Rogue Waves Research Articles

Evaluation of Conditions for the Development of Cryogenic Spinal Cord Injury Using a Canine Model: An Experimental Study on the Safety of Cryoablation for Metastatic Spinal Tumors.

Although the application of cryoablation to metastatic spinal tumors has been attempted, spinal cryoablation has the unique complication of cryogenic spinal cord injury. This study aimed to elucidate the conditions for the development of cryogenic spinal cord injury. Fifteen canines were used in this study. A metal probe was inserted into the 13th thoracic vertebral body. Cryoablation was performed for 10 minutes by freezing the probe in liquid nitrogen. The control canine underwent probe insertion only. Spinal cord monitoring, epidural temperature measurement, motor function assessment, and pathologic examination of the spinal cord were performed. During the 10 minutes of cryoablation, the epidural temperature decreased and reached the lowest epidural temperature (LET) at the end of cryoablation. The LETs (degrees celsius [°C]) of each canine were -37, -30, -27, -8, -3, -2, 0, 1, 4, 8, 16, 18, 20, and 25, respectively. As the epidural temperature decreased, waveform amplitudes also decreased. At the end of cryoablation (10 minutes after the start of cryoablation), abnormal waves were observed in 92.9% (13/14) of canines. With epidural rewarming, the amplitude of the waveforms tended to recover. After epidural rewarming (2 hours after the start of cryoablation), abnormal waves were observed in 28.6% (4/14) of canines. The LETs (°C) of the canines with abnormal waves after epidural rewarming were -37, -30, -27, and -8. None of the canines with normal waves after epidural rewarming had any motor impairment. In contrast, all canines with remaining abnormal waves after epidural rewarming had motor impairment. In the pathologic assessment, cryogenic changes were found in canines with LETs (°C) of -37 -30, -27, -8, 0, and 1. This study showed that 10-minute spinal cryoablation with LETs (°C) of -37, -30, -27, -8, 0, and 1 caused cryogenic spinal cord injury. There was no evidence of cryogenic spinal cord injury in canines with LET of ≥4°C. The epidural temperature threshold for cryogenic spinal cord injury is between 1 and 4°C, suggesting that the epidural temperature should be maintained above at least 4°C to prevent cryogenic spinal cord injury.

Analysis of modulational instability of waves in the fluid-filled elastic tube

In this paper, we investigate the modulational instability of waves within a fluid-filled elastic tube considering the fluid viscosity. Through the reductive perturbation method, we derived a modified Nonlinear Schrödinger Equation (NLSE) with damping term for the system. Based on the modified NLSE, we found that modulational instability exists in this system though the fluid viscosity is considered. Both the instability and the stability region are given. Furthermore, the growth rate of the instability is obtained. The dependence of the growth rate on the system parameters such as perturbation wave length, the background wave length, the viscosity of the fluid and the elasticity of the tube is given in the present paper. It indicates that the fluid viscosity refrain the modulational instability. The results may be used to predict the production of the rogue wave in this system, as well as to the other similar system.

Higher rogue and rogue-soliton interaction solutions of a (2 + 1) dimensional nonlinear model in fluid mechanics

In this study, two new theorems are generalized. We obtained a new paradigm about the second order rogue wave and multiple exponential functions, and a new paradigm about the second order rogue wave and multiple hyperbolic cosine functions. Six sets of interaction solutions of the model are solved by means of symbolic calculation and two new theorems. Meaningful graphs of the propagation processes along time demonstrated the interaction phenomena for these solutions. The energy transfer process can be observed when the second order rogue waves interact with multiple exponential functions or multiple hyperbolic cosine functions. As a conclusion from our paper, the solitons’ energy transfers to the second order rogue wave at beginning, the rogue wave’s energy dissipates and transfers to the solitons along the time moving. It will contribute to the research on the generation of rogue waves.

Rogue waves on the periodic background of the Kuralay-II equation

We derive the rogue wave solutions of the Kuralay-II equation by applying the Darboux transformation method with the Lax pair on the periodic background. These solutions are represented using Jacobian elliptic functions: dnoidal and cnoidal. The rogue wave solutions can be obtained on the periodic background while the dnoidal travelling periodic wave and cnoidal travelling periodic wave are modulationally unstable with respect to the long-wave perturbations.

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

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

Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque.

In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.

The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in data-driven higher-order rogue waves.

Due to the dynamic characteristics of instantaneity and steepness, employing domain decomposition techniques for simulating rogue wave solutions is highly appropriate. Wherein, the backward compatible physics-informed neural network (bc-PINN) is a temporally sequential scheme to solve PDEs over successive time segments while satisfying all previously obtained solutions. In this work, we propose improvements to the original bc-PINN algorithm in two aspects based on the characteristics of error propagation. One is to modify the loss term for ensuring backward compatibility by selecting the earliest learned solution for each sub-domain as pseudo-reference solution. The other is to adopt the concatenation of solutions obtained from individual subnetworks as the final form of the predicted solution. The improved backward compatible PINN (Ibc-PINN) is applied to study data-driven higher-order rogue waves for the nonlinear Schrödinger (NLS) equation and the AB system to demonstrate the effectiveness and advantages. Transfer learning and initial condition guided learning (ICGL) techniques are also utilized to accelerate the training. Moreover, the error analysis is conducted on each sub-domain, and it turns out that the slowdown of Ibc-PINN in error accumulation speed can yield greater advantages in accuracy. In short, numerical results fully indicate that Ibc-PINN significantly outperforms bc-PINN in terms of accuracy and stability without sacrificing efficiency.

Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation.

This work explores diverse novel soliton solutions due to fractional derivative, dispersive, and nonlinearity effects for the nonlinear time M-fractional paraxial wave equation. The advanced exp [-φ(ξ)] expansion method integrates the nonlinear M-fractional Paraxial wave equation for achieving creative solitonic and traveling wave envelopes to reconnoiter such dynamics. As a result, trigonometric and hyperbolic solutions have been found via the proposed method. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. For any chosen set of the allowed parameters 3D, 2D and density plots illustrate, this inquisition achieved kink shape, the collision of kink type and rogue wave, periodic rogue wave, some distinct singular periodic soliton waves for time M-fractional Paraxial wave equation. As certain nonlinear effects cancel out dispersion effects, optical solitons typically can travel great distances without dissipating. We have constructed reasonable soliton solutions and managed the actual meaning of the acquired solutions of action by characterizing the particular advantages of the summarized parameters by the portrayal of figures and by interpreting the physical occurrences. New precise voyaging wave configurations are obtained using symbolic computation and the previously described methodologies. However, the movement role of the waves is explored, and the modulation instability analysis is used to describe the stability of waves in a dispersive fashion of the obtained solutions, confirming that all created solutions are precise and stable.

Dark Localized Waves in Shallow Waters: Analysis within an Extended Boussinesq System

We study dark localized waves within a nonlinear system based on the Boussinesq approximation, describing the dynamics of shallow water waves. Employing symbolic calculus, we apply the Hirota bilinear method to transform an extended Boussinesq system into a bilinear form, and then use the multiple rogue wave method to obtain its dark rational solutions. Exploring the first- and second-order dark solutions, we examine the conditions under which these localized solutions exist and their spatiotemporal distributions. Through the selection of various parameters and by utilizing different visualization techniques (intensity distributions and contour plots), we explore the dynamical properties of dark solutions found: in particular, the first- and second-order dark rogue waves. We also explore the methods of their control. The findings presented here not only deepen the understanding of physical phenomena described by the (1+1)-dimensional Boussinesq equation, but also expand avenues for further research. Our method can be extended to other nonlinear systems, to conceivably obtain higher-order dark rogue waves.

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

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

Propagation of collisional among multi-soliton, multi-singular solition and rogue wave around the critical values in an unmagetized plasma

Progress in understanding the propagation characteristics of (i) collisional acoustic among multi-soliton and multi-singular soliton around the critical values and their corresponding phase shifts and (ii) collision between two rogue waves (RWs) propagating toward each in a plasma environment is presented. The considered plasma environment consists of mobile cold positrons, immobile positive ions and (r, q)-distributed hot positrons, and electrons. To accomplish our goal, the coupled modified Korteweg–de Vries equations (mKdVEs) and nonlinear Schrödinger equations (NLSEs) are derived from the considered plasma environment. Based on the concept of Hirota's bilinear method, the multi-soliton and multi-singular soliton solutions of the coupled mKdVEs are determined directly. In addition, the analytical unstable RWs solutions of the coupled NLSEs are determined. With the impact of physical parameters, (i) the trajectories are described for double, triple, quadruple and quintuple positron acoustic bi-directional multi-soliton and (ii) the variation of collisional RWs profiles are displayed with physical interpretation. The results described by the coupled mKdVEs also show that the scattered double-, triple-, quadruple-, and quintuple-soliton are elastic and preserved their original features oppositely after the collision around the critical value of any specific plasma parameter.

Rogue waves and their dynamics in the Ito’s system with the nonzero constant background

Rogue waves and their dynamics in the Ito’s system with the nonzero constant background

Rogue wave solutions and rogue-breather solutions to the focusing nonlinear Schrödinger equation

Based on the long wave limit method, the general form of the second-order and third-order rogue wave solutions to the focusing nonlinear Schrödinger equation are given by introducing some arbitrary parameters. The interaction solutions between the first-order rogue wave and one-breather wave are constructed by taking a long wave limit on the two-breather solutions. By applying the same method to the three-breather solutions, two types of interaction solutions are obtained, namely the first-order rogue wave and two breather waves, the second-order rogue wave and one-breather wave, respectively. The influence of the parameters related to the phase on the interaction phenomena is graphically demonstrated. Collisions occur among the rogue waves and breather waves. After the collisions, the shape of them remains unchanged. The abundant interaction phenomena in this paper will contribute to a better understanding of the propagation and control of nonlinear waves.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

Dynamic behavior of explicit elliptic and quasi periodic-wave solutions to the generalized [formula omitted]-dimensional Kundu–Mukherjee–Naskar equation

This paper introduces novel explicit traveling solutions to the generalized (2+1)-dimensional Kundu–Mukherjee–Naskar equation, a model relevant in optical fiber pulses, fluid dynamics, medical imaging, and ocean rogue waves. We propose rational form functions that incorporate both sine and cosine variables or sinh and cosh variables as potential solutions to the model. Subsequently, we utilize 3D diagrams to visually illustrate the characteristics of the identified propagating solutions, which exhibit elliptic-periodic, convex-periodic, cusp-soliton, and quasi-periodic patterns. Furthermore, a graphical analysis is conducted to explore the impact of the model’s parameters; nonlinearity and dispersion. Additionally, we introduce a comparative analysis to assess the superiority of the proposed approach over other existing methods such as Kudryashov-expansion, sine-function, and tanh(coth)-expansion methods. We anticipate that the insights gained from this analysis will prove valuable for understanding and studying other nonlinear models.

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

We 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.