Interaction Of Solitary Waves Research Articles

In-depth analysis and exploration of rogue wave, lump wave, and different solitonic patterns to the Painlevé-integrable (3+1)D nonlinear evolution equations using a new extended methodology

This paper proposes a novel extended methodology, establishing the new extended Generalized Exponential Rational Function Method (GERFM), which is known as a “new extended GERFM” for solving the Painleve-integrable [Formula: see text]D generalized nonlinear evolution equation. We enhance GERFM by incorporating additional terms that combine functions and derivatives, making it a more generalized and efficient approach for solving nonlinear partial differential equations. Moreover, the unified method is applied to thoroughly delve deeper into the equation’s properties to understand its mathematical characteristics and potential solutions. This approach aims to systematically investigate and elucidate the equation’s inherent features, providing a more complete insight into its behavior and the nature of its solutions. Utilizing these methods, we have derived new exact solutions that involve exponential, trigonometric, hyperbolic, polynomial, and rational functions with additional parameters to the model. To emphasize the physical importance of these solutions, we present three-dimensional (3D) and contour plots, investigating different parameter choices. The graphs illustrate various wave patterns such as irregular periodic solitons, rogue waves, lumps, and the interaction of solitary and lump waves based on different parameter values. This visualization method enriches our comprehension of the solutions and facilitates a thorough exploration of how they could be applied to the generation of prolonged waves with small magnitudes in plasma physics, fluid dynamics, and other media with weak dispersion.

Laboratory study of the breaking and energy distribution of internal solitary waves over a ridge

The breaking and energy distribution of mode-1 depression internal solitary wave interactions with Gaussian ridges are examined through laboratory experiments. A series of processes, such as shoaling, breaking, transmission and reflection, are captured completely by measuring the velocity field in a large region. It is found that the maximum interface descent ( $a_{max}$ ) during wave shoaling is an important parameter for diagnosing the type of wave–ridge interaction and energy distribution. The wave breaking on the ridge depends on the modified blockage parameter $\zeta _m$ , the ratio of the sum of the upper layer depth and $a_{max}$ to the water depth at the top of the ridge. As $\zeta _m$ increases, the interaction type transitions from no breaking to plunging and mixed plunging–collapsing breaking. Within the scope of this experiment, the energy distribution can be characterized solely by $\zeta _m$ . The transmission energy decreases monotonically with increasing $\zeta _m$ , and there is a linear relationship between $\zeta _m^2$ and the reflection coefficient. The value of $a_{max}$ can be determined from the basic initial parameters of the experiment. Based on the incident wave parameters, the depth of the upper and lower layers, and the topographic parameters, two new simple methods for predicting $a_{max}$ on the ridge are proposed.

Numerical investigation of the Klein-Gordon and sine-Gordon equations using the Christov expansion

Abstract The numerical interactions of solitary waves given by the Klein-Gordon and sine-Gordon equations are demonstrated. To do that we apply the “Christov expansion method”, which is a Galerking type expansion. The Christov expansion is based on the Christov functions which are functions that form a complete orthonormal set of functions on L 2(−∞, ∞) that allow us to expand derivatives, nonlinear products back into the same basis. Here we show how can implement this method to solitary wave equations that form kink or anti-kink type of solutions. We also show how can speed up the algorithms by rewriting the expressions for the nonlinear terms more efficiently. To this end we need to mention that the numerical integration is performed using a Crank–Nicolson type scheme. Our results are in a very good agreement with other published works.

On the study of solitary wave dynamics and interaction phenomena in the ultrasound imaging modelled by the fractional nonlinear system

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}-fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.

Erratum to: “Solitary Wave Interactions in the Cubic Whitham Equation” [RJMP 31 (2), 199–208 (2024)

Erratum to: “Solitary Wave Interactions in the Cubic Whitham Equation” [RJMP 31 (2), 199–208 (2024)

Dynamical property of interaction solutions to the Chafee-Infante equation via NMSE method

In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18.

Tsunami Squares: Leapfrog scheme implementation and benchmark study on wave–shore interaction of solitary waves

Impulse waves are generated by rapid subaerial mass movements including landslides, avalanches and glacier break-offs, which pose a potential risk to public facilities and residents along the shore of natural lakes or engineered reservoirs. Therefore, the prediction and assessment of impulse waves are of considerable importance to practical engineering. Tsunami Squares, as a meshless numerical method based on a hybrid Eulerian–Lagrangian algorithm, have focused on the simulation of landslide-generated impulse waves. An updated numerical scheme referred to as Tsunami Squares Leapfrog, was developed which contains a new smooth function able to achieve space and time convergence tests as well as the Leapfrog time integration method enabling second-order accuracy. The updated scheme shows improved performance due to a lower wave decay rate per unit propagation distance compared to the original implementation of Tsunami Squares. A systematic benchmark testing of the updated scheme was conducted by simulating the run-up, reflection and overland flow of solitary waves along a slope for various initial wave amplitudes, water depths and slope angles. For run-up, the updated scheme shows good performance when the initial relative wave amplitude is smaller than 0.4. Otherwise, the model tends to underestimate the run-up height for mild slopes, while an overestimation is observed for steeper slopes. With respect to overland flow, the prediction error of the maximum flow height can be limited to ± 50% within a 90% confidence interval. However, the prediction of the front propagation velocity can only be controlled to ± 100% within a 90% confidence interval. Furthermore, a sensitivity analysis of the dynamic friction coefficient of water was performed and a suggested range from 0.01 to 0.1 was given for reference.

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.