There were 6 cases (33%) of hydrocephalus, 9 (50%) of extracerebral space widened, 5 (27%) of corpus callosum thinning, 3 (17%) of ventricular dilation, 3 (17%) of abnormal signals in the brain parenchyma (frontal lobe, basal ganglia region, and brain stem), and 3 (17%) of abnormal signals in the lateral paraventricular. In addition, there were 3 cases (17%) of cerebral white matter atrophy and 1 (5%) of cytotoxic edema in the basal ganglia and cerebral peduncle. EEG data displayed 2 cases (11%) of hypsarrhythmia, 3 (17%) of voltage reduction, 12(67%) of abnormal discharge, 13 (72%) of abnormal sleep physiological waves or abnormal sleep structure, 1 (5%) of immature (delayed) EEG development, and 8 (44%) of slow background. There were 2 cases (11%) of spasms, 1 (5%) of atonic seizures, and 1 (5%) of myoclonic seizures. There were 16 patients (89%) with hyperhomocysteinemia. During follow-up, 1 patient was lost to follow-up, and 1 died. In total, 87.5% (14/16) of the children had varying developmental delays. EEG was re-examined in 11 cases, of which 8 were normal, and 3 were abnormal. Treatments included intramuscular injections of vitamin B12, L-carnitine, betaine, folic acid, and oral antiepileptic therapy. Acute treatment included anti-infective, blood transfusion, fluid replacement, and correcting acidosis. The other treatments included low-protein diets and special formula milk powder.ConclusionMethylmalonic acidemia can affect the central nervous system, leading to structural changes or abnormal signals on brain MRI. Metabolic screening and genetic testing help clarify the diagnosis. EEG can reflect changes in brain waves during the acute phase.

Rogue waves and instability arising from long-wave-short-wave resonance beyond the integrable regime.

We consider instability and localized patterns arising from the long-wave-short-wave resonance in the nonintegrable regime numerically. We study the stability and instability of elliptic-function periodic waves with respect to subharmonic perturbations, whose period is a multiple of the period of the elliptic waves. We thus find the modulational instability (MI) of the corresponding dnoidal waves. Upon varying parameters of dnoidal waves, spectrally unstable ones can be transformed into stable states via the Hamiltonian Hopf bifurcation. For snoidal waves, we find a transition of the dominant instability scenario between the MI and the instability with a bubblelike spectrum. For cnoidal waves, we produce three variants of the MI. Evolution of the unstable states is also considered, leading to formation of rogue waves on top of the elliptic-wave and continuous-wave backgrounds.

Solitonic rogue and modulated wave patterns in the monoatomic chain with anharmonic potential

Modulation instability and rogue wave structures have been investigated in this work. This study is an extension of the work in Abbagari (2023), where a nonlinear Schrödinger equation with higher-order dispersion is derived to show only the development of the modulated waves bounded to bright soliton as a nonlinear exhibition of modulation instability. Here, the coupled nonlinear Schrödinger equation is derived by using the multi-scale scheme. An overview of the analytical calculations of the perturbed plane wave is carried out to show the effects of the nonlinear chain parameters on the modulation instability growth rate and bandwidths. The interest of this study lies equally in the nonlinear modes of excitation, where solitonic waves are generated under certain conditions in lower and upper frequency bands. On the other hand, relevant results have been developed to show the features of the type I and type II rogue waves of the Manakov system. Such investigations are obtained under the variation of the interaction potential parameters and the free parameter of the similarity method. Via a numerical simulation, rogue wave structures have been generated as a consequence of the long-time evolution of the perturbed plane wave. At a specific time of propagation, another localized object has been obtained to show the Akhmediev breathers and Kuznetsov-Ma solitons clusters under a strong perturbed wave number. These results have opened up new features, and many applications could follow in the future.

Triad resonance for internal waves in a uniformly stratified fluid: Rogue waves and breathers.

Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined. Exact solutions with four free parameters (two related to the amplitudes of the background plane waves, two related to the frequencies of slowly varying envelopes) describe motions localized in both space and time. The differences between rogue waves of the degenerate versus the nondegenerate cases are highlighted. The phase and profile of the degenerate case rogue waves are correlated. The volume or energy of the rogue wave (defined as the total extent or energy contents of the fluid set in motion for the duration of the rogue wave) may change drastically, if the wave envelope parameters vary. Pulsating modes (breathers) have been studied previously by layered-fluid and modified Korteweg-de Vries models. Here we extend the consideration to stratified fluids but for the simpler case of nondegenerate triads. Instabilities of fission and fusion of breathers are confirmed computationally with Floquet analysis. This knowledge should prove useful for energy transfer processes in the oceans.

Hybrid structures of the rogue waves and breather-like waves for the higher-order coupled nonlinear Schrödinger equations

Under investigation in this paper is the higher-order coupled nonlinear Schrödinger system, which describe the ultrashort optical pulses in nonlinear medium. Based on Lax pair, choosing nonzero seed solutions, Nth-order hybrid solutions for the higher-order coupled nonlinear Schrödinger equations are constructed via the generalized Darboux transformation. The profile and energy distribution of three kinds of hybrid structures of the rogue waves and breather-like waves were exhibited: (I) the first-order rogue waves and one breather-like waves; (II) the second-order rogue waves and two breather-like waves; (III) the third-order rogue waves and three breather-like waves. In addition, the influences of parameters for the properties of hybrid structures are discussed.

Quasi-periodic breathers and rogue waves to the focusing Davey–Stewartson equation

The Davey–Stewartson equation has been instrumental in describing various physical phenomena, especially (2+1)-dimensional breathers and rogue waves. In this paper, we present a direct approach to studying the quasi-periodic breathers of the Davey–Stewartson equation. By employing Hirota’s bilinear method and leveraging certain identities of theta functions, the problem is transformed into an over-determined nonlinear algebraic system, which can be formulated as a nonlinear least square problem and solved by classical numerical iterative algorithms. Through asymptotic analysis and numerical experiments, we categorize these solutions into three cases: quasi-periodic breathers, quasi-periodic stationary breathers, and quasi-periodic homoclinic orbits. The latter exhibit behavior reminiscent of quasi-periodic rogue waves, which are often observed in oceanic rogue wave phenomena. This work advances our understanding of the dynamics and properties of (2+1)-dimensional quasi-periodic breathers and rouge waves.

Electron acoustic counterpropagating multi-solitons and rogue waves collision in an unmagnetized plasma in the presence of critical density ratios

In this paper, the nonlinear propagation of electrostatic collisional among multi-solitons around the critical values along with their corresponding phase shifts and collision between two rouge waves propagating toward each other is studied in an unmagnetized plasma environment. Using the concept of Hirota’s bilinear method, the useful forms of multi-solitons solutions of the coupled modified Korteweg–de Vries equations (mKdVEs) are determined. Furthermore, the coupled nonlinear Schrödinger equations (NLSEs) are derived from mKdVEs using the appropriate starching coordinates. The analytic solutions of different orders for the coupled NLSEs are also presented. The effects of the parameters related to the plasma environment on the electron acoustic scattered solitons, phase shifts and scattered rouge waves are analyzed. The proposed results provide the theoretical guidance to understand the propagation characteristics of collisional solitons, and their phase shifts around the critical values and collisional rouge waves in the modulated ranges.

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

The Boussinesq and Camassa–Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physics-informed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa–Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision time-splitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shapedsoliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method.

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A Novel Method for Forecasting Freak Wave Occurrence in A Short Crested Coastal Sea

A Novel Method for Forecasting Freak Wave Occurrence in A Short Crested Coastal Sea

Lump solution, lump-stripe solution, rogue wave solution and periodic solution of the (2 + 1)-dimensional Fokas system

Lump solution, lump-stripe solution, rogue wave solution and periodic solution of the (2 + 1)-dimensional Fokas system

Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas

An asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct (carrier) wavenumbers (k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_2$$\\end{document}) and amplitudes (Ψ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _1$$\\end{document} and Ψ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Psi _2$$\\end{document}) are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}-distributed electron background in one dimension. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the two carrier wavenumbers. These coefficients present no symmetry whatsoever, in the general case (of different wavenumbers). The possibility for coupled envelope (vector soliton) solutions to occur has been investigated. Although the CNLS equations are asymmetric and non-integrable, in principle, the system admits various types of vector soliton solutions, physically representing nonlinear, localized electrostatic plasma modes, whose areas of existence is calculated on the wavenumbers’ parameter plane. The possibility for either bright (B) or dark (D) type excitations for either of the (2) waves provides four (4) combinations for the envelope pair (BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the soliton parameters (maximum amplitude, width) are also calculated for each type of vector soliton solution, in its respective area of existence. The dependence of the vector soliton characteristics on the (two) carrier wavenumbers and on the spectral index κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} characterizing the electron distribution has been explored. In certain cases, the (envelope) amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among copropagating wavepackets. As κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} decreases from large values, modulational instability occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of vector solitons on the parameter plane(s) also varies significantly with decreasing κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document}, and in fact dramatically for κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa$$\\end{document} between 3 and 2. Deviation from the Maxwell-Boltzmann picture therefore seems to favor modulational instability as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

Data-driven high-order rogue waves and parameters discovery for Gardner equation using deep learning approach

In this paper, an adaptive mix training physics-informed neural networks (A-MTPINNs) model is proposed to study the high-order rogue waves of the famous Gardner equation (GE). Through a series of numerical experiments, we can conclude that this A-MTPINNs model can not only recover the dynamic behavior of the high-order rogue waves of the GE well, but also can guarantee higher learning ability and prediction accuracy than the original PINNs model, and the prediction accuracy is improved by two orders of magnitude. By testing that the A-MTPINNs model can maintain high robustness under different noises for the data-driven solutions of the GE. Furthermore, this model also has good performance for the inverse problem of the Gardner equation by a data-driven discovery approach and also remains robust in noise experiments.

Localized wave solutions and interactions of the (2+1)-dimensional Hirota–Satsuma–Ito equation

This paper studies the (2+1)-dimensional Hirota–Satsuma–Ito equation. Based on an associated Hirota bilinear form, lump-type solution, two types of interaction solutions, and breather wave solution of the (2+1)-dimensional Hirota–Satsuma–Ito equation are obtained, which are all related to the seed solution of the equation. It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons, and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton. Furthermore, the breather wave solution is also obtained by reducing the two-soliton solutions. The trajectory and period of the one-order breather wave are analyzed. The corresponding dynamical characteristics are demonstrated by the graphs.

Novel dynamical behaviors of interaction solutions of the new (3+1)-dimensional integrable fourth-order nonlinear equation

The breather, rogue and interaction waves of the (3+1)-dimensional integrable fourth-order nonlinear equation are reviewed. The breather and rouge waves are attained with the aid of the extended homoclinic test. Thereafter, the interaction solutions between a lump wave and a 1-kink or 2-kink soliton are researched. Additionally, four kinds of interaction solutions between lump, kink, and periodic waves through a ” rational-cosh-cos” test function are established. Furthermore, the dynamic attributions of the attained solutions are presented utilizing the graphical analysis.

Generalized perturbation (n, N − n) fold Darboux transformation for a nonlocal Hirota equation with variable coefficients

In this paper, a nonlocal Hirota equation with variable coefficients is investigated by applying the generalized perturbation (n, N − n) fold Darboux transformation method and Taylor expansion method. Multi-soliton solutions are obtained when the seed solution is trivial, and multi-soliton solutions, multi-breather solutions, high-order rogue wave solutions and their interaction solutions are obtained when the seed solution is a plane wave solution. Especially, we get the interaction solution of soliton, breather and rogue wave solution. In addition, by choosing appropriate parameters, the dynamic behaviors of the obtained solution are explored.

Formation of solitons for the modified nonlinear Schrödinger equation

The study of modified nonlinear Schrödinger equation plays a significant role in the description of wave propagation through optical waveguides and rogue waves in ocean. The main aim of this work is to obtain solitons as well as other types of exact wave solutions to the modified nonlinear Schrödinger equation. The accurate traveling wave solutions are obtained using the [Formula: see text]-exponential expansion method and the modified auxiliary equation method. Consequently, soliton solutions and periodic wave solutions have been retrieved. The graphical illustrations of the results are provided for suitable values of the parameters involved in the solutions to explain the dynamical nature of the considered equation.

New type of rogue waves

New type of localized solutions for the two-dimensional multicomponent Yajima–Oikawa system is presented. The dynamics of solutions of this type occurs on the zero background and is similar to that of rogue waves.