Interaction Of Solitary Waves Research Articles

Solitary Wave Interactions in the Cubic Whitham Equation

Solitary Wave Interactions in the Cubic Whitham Equation

Higher order Galerkin finite element method for [formula omitted]-dimensional generalized Benjamin–Bona–Mahony–Burgers equation: A numerical investigation

In this article, we study solitary wave solutions of the generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation using higher-order shape elements in the Galerkin finite element method (FEM). Higher-order elements in FEMs are known to produce better results in solution approximations; however, these elements have received fewer studies in the literature. As a result, for the finite element analysis of the gBBMB equation, we consider Lagrange quadratic shape functions. We employ the Galerkin finite element approximation to derive a priori error estimates for semi-discrete solutions. For fully discrete solutions, we adopt the Crank–Nicolson approach, and to handle nonlinearity, we utilize a predictor–corrector scheme with Crank–Nicolson extrapolation. Additionally, we perform a stability analysis for time using the energy method. In the space, O(h3) convergence in L2(Ω) norm and O(h2) convergence in H1(Ω) norm are observed. Furthermore, an optimal O(Δt2) convergence in the maximum norm for the temporal direction is also obtained. We test the theoretical results on a few numerical examples in one- and two-dimensional spaces, including the dispersion of a single solitary wave and the interaction of double and triple solitary waves. To demonstrate the efficiency and effectiveness of the present scheme, we compute L2 and L∞ normed errors, along with the mass, momentum, and energy invariants. The obtained results are compared with the existing literature findings both numerically and graphically. We find quadratic shape functions improve accuracy in mass, momentum, energy invariants and also give rise to a higher order of convergence for Galerkin approximations for the considered nonlinear problems.

Oceanic internal solitary wave interactions via the KP equation in a three-layer fluid with shear flow

Oceanic internal solitary wave interactions via the KP equation in a three-layer fluid with shear flow

On the dynamics of soliton interactions in the stellar environments

The effects of trapping of relativistically degenerate electrons are studied on the formation and interaction of nonlinear ion-acoustic solitary waves (IASWs) in quantum plasmas. These plasmas are detected in high-density astrophysical entities and can be created in the laboratory by interacting powerful lasers with matter. The formula for the number density of electrons in a state of relativistic degeneracy is provided, along with an analysis of the non-relativistic and ultra-relativistic scenarios. While previous studies have delved into specific aspects of relativistic effects, there needs to be a more detailed and systematic examination of the fully relativistic limit, which is essential for gaining a holistic perspective on the behavior of solitons in these extreme conditions. The aim of this work is to comprehensively investigate the fully relativistic limit of the system to fill this gap. The reductive perturbation technique is utilized to deduce the Korteweg–de Vries (KdV) equation, which is used to analyze the properties of the IASWs. Hirota bilinear formalism is applied to obtain single- and multi-soliton solutions for the KdV equation. The numerical analysis is focused on the plasma properties of the white dwarf in the ongoing investigation. The amplitude of the IASWs is found to be maximum for the non-relativistic, intermediate for the ultra-relativistic, and minimum for the fully relativistic limit. Most importantly, it is found that the fastest interaction occurs in the non-relativistic limit and the slowest in the fully relativistic limit.

Physics-Informed Neural Networks with Two Weighted Loss Function Methods for Interactions of Two-Dimensional Oceanic Internal Solitary Waves

Physics-Informed Neural Networks with Two Weighted Loss Function Methods for Interactions of Two-Dimensional Oceanic Internal Solitary Waves

Nonlinear interaction of head-on solitary waves in integrable and nonintegrable systems

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain (a nonintegrable system) and compares the simulation results with the theoretical results in fluid (an integrable system). Three stages (the pre-in-phase traveling stage, the central-collision stage, and the post-in-phase traveling stage) are identified to describe the nonlinear interaction processes in the granular chain. The nonlinear scattering effect occurs in the central-collision stage, which decreases the amplitude of the incident solitary waves. Compared with the leading-time phase in the incident and separation collision processes, the lagging-time phase in the separation collision process is smaller. This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage. We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain. The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude. The results are reversed in the fluid. An increase in solitary wave amplitude leads to decreased attachment, detachment, and residence times for granular chains and fluid. For the immediate time-phase shift, leading and lagging phenomena appear in the granular chain and the fluid, respectively. These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Bipolar Solitary Wave Interactions within the Schamel Equation

Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we investigate the features and properties of bipolar solitary wave interactions within the framework of the non-integrable Schamel equation, contrasting them with the integrable modified Korteweg-de Vries (mKdV) equation. We show that in bipolar solitary wave interactions involving two solitary waves with significantly different amplitudes in magnitude, the behavior closely resembles what is observed in the mKdV equation. However, when solitary waves have similar amplitudes in modulus, the maximum value of their interaction remains less than the sum of their initial amplitudes. This distinguishes these interactions from integrable models, where the resulting impulse amplitude equals the sum of the soliton amplitudes before interaction. Furthermore, in the Schamel equation, smaller solitary waves can transfer some energy to larger ones, leading to an increase in the larger soliton amplitude and a decrease in the smaller one amplitude. This effect is particularly prominent when the initial solitary waves have similar amplitudes. Consequently, large solitary waves can accumulate energy, which is crucial in scenarios involving soliton turbulence or soliton gas, where numerous solitons interact repeatedly. In this sense, non-integrability can be considered a factor that triggers the formation of rogue waves.

SPH Simulation of Solitary Wave Interaction with Cylinder

The wave impact on marine structures is concerning in ocean and coastal engineering. Cylinders are important components of various marine structures such as piers of sea-crossing bridge, columns of oil and gas platforms and subsea pipelines. In this study, the interaction of solitary wave with a submerged horizontal cylinder and a surface-piercing vertical cylinder are numerically studied by the Smoothed Particle Hydrodynamics (SPH) code SPHinXsys. SPHinXsys is an open-source multi-physics library based on the weakly compressible SPH and invokes the low-dissipation Riemann solver for alleviating numerical noises in the simulation of fluid dynamics. The capability of SPHinXsys in reproducing the fluid fields of solitary wave propagating through cylinders is demonstrated by comparing with the experimental data. With the validation in hand, the features of the wave–structure process are examined.

Interaction of Schamel type solitary waves with superthermal trapped electrons

The expanded Poincar'e-Lighthill-Kuo approach is used to explore the collision of Schamel type ion acoustic solitary waves. The phase shift, the most crucial characteristic, is calculated using specific solutions on the basis of the Schamel type Korteweg-de Vries equations. It has been revealed that, in given regime of physically acceptable conditions, the effects of electron confinement and superthermality have a significant influence in altering the phase shift. The results will be a valuable contribution to ongoing research on plasma waves interactions as well as to inquiries into a number of laboratory plasma experiments.

Analysis of soliton interactions of modified Korteweg-de Vries equation using conserved quantities

In this paper, the conservative quantities are used to develop an approximate method to calculate the merged waveform shape of the solitary waves described by modified Korteweg–de Vries (mKdV) equation. With this method, we can efficiently and effectively capture the physics of the complicated merging phenomena when two solitary waves described by the nonlinear evolution partial differential equation merge at the maximum without the need to solve the equation in detail. This offers a simple and robust tool to analyse the interactions between solitons and to benchmark the results obtained by the asymptotic and numerical methods. It is expected that the approximate analysis demonstrated in this paper can be applied to a series of nonlinear evolution equations to simulate various solitary wave interaction problems. In future, our goal is to extend this simple method to other nonlinear wave evolution phenomena.

Oblique interactions of internal solitary waves in the lower atmosphere

Internal solitary waves frequently occur in the atmosphere. On rare occasions, they create the awe-inspiring spectacle known, for example, as the Morning Glory Clouds, a spectacular roll cloud, or series of roll clouds predictably appearing in the southern part of the Gulf of Carpentaria. Nevertheless, solitary wave–wave interactions have rarely been studied and documented; thus, we here focus on the long-time evolution of the superposition of two solitons featuring an X-shape and, more complicated, the interactions between three solitons initially posing as a Y-shape. To better understand the underlying dynamics of these phenomena, we derive a bidirectional and isotropic theoretical equation in a two-layer fluid system with variable bottom topography. This is accomplished by using its Hamiltonian structure and the Taylor expansion of the Dirichlet–Neumann operator for the potential theory. Essentially, the derived equation is an extension of the widely recognized Benjamin–Ono equation at two horizontal dimensions, and thereby, it possesses plane soliton solutions propagating in any horizontal direction. It is noted that the initial angles play an essential role in the oblique wave–wave interactions, manifested as the determination of waveforms, amplitudes, and the emergence of the Mach stem. In addition, the wave evolution is slightly modulated by the topographic effects, partly due to invoking the assumption of small topography.

Two-dimensional numerical simulations of solitary wave interaction with a vertical elastic plate

In this study, we numerically investigated the interaction between a solitary wave and a bottom-mounted, submerged elastic plate. The fluid structure interaction (FSI) solver (solids4Foam), the stabilized k-ω SST turbulence closure model, and the geometric volume of fluid approach (isoAdvector) were combined to simulate the dynamic process, including the plate movement and the flow field. The model package was first validated by using experimental data, and the effects of plate height, plate thickness, and the Young's modulus of plate material on the hydrodynamics were further investigated. Moreover, the Euler-Bernoulli beam theory was used to calculate the plate displacement. Lengthening the plate height, reducing the plate thicknesses, or reducing Young's modulus can all increase the plate displacement. Among them, lengthening the plate height has the greatest impact. Finally, the effects of the wave nonlinearity and plate material property on wave reflection, transmission, and dissipation were also analyzed. The results show that while the height of the plate has the most significant effect on the reflection coefficient, the plate thickness and the plate material have little influence on the reflection coefficient. The more flexible the plate is, the smaller the transmitted coefficient, and the larger the dissipation coefficient are.

On the propagation and interaction of ion-acoustic solitary, periodic, shock, and breather waves in a non-Maxwellian electron–positron–ion magnetoplasma

This investigation analyzes the propagation of nonlinear ion-acoustic waves (IAWs) in an unmagnetized, collisionless plasma composed of inertial positive ions and inertialess Maxwellian positrons as well as the inertialess non-Maxwellian electrons that obey (r, q)-distribution. To observe the impact of particle trapping on the nonlinear IAWs in an electron–positron–ion plasma, the Korteweg–De Vries (KdV) and modified KdV (mKdV) equations are derived using a reductive perturbation method. In the distribution function, the spectral parameters (r, q) put up their contribution to the flatness and high-energy tails, respectively. An important aspect of this investigation is the determination of well-known quasi-periodic solutions, multi-soliton solutions, breathers, and shocks under the variation of different physical parameters, especially spectral indices (r, q). Finally, the interaction of solitons is also presented for discussion of the complete profile. In addition, a detailed comparison, especially in a periodic wave, is made between the generalized (r, q)-distribution and the limiting cases of Kappa and Maxwellian distributions. The results presented in this study contribute to a better understanding of the characteristics of both high- and low-energy parts of the electron distribution function as well as the formation of periodic, soliton, multi-soliton, breathers, and shocks in space and astrophysical plasmas.

Experimental and numerical studies of solitary wave interaction with perforated caisson breakwaters

This study experimentally and numerically investigated interactions between solitary waves and the perforated caisson breakwaters. By caisson, we mean a sealed chamber filled with sand and rocks inside, and it is a common structure used for the construction of vertical breakwater. In the laboratory, the solitary waves with larger relative wave heights were well generated based on the “collapsing water column” technique and successfully acted on the perforated caisson models. Using the volume of fluid method and the k–ε model, combined with the ideal gas equation at a constant temperature, the wave transformation and vortex evolution in the vicinity of the perforated caisson breakwaters were simulated. A reasonable agreement was observed between the numerical and the experimental results. By comparing with the non-perforated caissons, the perforated caissons effectively reduced the reflected and transmitted wave heights, and the occurrence of the reflected waves was found to be delayed due to the existence of the wave chamber. Based on the numerical results, distributions of the fluid velocity and turbulence kinetic energy (TKE) near the perforated caissons were examined. The wave dissipation mechanism of perforated caisson under the solitary wave was different from that under the periodic wave. The results showed that vortices and TKE were mainly concentrated near the perforated front wall. The incident wave energy was dissipated in the generating vortices formed by fluids jetting through perforations. Additionally, variations of the wave reflection, transmission, dissipation coefficients, and wave overtopping volumes were investigated against different relative crest freeboards, relative wave chamber widths, caisson porosities, and relative wave heights under the solitary waves. Valuable results were presented for practical engineering applications.

Interaction of mode-one internal solitary waves of opposite polarity in double-pycnocline stratifications

Numerical simulations of the interaction of internal solitary waves (ISWs) of opposite polarity are conducted by solving the incompressible Euler equations under the Boussinesq approximation. A double-pycnocline stratification is used. A method to determine when ISWs of both polarities exist is also presented. The simulations confirm previous work that the interaction of waves of the same polarity are soliton-like; however, here it is shown that when a fast ISW with the same polarity as a Korteweg–de Vries (KdV) solitary wave catches up and interacts with a slower ISW of opposite polarity, the interaction can be far from soliton-like. The energy in the fast KdV-polarity wave can increase by more than a factor of 5 while the energy in the slower negative-KdV-polarity wave can decrease by 50 %. Large trailing wave trains may be generated and in some cases multiple ISWs with KdV polarity may be formed by the interaction.

Numerical study of the interaction between an internal solitary wave and a submerged extended cylinder using OpenFOAM

To investigate the impacts of cylinder extended length, rotation angle and wave incidence angle on hydrodynamic characteristics, the numerical simulation of an internal solitary wave (ISW) interaction with a submerged extended cylinder is conducted by using OpenFOAM. The accordance between the experimental data and numerical simulation forces on the isolated cylinder shows that the OpenFOAM-based numerical model can deal with the ISW-submerged cylinder interaction. The distributions of forces, maximum dynamic pressure forces and flow field around the submerged extended cylinder are discussed to reveal the hydrodynamic characteristics of submerged cylinder under the ISW action. The horizontal forces on an extended cylinder are found to change from increasing to decreasing with the increase in extended length when the extended length is 2.5 times the cylinder diameter. The forces on the cylinder are sensitive to the change in the rotation angle. Both the horizontal loads and the evolution of the wave profile are easily affected by the change in incidence angle. The vortices shedding around the cylinder varies with the change of wave incidence angle. Investigation of the ISW and a submerged cylinder interactions can help to understand the mechanism of structural damage in internal wave ocean environment.

Solitary Waves and Their Interactions in the Cylindrical Korteweg–De Vries Equation

We consider approximate, exact, and numerical solutions to the cylindrical Korteweg–de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the interaction of solitary waves of different types.

Solitary wave interaction with an impervious house in the coastal community using a parallel three-dimensional incompressible CIP-based model

Three-dimensional numerical investigations on solitary wave interaction with an impervious house in the coastal community using a parallel incompressible CIP-based model are carried out. Simple and effective parallel computation will be realized based on algorithm of domain decomposition method and hybrid OpenMPI–OpenMP computational framework. The convection term is discretized by a constrained interpolation profile (CIP) method and an accurate interface capturing scheme, the Tangent of Hyperbola for Interface Capturing with Slope Weighting (THINC/SW), is applied to reconstruct the free surface. The reliability of the present model is validated by a series of benchmarks, including regular wave interaction with a single cylinder, regular wave reflection and diffraction on a vertical structure and interactions between a solitary wave and an impervious structure. The present results are in good agreement with the published results. The present model is applied to investigate the solitary wave reflection and diffraction on the house with different roof types. The variations of wave force, wave elevation and free surface are investigated, which considers the effect of natural disasters such as tsunamis on the safety of the house with different roof types in the coastal community. These results provide engineering references for the design of tsunami-resilient houses.