Rogue Waves Research Articles

Adaptive sampling physics-informed neural network method for high-order rogue waves and parameters discovery of the (2 + 1)-dimensional CHKP equation.

Rogue waves are important physical phenomena, which have wide applications in nonlinear optics, hydrodynamics, Bose-Einstein condensates, and oceanic and atmospheric dynamics. We find that when using the original PINNs to study rogue waves of high dimensional PDEs, the prediction performance will become very poor, especially for high-order rogue waves due to that the randomness of selection of sample points makes insufficient use of the physical information describing the local sharp regions of rogue waves. In this paper, we propose an adaptive sampling physics-informed neural network method (ASPINN), which renders the points in local sharp regions to be selected sufficiently by a new adaptive search algorithm to lead to a prefect prediction performance. To valid the performance of our method, the (2+1)-dimensional CHKP equation is taken as an illustrative example. Experimental results reveal that the original PINNs can hardly be able to predict dynamical behaviors of the high-order rogue waves for the CHKP equation, but the ASPINN method can not only predict dynamical behaviors of these high-order rogue waves, but also greatly improve the prediction efficiency and accuracy to four orders of magnitude. Then, the data-driven inverse problem for the CHKP equation with different levels of corrupted noise is studied to show that the ASPINN method has good robustness. Moreover, some main factors affecting the neural network performance are discussed in detail, including the size of training data, the number of layers of the neural network, and the number of neurons per layer.

Breathers for the sixth-order nonlinear Schrödinger equation on the plane wave and periodic wave background

In this paper, we study the breathers in the framework of the sixth-order nonlinear Schrödinger equation by using the Darboux transformation. The primary objective of this research is twofold. First, we consider the nonlinear superposition of breathers on the plane wave background. Based on the concept that rogue waves are formed from colliding Akhmediev breathers, we obtain rogue wave sequences and a first-order Akhmediev breather with a central second-order rogue wave peak. Second, we consider the formation of breathers on the periodic wave background. The difficulty of solving the Lax pair is overcome, and we successfully construct the breathers on the cn- and dn-periodic wave background.

Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

Dynamic nature of analytical soliton solutions of the nonlinear ZKBBM and GZKBBM equations

A solitary wave, characterized as a localized perturbation in a medium, emerges as a result of a delicate equilibrium between nonlinear and dispersive phenomena. Solitons, a subtype of solitary waves, exhibit persistent shape and velocity during propagation, representing a fundamental phenomenon observed widely in natural systems and possessing various applications in nonlinear dynamics. This investigation focuses on two nonlinear evolution equations (NEEs), specifically the Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation and the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony (GZKBBM) equation, which find relevance in domains such as fluid dynamics and ocean engineering. Utilizing an ansatz-based methodology, soliton solutions of both bright and dark characteristics are derived, alongside exploration of rogue wave-type solutions. Notably, the manifestation of dark, bright, and rogue waves aligns with the physical interpretation of the generated solitons. Computational simulations conducted using Wolfram Mathematica aim to provide a comprehensive description of the physical phenomena. The novelty of this study lies in its unreported investigation, contributing new insights into the solitonic dynamics within the considered models.

PT-symmetric PINN for integrable nonlocal equations: Forward and inverse problems.

Since the PT-symmetric nonlocal equations contain the physical information of the PT-symmetric, it is very appropriate to embed the physical information of the PT-symmetric into the loss function of PINN, named PTS-PINN. For general PT-symmetric nonlocal equations, especially those equations involving the derivation of nonlocal terms due to the existence of nonlocal terms, directly using the original PINN method to solve such nonlocal equations will face certain challenges. This problem can be solved by the PTS-PINN method, which can be illustrated in two aspects. First, we treat the nonlocal term of the equation as a new local component so that the equation is coupled at this time. In this way, we successfully avoid differentiating nonlocal terms in neural networks. On the other hand, in order to improve the accuracy, we make a second improvement, which is to embed the physical information of the PT-symmetric into the loss function. Through a series of independent numerical experiments, we evaluate the efficacy of PTS-PINN in tackling the forward and inverse problems for the nonlocal NLS equation, the nonlocal derivative NLS equation, the nonlocal (2+1)-dimensional NLS equation, and the nonlocal three-wave interaction systems. The numerical experiments demonstrate that PTS-PINN has good performance. In particular, PTS-PINN has also demonstrated an extraordinary ability to learn large space-time scale rogue waves for nonlocal equations.

Large-time lump patterns of Kadomtsev-Petviashvili I equation in a plasma analyzed via vector one-constraint method

In plasma physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for investigating the evolution characteristics of nonlinear waves. For the KPI equation, the constraint method is an effective tool for generating solitonic or rational solutions from the solutions of lower-dimensional integrable systems. In this work, various nonsingular, rational lump solutions of the KPI equation are constructed by employing the vector one-constraint method and the generalized Darboux transformation of the (1 + 1)-dimensional vector Ablowitz–Kaup–Newell–Segur system. Furthermore, we investigate the large-time asymptotic behavior of high-order lumps in detail and discover distinct types of patterns. These lump patterns correspond to the high-order rogue wave patterns of the (1 + 1)-dimensional vector integrable equation and are associated with root structures of generalized Wronskian–Hermite polynomials.

Tolerance of Tourists in Praying in the Prayer Well Room of the Aceh Tsunami Museum

Indonesia is a maritime country located between three world plates that move together, so Indonesia is a country that has a high earthquake rate. Earthquakes that occur at sea can cause giant waves (tsunami). One of the tsunamis that occurred in Indonesia was the Aceh tsunami in 2004. To commemorate this incident, the Aceh Tsunami Museum was built. This museum was built on December 26 2009, so that the victims of the Tsunami disaster would not be forgotten, then as a safety education center, and an evacuation center if another Tsunami occurred. Inside the Tsunami Museum there is a room called the wishing well. On the walls of the prayer well room are the names of the victims of the Aceh Tsunami. Every visitor who enters here is encouraged to pray for the victims according to their respective beliefs. This is where we can see religious tolerance. The formulation of the research problem is how is the tolerance of tourists in praying in the space of sorrow (praying well room) of the Aceh Tsunami Museum? This research aims to determine the tolerance of tourists in praying in the space of sorrow (praying well room) of the Aceh tsunami museum. The method in this research is a qualitative approach where interviews and observations are carried out. The results obtained from the field will be analyzed and described in detail by researchers. The results of this research are that every visitor has a high level of tolerance

Rogue waves for the (2+1)-dimensional Myrzakulov–Lakshmanan-IV equation on a periodic background

In this paper, the rogue wave solutions of the (2+1)-dimensional Myrzakulov–Lakshmanan (ML)-IV equation, which is described by five component nonlinear evolution equations, are studied on a periodic background. By using the Jacobian elliptic function expansion method, the Darboux transformation (DT) method and the nonlinearization of the Lax pair, two kinds of rogue wave solutions which are expressed by Jacobian elliptic functions dn and cn, are obtained. The relationship between these five kinds of potential is summarized systematically. Firstly, the periodic rogue wave solution of one potential is obtained, and then the periodic rogue wave solutions of the other four potentials are obtained directly. The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.

Mixed localized waves in the coupled nonlinear Schrödinger equation with higher-order effects

In this work, the coupled nonlinear Schrödinger equation with higher-order effects is studied with the aid of Darboux transformation and an asymptotic analysis. We report mixed localized waves, rogue wave coexisting with breathers in the system. Under certain constraints, the structures and evolutionary processes of solutions have been influenced. For example, The phase, amplitude, and propagating direction of mixed localized waves change when rogue wave collides with breathers. Moreover, the localized waves characteristics together with collision dynamic behaviors of these explicit rogue wave and breathers are exhibited graphically and discussed in detail. These novel results may be meaningful to study integrable systems better.

Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission

An active elastic metasurface has more flexibility than a passively modulated elastic metasurface, owing to the manipulation of the phase gradient that can be realized without changing the geometrical configuration. In this study, a negative proportional feedback control system was employed to provide positive active control stiffness for adaptive unit cells, with the aim of achieving the active modulation of the phase gradient. The relationship between the control gain and the phase velocity of the flexural wave was derived, and the transfer coefficients and phase shifts of the flexural wave through the adaptive unit cells were resolved using the transfer matrix method. Finite element simulations for wave propagations in the adaptive unit cells were conducted, and they verified the analytic solutions. Based on this theoretical and numerical work, we designed active elastic metasurfaces with adaptive unit cells with sub-wavelength thicknesses according to the generalized Snell’s law. These metasurfaces show flexibility in achieving abnormal functions for transmitted waves, including negative refraction and wave focusing, and transforming guided waves at different operating frequencies by manipulating the control gain. Therefore, the proposed active metasurface has great potential in the fields of the tunable manipulation of elastic waves and the design of smart devices.

Interaction solutions of (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation via bilinear method

Using the bilinear neural network method (BNNM) and the symbolic computation system Mathematica, this paper explains how to find an exact solution for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation. In terms of activation function and weight coefficient, BNNM is a more appealing option for users than traditional symbolic computation methods. It is possible to develop a wide range of solutions and expand the classes of exact solutions by modifying the activation function. The activation function’s versatility allows it to generate a wide range of solutions with several theoretical and practical uses. The analytical solution is obtained by using a double layer type, while the rogue wave solution and mixed solutions are obtained by using a single layer type. The evolution of these waves is then illustrated using various 3D graphs, 2D graphs, and density plots.

Optical Darboux Transformer.

The optical Darboux transformer for solitons is introduced as a photonic device that performs the Darboux transformation directly in the optical domain. This enables two major advances for optical signal processing based on the nonlinear Fourier transform: (i)the multiplexing of solitonic waveforms corresponding to different discrete eigenvalues of the Zakharov-Shabat system, and (ii)the selective filtering of an arbitrary number of individual solitons too. The optical Darboux transformer can be built using existing commercially available photonic technology components and constitutes a universal tool for signal processing, optical communications, optical rogue waves generation, and waveform shaping and control in the nonlinear Fourier domain.

Numerical study on wave run-up and forces on a fixed cylinder under linear and nonlinear focused waves

To study the difference between wave run-up and wave forces on a fixed column under the action of focused waves formed with different physical mechanisms, a 3D two-phase numerical wave tank is developed based on the Navier-Stokes equation and the Finite Volume Method. The influences of wave nonlinearity, column radius and water depth on the run-up and force discrepancies are systematically investigated in four sets of numerical experiments. For focused waves with identical crest height and trough-to-trough period, the run-up height and dynamic pressure around the circumference of a cylinder are mostly larger in linear focused waves than in nonlinear focused waves, due to their different hydrodynamic characteristics. With the increased wave nonlinearity either by changing the period or crest height of the focused waves, the discrepancies become greater for run-up height while smaller for dynamic pressure. Moreover, both of them, less affected by water depth, are more sensitive to the variation of column radius in linear focused waves, showing a steeper gradient in space. Therefore, in practical engineering, the physical generation mechanism must be considered in evaluating the safety of columns of wind turbines hit by freak waves.

Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids

Rogue waves and breathers emerge as significant solitons, result from ubiquitous nonlinear effects of dynamics in the natural world, science and engineering. In this paper, we study the AB system, a model derived from a two-layer fluid that is widely used to investigate nonlinear fluid dynamics. We use the Darboux transformation to construct analytical solutions for higher-order hybrid rogue waves and breathers, which are expressed by symmetric algebraic matrices. We use these solutions and the spectrum parameters to explore the complex interaction dynamics of the hybrid rogue waves and breathers. We find that the interactions cause significant changes in the phases and shapes. Moreover, we discover an interesting phenomenon: the collisions between the rogue waves and breathers are semi-elastic, meaning that only the phases of the rogue waves change, while the speeds, shapes and phases of the breathers remain the same, before and after the collisions. This study provides new insights into the dynamics of the AB system, and helps us understand nonlinear phenomena in various two-layer fluid systems.

How Currents Trigger Extreme Sea Waves. The Roles of Stokes Drift, Eulerian Return Flow, and a Background Flow in the Open Ocean

AbstractA deterministic system of ocean surface waves and flow in the oceanic boundary layer is key to understanding the dynamics of the upper ocean. For the description of such complex systems, a higher‐order shear‐current modified nonlinear Schrödinger equation is newly derived and then used to physically interpret the interplay between Stokes drift, Eulerian return flow due to a passing wave group, and an open‐ocean vertically sheared flow in the extreme sea wave generation. The conditions for the suppression or enhancement of the modulation instability in the rogue wave dynamics in the presence of a background flow are reported, whose relevance and influence to the Craik‐Leibovich type 2 instability in triggering a Langmuir‐type circulation is discussed. The findings highlight the need for future studies to establish and assess the energy transfer from waves to currents or in the reversing order, asserting a plausible physical mechanism for the dissipation of the surface wave energy through wave‐current interactions in the open ocean.

Unattended automatic blood pressure measurements vs conventional office readings in predicting hypertension-mediated organ damage.

Hypertension is a leading cardiovascular risk factor. Accurate blood pressure (BP) measurement is pivotal in hypertension diagnosis and management. Conventional office blood pressure measurements (OBPMs) are error‑prone, exacerbated by the white‑coat effect. Unattended automated office blood pressure measurement (UAOBPM) is emerging as an alternative, mitigating the white‑coat effect. However, its ability to predict hypertension‑mediated organ damage (HMOD) remains disputable. This study compares UAOBPM with OBPM in terms of their association with various types of HMOD, including left ventricular hypertrophy, left atrial enlargement, left ventricular systolic and diastolic dysfunction, intima‑media complex thickening, microalbuminuria, and abnormal pulse wave velocity. A total of 219 hypertensive patients were recruited, interviewed, and examined. Subsequently, BP measurements were conducted in a randomized manner: 1) UAOBPM, after 5 minutes of solitary rest in an examination room, BP was automatically measured 3 times at 1‑minute intervals; 2) OBPM, after 5 minutes of rest, a physician performed 3 consecutive BP measurements at 1‑minute intervals. Subsequent evaluations aimed to detect HMOD and included echocardiography, carotid artery ultrasound, pulse wave velocity assessment, and laboratory tests. UAOBP values were lower than the OBP ones (mean [SD], 124.7 [14.4] vs 128.2 [14.2] mm Hg; P <0.001 for systolic BP, and 73.3 [10.2] vs 75.2 [10.6] mm Hg; P <0.001 for diastolic BP). Correlation and receiver operating characteristic curve analyses revealed no superiority of either method in predicting HMOD. The UAOBPM did not prove superior to OBPM in predicting HMOD. Further research is warranted to determine the role of UAOBPM in clinical practice.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

The Hakcheon-Chogok Megaturbidite in the Miocene Pohang-Youngduk Basin, SE Korea: high-gradient slope failure probably triggered by a giant tsunami wave

The Hakcheon-Chogok Megaturbidite in the Miocene Pohang-Youngduk Basin, SE Korea: high-gradient slope failure probably triggered by a giant tsunami wave

Novel interaction solutions to the (3+1)-dimensional Hirota bilinear equation by bilinear neural network method

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the ([Formula: see text])-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields.

Rational solutions consisting of multiple lump waves and line rogue waves in different spaces of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation in incompressible fluid

Rational solutions consisting of multiple lump waves and line rogue waves in different spaces of the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation in incompressible fluid