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ABSTRACTThe interplay of dust acoustic waves (DAWs) and it's amplitude modulation is investigated in a magnetized five‐component cometary plasma comprising positively charged hydrogen and oxygen ions, negatively charged dust grains, along with superthermal electrons (solar and cometary). The interplay dynamics of the dust acoustic waves (DAWs) is described by the Kortweg–de Vries (KdV) equation. The appearance of multi‐soliton profiles are derived using Hirota's method. The amplitude modulation of the DAWs is investigated through nonlinear Schrodinger equation. In the numerical study, it appears that the dark‐envelope soliton may exist in such cometary plasmas, but the emergence of bright‐envelope soliton is not possible at any condition. Superthermality, magnetic field strength, ion and electron concentration, and other physical factors have all been explored for their impacts. The findings of this research may contribute to our understanding of the nonlinear characteristics and wave propagation stability of electrostatic wave excitations in cometary coma environments. [ABSTRACT FROM AUTHOR]

We study a spherical, self‐gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity‐induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small‐amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schrödinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a "nonlinear instability" (occurring in the nonlinear regime of the system), namely, the modulational instability, which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature‐induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies. [ABSTRACT FROM AUTHOR]

This study delves into the soliton solutions of the (1+1)-dimensional perturbed complex Ginzburg–Landau equation characterized by Kerr law, under the condition of absent chromatic dispersion. The complex Ginzburg–Landau equation, a basic equation in nonlinear science, describes various complex systems including optical fibers, Bose–Einstein condensates, and fluid dynamics. By applying the Auxiliary equation scheme, we explore the bright and kink soliton solutions and modulation instability of the considered model by incorporating Kerr nonlinearity, which describes the intensity-dependent refractive index in optical media. Furthermore, detailed 3D-surface and 2D graphs of obtained soliton solutions have been added. Additionally, graphs illustrating the impacts of various parameters for the considered model on soliton solutions are presented in this study. Our findings provide deeper inspirations into the structures of soliton formation and propagation in nonlinear media, contributing to the theoretical understanding and potential implementations in optical communication systems and beyond. This work underscores the critical role of nonlinearity in forming solitons and offers a framework for further exploration of perturbed nonlinear systems devoid of chromatic dispersion effects. [ABSTRACT FROM AUTHOR]

The article explores the spatial Whitham equation as a model for water wave evolution, comparing it to other equations like the temporal Korteweg–de Vries and Whitham equations. It demonstrates that the spatial Whitham equation accurately predicts various water wave phenomena and compares numerical simulations with laboratory experiments on waves of depression and elevation in shallow and deep water. While the spatial Whitham equation provides accurate predictions for most wave phenomena, it faces challenges in accurately modeling wave packets on deep water. The text also delves into conserved quantities and stability properties of the spatial Whitham equation. [Extracted from the article]

The article "Evolution of water wave packets by wind in shallow water" explores the use of the Korteweg– de Vries (KdV) equation to study wind-driven water wave packets in shallow water. It discusses the impact of forcing/friction terms on wave growth and decay, utilizing the Whitham modulation theory to predict these rates. Numerical simulations validate the theoretical predictions, highlighting the formation of solitary waves and the emergence of a soliton gas in various scenarios. The study provides insights into the evolution of water wave packets by wind in shallow water, shedding light on the complex dynamics involved. [Extracted from the article]

Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an "EP comb" has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium. An ab initio laser theory is developed to show that exceptional-point lasers can self-generate frequency combs through dynamic population inversion. Notably, the repetition rate of these EP combs is independent of the cavity's free spectral range. [ABSTRACT FROM AUTHOR]

Last December 27th, 2023, Ebru Cavlak Aslan et al. have published the article in the Journal "Optical and Quantum Electronics" the article titled "Analysing of different wave structures to the dissipative NLS equation and modulation instability" (Opt Quantum Electron 56:254, 2024. https://doi.org/10.1007/s11082-023-06035-6) where they tried to present analytical solutions of a dissipative nonlinear Schrödinger equation and study the modulational instability of the continuous wave solution of that equation. After analyzing their results, we found a number of shortcomings about both the modulational instability study and the analytical solutions. For analytical solutions, we found that all their found exact solutions were about the standard nonlinear Schrödinger equation and may contain errors/mistakes since at least one of their found dark soliton solutions was done in the region of modulational instability of the standard nonlinear Schrödinger equation. We also found that all their results about the modulational instability were obtained on an erroneous plane wave solution of their model equation, leading thus to obsolete results. It is the aim of the present comment to correct all shortcomings found in that study on the modulational instability. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ , the Lévy index α , and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W -shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media. [ABSTRACT FROM AUTHOR]

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280]. [ABSTRACT FROM AUTHOR]

The article "Self-similarity and recurrence in stability spectra of near-extreme Stokes waves" delves into the stability of high-amplitude waves near the steepest wave with a 120° corner at the crest in an ideal fluid. It explores the self-similar recurrence of the stability spectrum near the origin of the spectral plane as the steepest wave is approached, building on previous research on Stokes waves and their instabilities. The study investigates the convergence to universal shapes of stability spectra as the steepness of the waves increases, discussing Benjamin-Feir instability branches, localized instabilities, and the modulational nature of eigenfunctions. The research offers valuable insights into the mechanisms of wave breaking and the intricate dynamics of near-extreme Stokes waves, shedding light on the significance of quasi-periodic eigenfunctions in understanding their stability. [Extracted from the article]

Unlike Zabusky and Kruskal who exploited the Korteweg–de Vries traveling solitons to explain the FPU-recurrence phenomenon, we consider a more robust time periodic Kuznetzov–Ma breather to resolve the paradox. The nonlinear Schrödinger equation is derived from the equation of motion of β -FPU chain, by using the method of multiple scales combined with a quasi-discreteness approximation. Modulational instability leads to the generation of a nonlinear wave of finite background, known as the Kuznetzov–Ma breather. The spatial localization and time periodic profile of the Kuznetzov–Ma breathers make it ideal in mimicking the FPU-recurrence phenomenon, as underscored by results of numerical simulations. [ABSTRACT FROM AUTHOR]

This research work focusses on examining the attributes of ion-acoustic waves (IA waves) in a plasma environment embodying superthermal-trapped electron distributions, characterised by the parameters κ and β representing superthermality and trapping respectively. The study utilises a modified nonlinear Schrödinger equation (m-NLSE) derived using the reductive perturbation method (RPM) to model the dynamics of modulated wave packets in this plasma framework. The study investigates the impact of superthermality (κ ) and trapping (β ) parameters on the envelopes of IA waves. The research explores the conditions for modulational instability (MI) of IA waves, characterised by the dispersion-nonlinearity product PQ in the m-NLSE. Bright and dark solitons are observed corresponding to different signs of PQ, i.e., P Q < 0 and P Q > 0 indicating the presence of modulationally unstable and stable regimes, respectively. The cutoff wavenumber k c at which MI enters increases with decreasing superthermality (increasing κ ). Interestingly, k c is observed to be independent of the trapping parameter (β ), suggesting that electron trapping does not significantly influence the onset of MI. Further, the effects of κ and β on the maximum growth rate ( Γ max ) and the modulational wavenumber ( K max ) associated with MI are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, the modulational instability (MI) of dust acoustic waves (DAWs) is examined in a strongly coupled dusty plasma system containing negatively charged dust grains, nonthermal ions, and superthermal electrons. To describe the MI criteria of DAWs, the nonlinear Schrödinger equation (NLSE) is derived using the Krylov–Bogoliubov-Mitropolsky (KBM) perturbation method. Our results show that both modulationally stable and unstable domains exist in such a strongly coupled dusty plasma system. It is found that modulational stable and unstable domains are significantly modified by the related dusty plasma parameters such as ion nonthermality, electron superthermality, and the temperatures of both electron and ion. Moreover, the MI growth rate was found to be strongly modified by those parameters. The results of this investigation may be important in understanding nonlinear phenomena in laboratory and space plasmas, particularly in the upper layer of the Earth’s atmosphere (viz., Earth’s mesosphere). [ABSTRACT FROM AUTHOR]

Most recently, Baber et al. (Optical solitons for 2D-NLSE in multimode fiber with Kerr nonlinearity and its modulation instability, Mod. Phys. Lett. B (2024) 2450341, doi:10.1142/S021798492450341X2450341-1) claimed to have obtained exact solutions of a two-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity that governs beam movement within a multimode optical fiber featuring a parabolic index shape, using the Jacobi Elliptic Function Expansion. Additionally, they have analyzed the modulation instability for the underlying model. It is the aim of this comment to point out that all results found in that work and published in the journal Mod. Phys. Lett. B (doi:10.1142/S021798492450341X2450341-1) are all erroneous, from the exact solutions of the considered equation to the modulation instability of the underlying model. [ABSTRACT FROM AUTHOR]

The rational solitons and other wave solutions demonstrate the existence of wave packets propagating through optical Kerr media, exhibiting non-diffractive and non-dispersive spatiotemporal localization. This paper focuses on constructing rational solitons, multi-wave solutions, and other solutions for a dimensionless time-dependent paraxial wave model (dt-pPWM). These solutions are derived through the application of symbolic computation and the ansatz function method, using both traveling wave and logarithmic transformations. Our approach encompasses three distinct methods: the positive quadratic function approach, the three-wave approach, and the double exponential approach. These wave results find practical applications and contribute to a deeper understanding of the physical phenomena described by this model. We have computed conservative quantities corresponding to the power, momentum, and energy of solitons. To assess the stability of the wave equation, we conducted modulational instability analysis, confirming the stability and accuracy of all soliton solutions. The structure of the solution provides a physical interpretation. By selecting appropriate parametric values, we generated 3D graphics illustrating various multi-wave solutions, including lump waves, rogue waves, and multipeak solitons. Additionally, we observed intriguing phenomena arising from the interaction of multi-waves, which have applications in telecommunications, quantum optics and other related fields. [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

The hydrodynamic stability behavior of a two-layer falling film is explored with a floating flexible plate on the top layer surface. The stress balance at the surface is modeled using a modified membrane equation. There is an insoluble surfactant at the liquid-liquid interface of the flow system. The linear instability of perturbation waves is captured by numerically solving the generalized Orr-Sommerfeld eigenvalue problem using the Chebyshev spectral collocation technique. Four different types of linear stability modes are identified: surface mode (SM), interface mode (IM), interface surfactant mode (ISM), and shear mode (SHM). The floating flexible plate has an inferior impact on all the existing modes in the longwave zone. However, a significant influence is noticed for the finite wave numbers. The surface and interface wave instabilities can be suppressed in the smaller wavenumber zone by imposing higher structural rigidity and uniform thickness of the flexible plate. The stabilizing nature of the surface mode becomes more powerful when the top layer viscosity is dominant. A new interfacial instability emerges when the top layer is less viscous than the lower one. At moderate Reynolds numbers, the behavior of interface mode is different in two different zones m r < 1 and m r > 1 , where m and r denote the viscosity and density ratio, respectively. Further, the unstable shear modes induced by the top and bottom layers are detected under the low inclination angles with a strong inertia force. The occurrence of the shear modes requires the viscosity and density of the lower layer to be much higher than those of the upper layer. The influence of characteristic parameters of the flexible plate on the lower layer shear mode is not very sensitive. Finally, the competition between the different modes for dominance of stability boundaries is also discussed. • The hydrodynamic stability of a two-layer falling film is explored with a floating flexible plate on the top surface. • The generalized Orr-Sommerfeld eigenvalue problem is solved using Chebyshev spectral collocation technique. • Four modes, known as surface, interface, interface surfactant, and shear modes are detected. • The flexible plate has an inferior impact in the longwave zone, but significant effect in the finite wave number zone. • The plate's rigidity and thickness suppress the surface and interface wave instability in the smaller wavenumber zone. [ABSTRACT FROM AUTHOR]

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with full dispersion (FDNLS) is considered in the semiclassical regime. The Whitham modulation equations are obtained for the FDNLS equation with general linear dispersion and a generalized, local nonlinearity. Assuming the existence of a four-parameter family of two-phase solutions, a multiple-scales approach yields a system of four independent, first-order, quasi-linear conservation laws of hydrodynamic type that correspond to the slow evolution of the two wavenumbers, mass, and momentum of modulated periodic traveling waves. The modulation equations are further analyzed in the dispersionless and weakly nonlinear regimes. The ill-posedness of the dispersionless equations corresponds to the classical criterion for modulational instability (MI). For modulations of linear waves, ill-posedness coincides with the generalized MI criterion, recently identified by Amiranashvili and Tobisch [New J. Phys., 21 (2019), 033029]. A new instability index is identified by the transition from real to complex characteristics for the weakly nonlinear modulation equations. This instability is associated with long wavelength modulations of nonlinear two-phase wavetrains and can exist even when the corresponding one-phase wavetrain is stable according to the generalized MI criterion. Another interpretation is that while infinitesimal perturbations of a periodic wave may not grow, small but finite amplitude perturbations may grow, hence this index identifies a nonlinear instability mechanism for one-phase waves. Classifications of instability indices for multiple FDNLS equations with higher-order dispersion, including applications to finite-depth water waves and the discrete NLS equation, are presented and compared with direct numerical simulations. [ABSTRACT FROM AUTHOR]

It is assumed that the low-frequency noise recorded on the surface of Mars may be associated with a charged dust component in its atmosphere and the occurrence of sound perturbations in such a dust system that modulate the electromagnetic wave from the Sun. It is also shown that it can be associated with plasma-dust processes in meteoroid tails. The mechanism for the excitation of modulational instability of an electromagnetic wave associated with a dust acoustic mode in the Martian atmosphere, namely in dust clouds at an altitude of 60 and 100 km, where the dusty plasma with particles of frozen carbon dioxide is detected, is described. It is shown that the development of modulational instability is due to the influence of high-frequency electromagnetic waves on the dusty plasma in the Martian atmosphere from both natural sources (solar radiation, lightning discharges) and anthropogenic nature (from equipment from space satellites and from stations on the surface of the planet). The parameters of electromagnetic pump waves, at which the active development of modulational instability of electromagnetic waves associated with the dust acoustic mode is expected, and the modulational instability growth rates are found. The development of the modulational instability in the dusty plasma of Martian clouds, in turn, can explain the occurrence of low-frequency noise recorded by equipment on the surface of Mars. The relation between observed radio noise in the range of 3 Hz–3 kHz and plasma-dust processes in the Martian atmosphere, in particular, in dust clouds at 60 and 100 km, as well as in dusty plasma meteroid tails, where the dust concentration is high, is discussed. [ABSTRACT FROM AUTHOR]

In the mid–high latitude atmosphere, the instability of planetary waves characterizes enhanced planetary wave activity or amplified large‐scale waviness leading to increased regional weather extremes. In this paper, a nonlinear Schrödinger equation is derived to describe the evolution of planetary waves. Then the consequences of Arctic amplification (AA)‐induced meridional background potential vorticity (PVy) changes on the modulational instability of planetary waves are examined. It is found that the modulational instability of uniform planetary wave trains mainly results from the presence of high‐order dispersion and nonlinearity, even though such an instability depends on the amplitude, vertical structure and zonal wavenumber of uniform planetary waves and the atmospheric stratification. Because the nonlinearity and high‐order dispersion depend on the magnitude of PVy, the modulational instability of planetary waves is significantly influenced by the variation of PVy associated with AA. It is also revealed that stronger modulational instability of planetary waves tends to occur in the smaller PVy region or in higher latitudes due to both stronger nonlinearity and weaker high‐order dispersion for fixed background and planetary wave parameters, which is conducive to more intense large‐scale waviness. However, because AA can reduce PVy in the mid–high latitudes mainly in the lower troposphere via reductions of winter zonal winds and meridional temperature gradients, the reduced PVy under AA can significantly enhance the modulational instability. Thus, the role of AA is to amplify planetary wave activity in mid–high latitudes through strengthening the modulational instability of planetary waves due to reduced PVy, which further enhances large‐scale waviness. [ABSTRACT FROM AUTHOR]

While the original Ait-Sahalia interest rate model has been found to be of considerable use for describing the time-series evolution of interest rates, it may not possess adequate specifications to explain the responses of interest rates to empirical phenomena such as volatility skew-smile effect, jumps, market regulatory lapses, economic crises, financial clashes, and political instability, among others collectively. In this article, we propose a modified version of this model by incorporating additional features to help collectively describe these empirical phenomena adequately. Since the proposed model does not have a closed-form formula, we construct new techniques of the truncated EM method to study it and justify the method within a Monte Carlo framework to compute some financial quantities. [ABSTRACT FROM AUTHOR]

In this work, we use symbolic computation and ansatz function schemes, to investigate the soliton solutions for the nonlinear Schrödinger equation (NLSE) along with Kudryashov's quintuple self-phase modulation system (KQSPMS) including dual-form of nonlocal nonlinearity (DFNLN). We initially determine the ordinary differential (OD) form for this model through a variable transformation. Then we introduce numerous new dynamical soliton types: the M-shaped rational soliton, the M-shaped interaction between one and two stripe solitons, the periodic cross-M-shaped rational (PCMR) soliton, the periodic cross-kink (PCK) soliton, multi-waves, and the homoclinic breather soliton. Secondly, we determine the partial differential (PD) form for this model through a variable transformation. Moreover, a lump soliton, a periodic wave, a rogue wave, a lump interaction with a periodic and kink wave, and three different types of breather soliton will obtain. We'll demonstrate these solutions' unique structure and extremely interesting interaction behavior. We'll also use graphs (3-D and contour plots) to discuss the dynamics of the results after setting the parameters to the proper values. [ABSTRACT FROM AUTHOR]

The features of nonlinear wave propagation in four-component collisional dusty plasma have been described by the damped nonlinear Schr ödinger equation (DNLSE). The stability and instability domains have been explored by modulational instability criteria. The physical impacts of collisional frequencies, time, and positive–negative grain density parameters on growth rate properties have been discussed using mesospheric data. It was reported that the model parameters modulated the plasma growth rate. Also, time becomes an important parameter in reducing the growth rate. [ABSTRACT FROM AUTHOR]

The article exclusively discusses how the cubic–quintic complex Ginzburg–Landau equation governs the dynamics of dissipative soliton in the FitzHugh–Nagumo model combined with linear self- and cross-diffusion terms. Then, based on the linear stability analysis, a set of equations describing the evolution of the perturbation amplitude is derived, and a detailed analysis of the modulational instability gain spectrum is presented. Moreover, the exact dissipative soliton of the cubic–quintic complex Ginzburg–Landau equation is found using Hirota's bilinear method. Furthermore, as an input condition to our simulations, we use the obtained dissipative soliton solution to check the stability of the moving pulse solution by solving the quintic FitzHugh–Nagumo model numerically with self- and cross-diffusion terms. As a result, numerical findings are in perfect correlation with analytical investigations, thus attesting that cardiac cell dynamics through the quintic FitzHugh–Nagumo model is the support of soliton-pulse-like solutions. [ABSTRACT FROM AUTHOR]

Optical rogue waves are a nonlinear phenomenon that offers a unique opportunity to gain fundamental insights into wave interaction and behavior, and the evolution of complex systems. Optical systems serve as a suitable testbed for the well-controlled investigation of this natural phenomenon, which cannot be easily studied in an ocean environment. Additionally, such systems offer practical applications in telecommunications and optical signal processing, making this topic a vital area of research. Fiber lasers are considered the best candidates for demonstrating and investigating the emergence of optical rogue waves. In particular, they offer significant advantages in nonlinear dynamics due to faster field evolution and a higher number of events that can be recorded within a relatively short time. In this paper, we present the development mechanisms of optical rogue wave events. It was found that multimode vector instability, pulse–pulse interaction, and soliton rain are the main nonlinear dynamics leading to the formation of optical rogue wave events. [ABSTRACT FROM AUTHOR]

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of 1 0 0 photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system. [ABSTRACT FROM AUTHOR]

The concern of this study is to investigate the optical solitons pluses. The (2 + 1) -dimensional Kundu–Mukherjee–Naskar (KMN) mathematical model in birefringent fibers is taken under consideration for this sake because this model has great importance in optics and delineate the propagation of soliton dynamics in optical fiber communication system and the rogue waves in the oceans and the bending of the light beam. Two newly developed approaches first extended rational sinh-Gordon method which is formulated from the well-known sinh-Gordon equation and second exp (− ϕ (ϖ)) -expansion function method are manipulated for the construction of optical solitons in distinct formats. Bright, dark, and singular along their combined forms, periodic and plane wave solutions are extracted with the aid of proposed methods. Moreover, the modulation instability of investigated model is also carried out via linear stability theory. To endorse the physical compatibility of the results, the 2D, 3D, contour, and density plots have been delineated using appropriate parametric values. According to our literature research, these two methods that we are working on have not been applied to the KMN equation in birefringent fibers before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field. The evaluated results suggested that the techniques employed in this research to recover inclusive and standard solutions are approachable, efficient, and speedier in computing and can be considered a handy tool in solving more complex phenomena with the assistance of symbolic computation. [ABSTRACT FROM AUTHOR]

Numerical simulations have been an increasingly important tool in space physics. Here, we introduce an open-source three-dimensional compressible Hall-Magnetohydrodynamic (MHD) simulation code LAPS (UCLA-Pseudo-Spectral, https://github.com/chenshihelio/LAPS). The code adopts a pseudo-spectral method based on Fourier Transform to evaluate spatial derivatives, and third-order explicit Runge-Kutta method for time advancement. It is parallelized using Message-Passing-Interface (MPI) with a "pencil" parallelization strategy and has very high scalability. The Expanding-Box-Model is implemented to incorporate spherical expansion effects of the solar wind. We carry out test simulations based on four classic (Hall)-MHD processes, namely, 1) incompressible Hall-MHD waves, 2) incompressible tearing mode instability, 3) Orszag-Tang vortex, and 4) parametric decay instability. The test results agree perfectly with theory predictions and results of previous studies. Given all its features, LAPS is a powerful tool for large-scale simulations of solar wind turbulence as well as other MHD and Hall-MHD processes happening in space. [ABSTRACT FROM AUTHOR]

Feature detection and matching is the key technique for remote sensing image processing and related applications. In this paper, a PDE-constrained optimization model is proposed to determine the scale levels advantageous for feature detection. A variance estimation technique is introduced to treat the observation optical images polluted by additive zero-mean Gaussian noise and determine the parameter of a nonlinear scale space governed by the partial differential equation. Additive Operator Splitting is applied to efficiently solve the PDE constraint, and an iterative algorithm is proposed to approximate the optimal subset of the original scale level set. The selected levels are distributed more uniformly in the total variation sense and helpful for generating more accurate and robust feature points. The experimental results show that the proposed method can achieve about a 30% improvement in the number of correct matches with only a small increase in time cost. [ABSTRACT FROM AUTHOR]

This paper analyzes the emergence of soliton trains in even higher-order dispersive media and explores the modulational instability phenomenon in optical media. The analysis considers quadratic, quartic, and sextic dispersions with weakly nonlocal Kerr nonlinearity. The results show that nonlocality enhances the MI gain and leads to soliton trains in response to different combinations of even dispersions and nonlocal nonlinearity. The study suggests that Kerr nonlocality can enhance the excitation of extreme events in even higher-order dispersive nonlinear media with potential applications in optical fibers and fiber lasers. [ABSTRACT FROM AUTHOR]

This paper aims to theoretically examine the phenomenon of modulation instability (MI) in the presence of a considerable nonlinear refractive index movement in a triad-core nonlinear coupler, including a bi-negative refractive material transmission medium. The formula for the instability gain is deduced with the help of the linear stability analysis. The significant role of cubic, quintic, septic, nonic, and stability of the front-to-reverse-transmitting wave nonlinearities in the phenomenon of MI in the nonlinear triad-core coupler is critically confirmed via MI gain spectra analysis. It is found that the cubic, quintic, septic, and nonic nonlinearities improve the MI in both dispersion regimes (normal and anomalous) by boosting the gain and width of the instability profile by taking account of reverse-to-front ratio power (f). On the other hand, quintic nonlinearity improves the MI in the cubic case by enhancing its width and gain in the normal and anomalous dispersion areas. Using 3D pictures, we analyze the dynamic behavior of the MI under an anomalous dispersion area and find that, in both normal and anomalous dispersion scenarios, the instability shape develops with septic and nonic nonlinearity. Furthermore, as the figures clearly show, neither the normal nor the anomalous dispersion scenario significantly changes due to septic and nonic nonlinearity alone. We aim to shed light on the role of contamination in saturation nonlinearity by examining soliton and MI, their production, and control in triad-core couplers, focusing on a Negative Index Meta Material (NIMM) transmission medium. [ABSTRACT FROM AUTHOR]

A theoretical investigation of nonplanar (cylindrical and spherical) amplitude modulation of dust acoustic waves (DAWs) in a four component dusty plasma including positive and negative dusts, Maxwellian ions and double spectral electron distribution (DSED) is carried out. The focusing modified nonlinear Schrödinger equation (MNLSE) is derived using the derivative expansion perturbation technique (DEPT). The limit of DSED to Maxwell distribution is discussed. The effects of the two spectral indices, $ r $ r and $ q $ q , on the modulational instability (MI) growth rate are examined and it is deduced that increasing the value of $ r $ r and $ q $ q raises the MI growth rate. Additionally, the structure of the first and second-order dust acoustic (DA) rogue waves in the modulational unstable region is analyzed. The influences of both of $ r $ r and $ q $ q on the rogue wave amplitude (energy) are investigated. We found that increasing the value of $ r $ r enhances the rogue wave amplitude while the opposite behavior is found by increasing $ q $ q values. The findings of the present study are useful to interpret some physical phenomena in various plasma situations that found in space regions, such as Titan (the largest moon) of Saturn, auroral region and asteroid zones, where the effectiveness of non-Maxwellian particles such as DSED is existed. [ABSTRACT FROM AUTHOR]

The time-fractional coupled Drinfel'd–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the Ω ′ Ω ,   1 Ω method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model's behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation's underlying dynamics and offer potential applications across various scientific fields. [ABSTRACT FROM AUTHOR]

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination. [ABSTRACT FROM AUTHOR]

Traditional visual detection methods that employ image data are often unstable due to environmental influences like lighting conditions. However, microfiber spectrometers are capable of capturing the specific wavelength characteristics of tail coal suspensions, effectively circumventing the instability caused by lighting variations. Utilizing spectral analysis techniques for detecting ash content in tail coal appears promising as a more stable method of indirect ash detection. In this context, this paper proposes a rapid detection method for the coal ash content in tailings suspensions based on absorption spectra and deep feature extraction. Initially, a preprocessing method, the inverse time weight function (ITWF), is presented, focusing on the intrinsic connection between the sedimentation phenomena of samples. This enables the model to learn and retain spectral time memory features, thereby enhancing its analytical capabilities. To better capture the spectral characteristics of tail coal suspensions, we designed the DSFN (DeepSpectraFusionNet) model. This model has an MSCR (multi-scale convolutional residual) module, addressing the conventional models' oversight of the strong correlation between adjacent wavelengths in the spectrum. This facilitates the extraction of relative positional information. Additionally, to uncover potential temporal relationships in sedimentation, we propose a CLSM-CS (convolutional long-short memory with candidate states) module, designed to strengthen the capturing of local information and sequential memory. Ultimately, the method employs a fused convolutional deep classifier to integrate and reconstruct both temporal memory and positional features. This results in a model that effectively correlates the ash content of suspensions with their absorption spectral characteristics. Experimental results confirmed that the proposed model achieved an accuracy of 80.65%, an F1-score of 80.45%, a precision of 83.43%, and a recall of 80.65%. These results outperformed recent coal recognition models and classical temporal models, meeting the high standards required for industrial on-site ash detection tasks. [ABSTRACT FROM AUTHOR]

Self-generated magnetic field and electron jet are observed when ultra-intense lasers (> 1 × 10 18 W / c m 2 ) interact with plasma. It is found that the self-generated magnetic field plays a significant role in the generation of electron jets. The generation mechanism of electron jets under the influence of a self-generated circular magnetic field is examined. It is revealed that magnetic modulation of self-generated magnetic fields can result in the collapse of these fields, consequently leading to the production of electron jets. Furthermore, it has been discovered that the velocity of the electron jets is associated with the maximum growth rate of the modulational instability. As the maximum growth rate of the modulational instability decreases, the velocity of the electron jets is reduced. The present work aids in getting a deeper understanding of the generation of electron jets in relativistic laser plasmas. [ABSTRACT FROM AUTHOR]

This paper addresses the fluid–structure interaction problem, with an interest on the interaction of a gravity wave with a flexible floating structure, anchored to a seabed of constant depth. To achieve this goal, we make use of the model equations, namely, the Navier–Stokes equations and the Navier–Lamé equation, as well as the associated the boundary conditions. Applying the multi-scale expansion method, these set of equations are reduced to a pair of nonlinearly coupled complex cubic Ginzburg–Landau equations (CCGLE). By applying the proposed modified expansion method, the group velocity dispersion and second-order dispersion relation are deduced. In the same vein, modulation instability (MI) is investigated as a mechanism of formation of pulse trains in fluid–structure system using a CCGLE. For the analytical analysis, we made use of the inverse scattering method to find analytical solutions to the coupled nonlinear equations. Through that method, the obtained solutions depict rogue-shaped waves. Our results suggest that uncontrolled MI within the interaction between a flexible body and gravity waves in viscous flow may be considered as the principal source of many structural ruptures, which are the first cause of critical damage due to the great energy and unpredictability of rogue waves. The present work aims to provide tools to model a wide range of physical problems regarding the interaction of surface gravity waves and an offshore-anchored structure, and it aims to be essential to our understanding of the nonlinear characteristics of offshore structures in real-sea states. [ABSTRACT FROM AUTHOR]

This work focuses on the study of the paraxial wave model with a space–time fractional form. This model has more importance for describing light propagation in nonlinear optical fibers and telecommunication lines. The main aim of this work is to observe the effect of the fractional parameters and compare the truncated M -fraction with the beta-fraction, conformable fraction form, and classical form of the PW model. For this observation, we applied the Simplest equation technique to acquire analytical solutions to the space–time (spatiotemporal) M -fractional paraxial wave model. We are able to acquire several new optical soliton solutions, including periodic waves, kink-type waves, rogue-type waves, and several novel periodic waves, by providing the appropriate fractional parametric values. These solutions have significance for shedding light on a number of physical phenomena in the realms of optical fiber and communication sciences. The diverse values of fractional parameters and the three-dimensional and contour plot graphs of certain chosen solutions are depicted, which are the most accurate physical characterizations of the outcomes. We also sketch the comparative graph of diverse fractional forms and the classical form of the paraxial wave equation in two-dimensional plots. Consequently, our findings represent an important breakthrough in this complex area and help further develop our comprehension of the behavior of solitons. [ABSTRACT FROM AUTHOR]

This paper examines the complex Ginzburg Landau equation, which describes pulse propagation inside a fiber with the triple power law of self-phase modulation. Since the effect of parameter selection has become very important in relevant model studies recently, self-phase modulation has been added to the complex Ginzburg Landau equation, which has been studied in the literature, and it is aimed at investigating the analytical solutions of the presented equation. Adding the triple power law of the self-phase modulation parameter to the model, in addition to existing studies in the literature, emphasizes the innovative aspect and importance of the study. The first aim is to reveal bright and singular solitons using the new Kudryashov method. The new Kudryashov method is a technique that is frequently used in the literature, is effective for generating analytical solutions, provides ease of operation, and can be applied to a wide class of nonlinear partial differential equations. The second goal is to show that the obtained solutions have modulation stability. By using modulation instability analysis, the gain spectrum is formed for different parameter values. Graphic presentations support the findings. Moreover, bright and singular soliton portraits are demonstrated with 3D and 2D graphs. The novelty of the study lies in the fact that the relevant model has not been studied before with an effective method such as the new Kudryashov method, and the modulation instability has been studied for the first time. [ABSTRACT FROM AUTHOR]

Transverse Mode Instability (TMI) that results from dynamic nonlinear thermo-optical scattering is the primary limitation to power scaling in high-power fiber lasers and amplifiers. It has been proposed that TMI can be suppressed by exciting multiple modes in a highly multimode fiber. We derive a semi-analytic frequency-domain theory of the threshold for the onset of TMI in narrowband fiber amplifiers under arbitrary multimode input excitation for general fiber geometries. Our detailed model includes the effect of gain saturation, pump depletion, and mode-dependent gain. We show that TMI results from the exponential growth of noise in all the modes at downshifted frequencies due to the thermo-optical coupling. The noise growth rate in each mode is given by the sum of signal powers in various modes weighted by pairwise thermo-optical coupling coefficients. We calculate thermo-optical coupling coefficients for all ∼ 1 0 4 pairs of modes in a standard circular multimode fiber and show that modes with large transverse spatial frequency mismatch are weakly coupled, resulting in a banded coupling matrix. This short-range behavior is due to the diffusive nature of the heat propagation, which mediates the coupling and leads to a lower noise growth rate upon multimode excitation compared to a single mode, resulting in significant TMI suppression. We find that the TMI threshold scales linearly with the number of modes that are excited asymptotically, leading to roughly an order of magnitude increase in the TMI threshold in an 82-mode fiber amplifier. [ABSTRACT FROM AUTHOR]

Pattern-illuminated Fourier ptychography (piFP) is an elegant combination of structured illumination imaging and a Fourier ptychographic algorithm with the ability to image beyond the diffraction limit of the employed optics. Artifact-free piFP super-resolution reconstruction requires a high level of stability in the illumination pattern. However, unpredictable pattern variation occurs in the presence of environment perturbation, intensity fluctuation, and pointing instability at the source, leading to declines in image reconstruction quality. To address this issue, we present an efficient and robust piFP algorithm based on low-rank approximation (LRA-piFP), which relaxes the requirement for the stability of illumination patterns. This LRA-piFP method can model frame-wise pattern variation during a full scan, thus improve the reconstruction quality significantly. We take numerical simulations and proof-of-principle experiments with both long-range imaging and microscopy for demonstrations. Results show that the LRA-piFP method can handle different kinds of pattern variation and outperforms other state-of-the-art techniques in terms of reconstruction quality and resolution improvement. Our method provides effective experimental robustness to piFP with a natural algorithmic extension, paving the way for its application in both macroscopic and microscopic imaging. [ABSTRACT FROM AUTHOR]

Segmentation methods based on convolutional neural networks (CNN) have achieved remarkable results in the field of medical image segmentation due to their powerful representation capabilities. However, for brain-tumor segmentation, owing to the significant variations in shape, texture, and location, traditional convolutional neural networks (CNNs) with limited convolutional kernel-receptive fields struggle to model explicit long-range (global) dependencies, thereby restricting segmentation accuracy and making it difficult to accurately identify tumor boundaries in medical imaging. As a result, researchers have introduced the Swin Transformer, which has the capability to model long-distance dependencies, into the field of brain-tumor segmentation, offering unique advantages in the global modeling and semantic interaction of remote information. However, due to the high computational complexity of the Swin Transformer and its reliance on large-scale pretraining, it faces constraints when processing large-scale medical images. Therefore, this study addresses this issue by proposing a smaller network, consisting of a dual-encoder network, which also resolves the instability issue that arises in the training process of large-scale visual models with the Swin Transformer, where activation values of residual units accumulate layer by layer, leading to a significant increase in differences in activation amplitudes across layers and causing model instability. The results of the experimental validation using real data show that our dual-encoder network has achieved significant performance improvements, and it also demonstrates a strong appeal in reducing computational complexity. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity ν>0$\nu >0$, when the perturbations are in the critical spaces Hx1Ly2$H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ν12−δ0$\nu ^{\frac{1}{2}-\delta _0}$ with any small δ0>0$\delta _0>0$, which implies that ν12$\nu ^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo‐spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues. [ABSTRACT FROM AUTHOR]

The formation and propagation dynamics of the finite‐amplitude ion‐acoustic wave (IAW) structures (e.g., soliton, breather, rogue wave, etc.) is theoretically investigated in a plasma comprising of kappa distributed solar and cometary electrons of different temperatures, a hot H3O+$$ {\mathrm{H}}_3{\mathrm{O}}^{+} $$ drift ion component, and a pair of oppositely charged oxygen ion components. The modified‐KdV (mKdV) equation is derived in order to study the propagation dynamics of the ion‐acoustic solitary wave (IASW). It is then converted into the nonlinear Schrödinger equation (NLS) through appropriate algebraic manipulation in order to observe the amplitude modulation of the IAWs. Also, the appearance of envelope soliton and the possibility of breather structure formation have been studied from the NLS equation. The dependence of plasma parameters on the formation and propagation of IAW structures has been briefly discussed. The modified‐KdV equation is reduced in a dynamical system through the application of coordinate transformation. The existence of finite‐amplitude nonlinear and supernonlinear IAWs is demonstrated by phase plane analysis. Due to the fact that the results are primarily associated with cometary plasma, they possibly provide greater insight of the nonlinear characteristics of cometary plasma. [ABSTRACT FROM AUTHOR]

Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained. [ABSTRACT FROM AUTHOR]