Solitary Waves Research Articles

Generation characteristics of internal solitary waves in the Northern Andaman sea based on MODIS observations and numerical simulations

The generation characteristics of internal solitary waves (ISWs) in the northern Andaman Sea are studied using remote sensing data and numerical simulations. The dataset comprises 230 images taken by the Moderate Resolution Imaging Spectroradiometer (MODIS) from 2020 to 2023, which reveal two distinct propagation directions of ISWs: southeastward (type-SE) and southwestward (type-SW). Generation hotspots are identified at the southern sill of the Preparis South Channel for type-SE ISWs and over the eastern shelf break for type-SW ISWs. Here, ISWs are intermittently generated, resulting in wave-free spatial gaps. The observed gaps motivate the hypothesis that ISWs are phase-locked with semidiurnal tides, resulting in specific generation time windows around slack tides. This study also reveals that the generation of ISWs is closely related to tidal ranges, with larger tidal ranges leading to more frequent ISWs. Specifically, type-SE and type-SW ISWs are detected only when the tidal range exceeds 1.3 m and 1.4 m, respectively. Numerical simulations with the MIT general circulation model further show that both types of ISWs are generated by nonlinear steepening of internal tides. The influence of background currents on ISW generation is also examined, which supports the proposed generation time window hypothesis. These findings highlight spatial gaps are attributable to the synergistic effects of MODIS observation times and intermittent generation of ISWs.

Solitons, Lumps, Breathers, and Interaction Phenomena for a (2+1)-Dimensional Variable-Coefficient Extended Shallow-Water Wave Equation

In this paper, soliton solutions, lump solutions, breather solutions, and lump-solitary wave solutions of a (2+1)-dimensional variable-coefficient extended shallow-water wave (vc-eSWW) equation are obtained based on its bilinear form. By calculating the vector field of the potential function, the interaction between lump waves and solitary waves is studied in detail. Lumps can emerge from the solitary wave and are semi-localized in time. The analytical solutions may enrich our understanding of the nature of shallow-water waves.

Traveling waves reflecting various processes represented by reaction–diffusion equations

The aim of this paper is to discover analytically the interactional responses of populations in a dynamic region where the reaction–diffusion process with forcing effects takes place through traveling wave solutions. An expansion method is considered here to properly capture the responses for the first time. In order to profoundly analyze the physical and mathematical discussions, some illustrative behavioral results are exhibited for various values of physical parameters. Especially for the different values of diffusion coefficients in the model under consideration, their effects on the behavior of the solitary wave are discussed and observationally supported by considering various illustrations. It is also seen that the solutions representing the diffusion seen to be in the form of the behavior of hexagonal Turing patterns in different time periods. The application of this study in mathematical biology is to analyze the relationship between the population density of certain species in any local region and the specific population density with invasion characteristics. In addition, the formation of the extinction vortex of the invading population, depending on the characteristics of the solutions presented, is also descriptively discussed.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Novel soliton solutions and phase plane analysis in nonlinear Schrödinger equations with logarithmic nonlinearities

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations.

Orbital stability of solitary wave solutions of a Hamiltonian PDE arising in magma dynamics

We consider a Hamiltonian PDE arising from a class of equations appearing in the study of magma dynamics in the Earth’s interior. Previously, it has been shown that the Hamiltonian PDE admits solitary wave solutions. Simpson et al. proved that the solitary wave solutions are orbitally stable for the case n=2. We verify the stability criterion analytically for the case n=3. Our results answer partially an open question proposed by Simpson et al. (2008).

Solitary wave-induced boulder transport on a beach

The influence of sediment on a slope on boulder transport is studied experimentally at the Hydrodynamics Laboratory of the University of Oslo. Experiments are conducted in a small 3 m long and 10 cm wide wave tank, filled with 5 cm of water. The transport of boulders is induced by breaking solitary waves with an amplitude normalized by water depth a/hw= 0.5. Generated solitary waves propagate towards a 1:10 beach. The experiments are conducted in two set-ups: (i) empty Polymethyl methacrylate (PMMA) bottom of the flume, (ii) the slope covered by a thin layer of 65μm sand. The boulders are represented by concrete blocks of different shape and size. They are placed alternately (one at a time) at different locations on the beach slope with respect to wave breaking point. Transport of boulders and their dynamics is studied with respect to boulder characteristics (size, orientation), their initial position and inclusion of sediment. It is shown that presence of sediment enhances boulder transport. In particular, in this set-up, the presence of sediment increased the boulder transport in 2–5 times. The maximum displacement increase was observed for boulders with the smallest length and height and the largest width initially located at the breaking position. Another interesting result regards the type of boulder motion. The boulders experienced either sliding or turning over. Boulders whose height was at least twice as large as their length exhibited turning-over. This held for boulders placed both on an empty PMMA slope and on a sedimentary slope. However, the largest boulder displacement on an empty PMMA slope occurred due to turning over, while on a sedimentary slope it occurred due to sliding.

Stochastic analysis and soliton solutions of the Chaffee–Infante equation in nonlinear optical media

The Chaffee–Infante (CI) equation is a nonlinear equation that may be used to predict the complex dynamics of soliton propagation in a nonlinear optical medium. In this study, we elucidate soliton solutions of the CI equation by stochastic differential equations (SDEs) with the Wiener process. In excess of the modified auxiliary equation (MAE) method, we obtain new exact soliton solutions. By combining stochastic differential equations with the Wiener process, we explain the stochastic processes directing the magnitude of solitons within the basis of the CI equation, providing dynamic views on their behavior. The acquired solitary waves are depicted via 3D and 2D graphs to show their dynamics with and without Brownian motion.

Generation of Solitary Waves with Analytical Solution for The (3+1)-dimensional pKP-BKP Equation and Reductions

In this study, new solitary wave solutions are obtained for the combination of the B-type Kadomtsev-Petviashvili (BKP) equation and the potential Kadomtsev-Petviashvili (pKP) equation, called the integrable (3+1)-dimensional coupled pKP-BKP equation, and its two reduced forms. For this purpose, the Bernoulli auxiliary equation method, which is an ansatz-based method, is used. As a result, kink, lump, bright soliton and breather wave solutions are observed. It is concluded that this method and the results obtained for the considered pKP -BKP equations are an important step for further studies in this field.

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

Multi-dimensional modeling of solitary wave–structure interaction problems by using a [formula omitted]-LES-SPH model

The interaction mechanisms between waves and marine structures are a popular research topic. This paper applies the weakly compressible smoothed particle hydrodynamics (WCSPH) method to study the dynamics of green water overtopping. To enhance the accuracy of the simulations, the SPH method coupled with the large eddy simulation (LES) model is employed for numerical investigations. Initially, we validate the effectiveness of the model by simulating the generation of solitary waves and irregular waves, as well as numerically reproducing the water surface morphology during the interaction between solitary waves and the deck. Subsequently, the validated model is used to study the dynamic characteristics of different types of waves overtopping, revealing significant variations in their motion. Furthermore, we investigate the effect of deck roughness during the entire green water overtopping process in terms of both protrusions extent and distribution, confirming that a reasonable setting of the protrusions can greatly reduce the wave impact loads on the deck, thereby protecting the structure. Additionally, a three-dimensional model is developed to study the green water problem, and we find that the turbulence phenomenon is more pronounced in the three-dimensional scenario.

Newly formed solitary wave solutions and other solitons to the (3+1)-dimensional mKdV–ZK equation utilizing a new modified Sardar sub-equation approach

In this paper, we are going to investigate the (3+1)-dimensional nonlinear modified Korteweg–de Vries-Zakharov–Kuznetsov (mKdV–ZK) equation, which governs the behavior of weakly nonlinear ion acoustic waves in magnetized electron–positron plasma. By taking advantage of the newly proposed modified Sardar sub-equation method, we derive a comprehensive set of exact soliton solutions to the mKdV–ZK equation. Additionally, we provide graphical representations of the solutions, including 2D, 3D, and contour plots, to visualize the characteristics and features of these nonlinear wave structures. These solutions encompass a diverse range of wave patterns, including traveling waves, bright solitons, periodic waves, dark–bright solitons, lump-type solitons, and multi-soliton solutions. The obtained solutions provide valuable insights into the nonlinear behaviors and dynamics exhibited by the mKdV–ZK equation. The success of the new modified Sardar sub-equation method in obtaining a diverse range of solutions for the [Formula: see text]-dimensional mKdV–ZK equation highlights its potential for applications in the analysis of various nonlinear systems in plasma physics and beyond. Also, the study reviewed the superiority of the modified method compared to the Sardar sub-equation method.

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.

Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation

Abstract In this study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.

Hydrodynamics and Sediment Transport Under Solitary Waves in the Swash Zone

Swash–swash interaction is a common natural phenomenon in the nearshore region, characterized by complex fluid motion. The characteristics of swash–swash interaction are crucial to sediment transport, subsequently affecting the beach morphology. This study investigates the hydrodynamics and sediment transport in swash–swash interaction under two successive solitary waves using a two-phase Smoothed Particle Hydrodynamics (SPH) model. The effects of the time interval between the two waves are examined. It is shown that the time interval has a minor effect on the breaking and swash–swash interacting patterns as well as the final beach morphology but influences the run-up of the second wave and the instantaneous sediment flux. Under wave breaking in the swash–swash interaction, there is significant sediment suspension due to strong vortices, and the suspended sediment forms a plume upward from the bed. The sediment plumes gradually settle down as the vortices decay. These insights enhance the understanding of sediment transport and beach morphology under complex swash–swash interaction.

A quasi-three-phase flow simulation of the interactions between solitary waves and a vertical seawall installed on a sandy beach

Local scour and wave loading are two key factors that affect the safety of seawalls under tsunami attacks. In this study, the scouring process at the toe of and wave impact force on a vertical seawall under consecutive solitary wave attacks are simulated using a quasi-three-phase (air, water, and sediment) flow model, which models the air and water as a fluid mixture phase and sediment as a solid phase. The air and water interface is modeled by a VOF method and the sediment–fluid interaction is modeled using the Eulerian two-phase flow method. A comparison between the beach profiles with and without the seawall shows that, with a seawall, a deeper scour hole is generated at the toe of the seawall. Besides, the presence of the seawall causes the eroded sediment to be deposited further offshore. The numerical results are then analyzed in detail regarding the flow field and sediment transport process to reveal the mechanisms of the above eroding/depositing patterns. The wave impact force on the seawall is also analyzed to understand the effect of scouring on the wave loading. It is shown that the wave impact force, despite being stochastic for a specific wave, increases in general with the deepening of the scour hole because the latter increases the exposure of the seawall to wave slamming.

Kudryashov-Sinelshchikov equation: phase portraits, bifurcation analysis and solitary waves

Kudryashov-Sinelshchikov equation: phase portraits, bifurcation analysis and solitary waves

Stability analysis and solitary wave solutions for Yu Toda Sasa Fukuyama equation

Stability analysis and solitary wave solutions for Yu Toda Sasa Fukuyama equation