Solitary Wave Theory Research Articles

Uniform solitary wave theory for viscous flow over topography

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions.

Supplementary optical solitonic expressions for Gerdjikov–Ivanov equations with three Kudryashov-based methods

The present manuscript uses three Kudryashov-based methods to analytically inspect the class of Gerdjikov–Ivanov equations, which comprises the standard Gerdjikov–Ivanov equation and the perturbed Gerdjikov–Ivanov equation. Various optical solitonic solutions have been constructed. Certainly, as the reported solitonic structures happened to be exponential functions, diverse true solitonic solutions can easily be resorted to upon suitably fixing the involving parameters, including mainly the bright and singular solitons. Lastly, the study graphically examined some of the constructed structures, which were then found to portray some interesting known shapes in the theory of solitary waves and nonlinear Schrödinger equations. Additionally, the Kudryashov-index d has been noted to play a significant role in the propagation of complex waves in the nonlinear media described by Gerdjikov–Ivanov equations.

Numerical Simulation of Subaerial Granular Landslide Impulse Waves and Their Behaviour on a Slope Using a Coupled Smoothed Particle Hydrodynamics–Discrete Element Method

Numerical simulations were conducted to investigate the wave features of subaerial granular landslide-generated impulse waves and their impact on slopes. A numerical solution was obtained by coupling smoothed particle hydrodynamics (SPH) and the discrete element method (DEM). Several predictive equations were tested for their applicability in predicting the maximum crest amplitude of impulse waves generated by slides of different shapes. The results indicated that the predictive model developed by Heller and Hager, utilising slide centroid impact velocity, showed favourable prediction accuracy for the maximum crest amplitude, almost independent of the slide shape at impact. Regarding the leading wave, although the wave profile and velocity distribution deviated significantly from a solitary wave of the same wave amplitude, the maximum run-up could be satisfactorily estimated using solitary wave theory. In addition, the increase in the maximum dynamic forces exerted by the impulse waves on the slope followed a power law with the incident wave amplitude.

Novel exact traveling wave solutions of the space-time fractional Sharma Tasso-Olver equation via three reliable methods

The dominant intention of this article is to extract the new exact traveling waves solutions of the nonlinear space-time fractional Sharma-Tasso-Olver equation in the sense of beta-derivative by using three integration schemes namely, Riccati-Bernoulli (RB) sub-ODE method, Generalized Bernoulli (GB) sub-ODE method and Generalized tanh (GT) method. By the virtue of employed techniques, different types of solutions are obtained in the form of trigonometric, hyperbolic, and exponential functions respectively. The obtained solutions are also verified for the aforesaid equation through symbolic soft computations. To promote the vital propagated features; some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the specific values to the parameters under the confined conditions. Further, based on the bifurcation theory, we examine the phase portrait of the proposed nonlinear equation. Furthermore, we ensure that all the solutions are innovative and have remarkable impacts on the prevailing solitary wave theory literature.

Partial dam-break wave characteristics due to partial gate opening

Understanding how to predict the characteristic height of a dam-break wave is crucial for effective risk management in dam-break scenarios. This study introduces an experimental framework to simulate partial dam-break waves triggered by partial gate openings. We conducted numerous experiments to explore how different factors, including water-level differences (hi), reservoir widths (L), dam-break openings (e), and submerged depths (hw) influence the characteristic heights of the main-wave peak (h1c), subwave peak (h2c), and main-wave trough (h1t). Our analysis led to the development of an equation for predicting h1c. Furthermore, we compared the partial dam-break wave profile with the theoretical profile of a solitary wave, identifying optimal conditions for solitary wave generation and examining the velocity characteristics of partial dam-break waves. The results reveal that: (1) h1c, h2c, and h1t have positive correlations with hi and L, but negative correlations with e and hw; (2) notably, the main-wave pre-peak profile aligns more closely with the solitary wave theory; and (3) the solitary wave theory accurately predicts the main-wave peak velocity. This study enhances our understanding of wave characteristic evolution during partial dam-breaks and can be used for validation in numerical simulations.

Numerical research on internal solitary waves in stratified fluid

Internal solitary waves are a common nonlinear large amplitude internal wave in nature, which can cause strong divergence and sudden strong currents in seawater during propagation, thus having great destructive effects on marine structures. The paper focuses on numerical wave generation based on three types of internal solitary wave theories. The waves are numerical simulated by RANS method in the paper. The numerical results of internal solitary wave theories are compared and analyzed. It shows that under the same numerical water tank scale and calculation conditions, the mKdV theory can obtain the largest wave amplitude, followed by the eKdV theory and the KdV theory. In addition, background currents will accelerate the propagation speed of internal solitary waves and increase their amplitude slightly.

Toward the response of coastal bridges with updated RC shear key properties under the multi-hazard action of earthquakes and tsunamis

Coastal bridges are significantly endangered by the combined effects of earthquakes and tsunamis. The accurate prediction of bridge responses in such scenarios is vital to ensure their security. Reinforced concrete (RC) shear keys, which serve as sacrificial bridge components, can effectively mitigate inertial loads from superstructures, and thus safeguard pier columns. Previous research has shown that their mechanical properties can differ considerably under cyclic and persistent impacts. Therefore, this study focused on the seismic-tsunami responses of a simple-support coastal bridge equipped with RC shear keys. A horizontal failure shear key was incorporated because of its distinct performance observed previously. The OpenSees platform was employed to conduct the numerical work, with an updated material framework developed to modify the shear key behavior before the tsunami analysis. Natural ground motion records were selected to demonstrate the dynamic analysis, and the records were scaled using spectral accelerations (Sa) to account for the various damage states of shear keys resulting from superstructure collisions. The tsunami wave impact was modeled using the second-order solitary wave theory, with different wave conditions employed to consider the immersed conditions. The calculation results were compared for the bridges with and without updated shear key properties, showing distinctions in terms of time-history and maximum responses at certain seismic levels. Based on the analysis results, this study is expected to provide a reliable estimation of the seismic-tsunami responses of coastal bridges.

Verification of solitary wave numerical simulation and case study on interaction between solitary wave and semi-submerged structures based on SPH model

Due to significant influence on the safety of marine structures, the interaction between extreme waves and structures is a crucial area of study in marine science. This paper focus on the verification of a solitary wave meshless SPH model and the application of the model on the interaction between solitary waves and semi-submersible structures. A solitary wave propagation model is established based on the SPH method combined with Rayleigh solitary wave theory, quintic kernel function, artificial viscosity, and Symplectic Method. The accuracy of the model is validated by comparing the calculated wave height with the theoretical value. The calculated results with relative particle spacing H0/d0 ≥ 20 are in good agreement with the analytical solution. The simulated solitary wave is also quite stable with a maximum L2 error 0.016. Therefore, the proposed SPH model can accurately simulate the propagation of the solitary waves. A case study on the interaction between solitary waves and semi-submersible platforms is conducted. The results show that the interaction between solitary waves and semi-submersible causes two double peaks with wave heights of 0.398 m and 0.410 m, respectively, induced by overtopping at the center of the platform. The wave transmission coefficient Kt is 0.880 due to that the solitary wave height reduces from 0.498 m to 0.438 m after the solitary wave propagates through the semi-submersible structure. In addition, the solitary wave induces significant vertical wave loads of the structure with a load amplitude of 0.688, while horizontal wave loads are relatively small with a load amplitude of 0.089. The solitary wave arrived the structure induces the upstream and downstream overtopping and forms a hydraulic jump leading to the complex flow field. The maximum velocity at the top and bottom of the structure is 2.2 m/s and 0.8 m/s respectively. Positive or negative vortex are formed at the bottom of the leading edge, top and downstream of the structure with the maximum intensity 28 s-1 and -40 s-1. In a word, the meshless SPH model can conveniently and accurately simulate the propagation of the solitary waves, and be applied to the investigation of the wave height, velocity, vorticity, wave load, and wave breaking of the interaction between solitary waves and structures in ocean engineering.

Spatial Dynamics and Solitary Hydroelastic Surface Waves

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

Modulation theory for solitary waves generated by viscous flow over a step

The effect of dissipation on the flow of a fluid over a step or jump is examined. This flow has been described theoretically and numerically by using the framework of the forced Korteweg–de Vries–Burgers equation. This model includes the influence of the viscosity of the fluid beyond the Korteweg–de Vries approximation. Modulation theory solutions for the undular bores generated upstream and downstream of the forcing are found. Predictions show that the effect of viscosity plays a key role in determining the properties of waves such as the upstream and downstream solitary wave amplitudes and the widths of the bores. The numerical simulations have shown that the unsteady flow consists of upstream and downstream undular bores, which are connected by a locally steady solution over a step. These numerical solutions are compared with modulation theory results with the excellent agreement obtained. Data from the undular bores are all tabulated. Also, the numerical solutions of the Euler equations are compared with the results of the model, which gave good comparisons. Moreover, the comparisons with the experimental measurements of Lee et al. (1989) together with the Navier–Stokes equations show evidence of the significance of this model.

Solitary wave characteristics on the fine structure of the mesospheric sporadic sodium layer

The so-called sporadic sodium layers (SSLs or NaS) are proposed to be strongly related to wave fluctuations. The solitary wave is a particular solution of the partial differential equation whose energy travels as a localized wave packet. A soliton, on the other hand, is a special type of solitary wave that exhibits a particle-like behavior with a strong stable form. For the first time, the solitary wave theory has been used in this research to study the fine structure of SSL/NaS. We performed soliton fitting processes on the observed data from the Andes Lidar Observatory and found out that 24/27 NaS events had exhibited similar features/characteristics to a soliton. Time series of the net anomaly of the NaS revealed the same variation process to the solution of a generalized five-order KdV equation. Our results, therefore, suggested that the NaS phenomenon would be a pertinent tracer for non-linear wave studies in the atmosphere.

Application of different internal solitary wave theories for SAR remote sensing inversion in the northern South China Sea

This study discusses the theoretical method of the inversion of internal solitary waves (ISWs) characteristic parameters using synthetic aperture radar (SAR) images and ocean reanalysis data. Based on the horizontal distribution characteristics of the pixel intensity of the internal wave fringe in the image, an imaging theoretical model based on the extended Korteweg-de Vries (eKdV) and Benjamin-Ono (BO) theories in SAR was constructed, and the peak-to-peak method and curve-fitting method for calculating the characteristic wavelength of internal solitary waves were determined. The results show that the eKdV theoretical model has the advantage of inverting solitary waves with large amplitudes under the same marine stratified structure, and it provides a more reasonable characteristic parameter estimation than the traditional Korteweg-de Vries (KdV) theory. Although the BO theory is theoretically applicable to deep-water internal waves whose wavelength is far less than the water depth, it can still provide good inversion results in practical applications. Compared to the peak-to-peak method, the curve fitting method fully considers the pixel intensity variation characteristics of the internal wave fringe profile of the image, and the calculated characteristic parameters are more reasonable. Using the maximum buoyancy frequency and internal wave vertical mode to construct the marine stratification model of the deep waters of the northeastern South China Sea may have great uncertainty, and it is better to calculate the interface depth using the relative error of the coefficients of two-layer and continuous internal wave equations. In addition, this study also proposes an internal wave inversion method using continuous SAR images, which provides more reference and support for the selection of internal wave inversion models and reliability test of inversion results in a single SAR image.

Fractional solitons: New phenomena and exact solutions

The fractional solitons have demonstrated many new phenomena, which cannot be explained by the traditional solitary wave theory. This paper studies some famous fractional wave equations including the fractional KdV–Burgers equation and the fractional approximate long water wave equation by a modified tanh-function method. The solving process is given in details, and new solitons can be rigorously explained by the obtained exact solutions. This paper offers a new window for studying fractional solitons.

New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System

This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature.

Numerical Simulation of Flow around A Cylinder in Hyperbolic Flow Field Induced by Internal Solitary Waves

To study the flow field and hydrodynamic characteristics of hyperbolic flow around a cylinder caused by an internal solitary wave, the hyperbolic flow around the cylinder induced by the inner solitary wave, inlet condition is the lower deep horizontal velocity induced by the inner solitary wave and the two-dimensional Navier-Stokes equation is used as the flow field control equation according to the applicability conditions of the eKdV inner solitary wave theory. A numerical simulation method for the nonlinear action of internal solitary wave-induced hyperbolic flow on a horizontal cylinder is established. A series of calculation results show that the numerical simulation results of horizontal force on the cylinder are in good agreement with experimental results. The wakefield of hyperbolic flow around a cylinder has distinct characteristics. In the process of flow around the cylinder, the vortex does not fall off alternately and the lateral lift force is neglected. The force of internal solitary wave-induced hyperbolic flow on the cylinder is non-centrosymmetric, and with the increase of maximum Reynolds number and characteristic period, the horizontal resistance of the cylinder becomes unidirectional. Therefore, the maximum Reynolds number and the characteristic period affect the horizontal force of internal solitary wave-induced hyperbolic flow on a two-dimensional cylinder.

Diverse optical solitons solutions of the fractional complex Ginzburg-Landau equation via two altered methods

<abstract> <p>This work evaluates the fractional complex Ginzburg-Landau equation in the sense of truncated M- fractional derivative and analyzes its soliton solutions and other new solutions in the appearance of a detuning factor in non-linear optics. The multiple, bright, and bright-dark soliton solutions of this equation are obtained using the modified $\left({{{G'} / {{G^2}}}} \right)$ and $\left({{1 / {G'}}} \right) - $expansion methods. The equation is evaluated with Kerr law, quadratic –cubic law and parabolic law non-linear fibers. To shed light on the behavior of solitons, the graphical illustrations in the form of 2D and 3D of the obtained solutions are represented for different values of various parameters. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. The various forms of solutions to the aforementioned nonlinear equation that arises in fluid dynamics and nonlinear processes are presented. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.</p> </abstract>

Bifurcation of some new traveling wave solutions for the time–space M- fractional MEW equation via three altered methods

In this work,1/G′, modified G′/G2 and new extended direct algebraic methods are proposed to construct the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions of the time-fractional modified equal-width (MEW) equation in the sense of M- truncated fractional derivative. These methods contribute a variety of exact solutions in terms of the hyperbolic, trigonometric and rational functions to the scientific literature. The obtained solutions are verified for aforesaid equation through symbolic soft computations. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions. Further, the dynamical behavior is investigated. Based on the bifurcation constrains on the system’s parameters, we constructed also some new wave solution which are assorted into solitary, kink, periodic, and super periodic wave solutions. The influence of the included parameters on the solution is clarified. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.

Experiment-based coupled dynamic analysis of submerged production platform-mooring system in internal solitary waves

As a competitive alternative to the conventional dry tree and wet tree concepts, the next generation subsea production system, with a focus on the critical issues of well access and riser design, is designed to utilize shallow-water rated subsea production systems for the development of large oil and gas fields in deep and ultra-deep water. This study focuses on the crucial component of the next generation subsea production system, referred to as the submerged production platform-mooring system, and investigates its nonlinear coupled dynamic responses to internal solitary waves. A time-domain coupled dynamic analysis method is developed. Specifically, based on the load experiment in a large gravitationally stratified fluid flume, the applicability conditions of internal solitary wave theories and the determination method of hydrodynamic coefficients are quantitatively characterized, and the internal solitary wave load model for the submerged production platform is developed and validated. The fully coupled dynamic analysis model for the submerged production platform-mooring system is established and solved by the Generalized-α integration algorithm in combination with the Newton iterative method. The coupling effects, dynamic response characteristics, and security weaknesses of the submerged production platform-mooring system are analyzed. The effects of various key factors on the dynamic responses of the submerged production platform-mooring system are systematically analyzed. The results demonstrate that the dynamic responses of the submerged production platform-mooring system to internal solitary waves are characterized by the large amplitude surge and heave motions, and the significant fluctuation in the mooring tension, particularly when the submerged production platform is located at the density pycnocline. The developed method also has its applicability to the dynamic analysis of the deepwater free standing hybrid riser system in internal solitary waves.

Novel exact solutions, bifurcation of nonlinear and supernonlinear traveling waves for M-fractional generalized reaction Duffing model and the density dependent M-fractional diffusion reaction equation

In the present work, we study the dynamical behavior of nonlinear traveling waves for the M-fractional generalized reaction Duffing model and density dependent M-fractional diffusion reaction equation. Novel exact solutions in the form of rational, hyperbolic and trigonometric functions are obtained using the new extended direct algebraic method. The obtained solutions are also verified for the aforesaid equations with the help of symbolic soft computations. Furthermore, some produced results are graphically illustrated using 3D surface graphs and 2D line plots, which provide useful information about the dynamical behavior. Moreover, we assurance that all the obtained solutions are accurate, efficient and an excellent contribution in the literature of solitary wave theory. Bifurcation behavior of some nonlinear traveling waves of the proposed equations is also studied with the help of bifurcation theory of planar dynamical systems through numerical simulation. It is observed that proposed equations supports nonlinear solitary waves, periodic waves and most important supernonlinear periodic waves.

Traveling water waves — the ebb and flow of two centuries

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.