Interaction Of Solitary Waves Research Articles

Decoupled conservative schemes for computing dynamics of the strongly coupled nonlinear Schrödinger system

Two types of first-order decoupled conservative schemes are firstly proposed for the strongly coupled nonlinear Schrödinger system by using pseudospectral method in space and coordinate increment discrete gradient method in time. And then, in order to improve the solution accuracy in time, the composition methods are employed to construct second- and fourth-order schemes. The proposed schemes are efficient for the system in d≥2 dimensions and also easy to code because of their decoupled feature. A fast solver is proposed to speed up the computation. Ample numerical examples including the motion of single soliton and interaction of multiple solitary waves are carried out to exhibit the performance of the schemes.

Numerical study of solitary waves past slotted breakwaters with a single row of vertical piles: Wave processes and flow behavior

Interactions of solitary waves with slotted breakwaters comprising a single row of vertical circular piles was studied using a numerical wave tank (NWT) based on the Navier–Stokes equations. The NWT combines the immersed boundary method to impose boundary conditions on solid surfaces, the level-set method to track the free surface, and a fast Poisson solver achieving efficient simulations. The NWT was validated by comparisons to experimental and numerical data in the literature. Subsequently, the assessment of the hydrodynamic behavior of the breakwater was carried out, examining different values of incident wave height, Hi, pile diameter, D, distance between piles, S, and spacing between piles, e=S−D. It was found that for constant D and decreasing e∕S, the wave run-up, Kru, reflection, Kr, and energy loss, Kd, coefficients increase, while the transmission coefficient, Kt, decreases. Also, for constant e∕S and D, increasing Hi, resulted into the increase of Kr, Kru, and Kd but the decrease of Kt. Moreover, for constant e∕S, smaller D leads to increased Kru and Kd and decreased Kr and Kt. These trends amplify when Hi increases. To explain the effect of the geometric parameters, wave breaking, air-entrapment, and vorticity generation were examined in the vicinity of the piles.

Propagation of Solitary Waves over a Submerged Slotted Barrier

In this article, the interaction of solitary waves and a submerged slotted barrier is investigated in which the slotted barrier consists of three impermeable elements and its porosity can be determined by the distance between the two neighboring elements. A new experiment is conducted to measure free surface elevation, velocity, and turbulent kinetic energy. Numerical simulation is performed using a two-dimensional model based on the Reynolds-Averaged Navier-Stokes equations and the non-linear k-ɛ turbulence model. A detailed flow pattern is illustrated by a flow visualization technique. A laboratory observation indicates that flow separations occur at each element of the slotted barrier and the vortex shedding process is then triggered due to the complicated interaction of those induced vortices that further create a complex flow pattern. During the vortex shedding process, seeding particles that are initially accumulated near the seafloor are suspended by an upward jet formed by vortices interacting. Model-data comparisons are carried out to examine the accuracy of the model. Overall model-data comparisons are in satisfactory agreement, but modeled results sometimes fail to predict the positions of the induced vortices. Since the measured data is unique in terms of velocity and turbulence, the dataset can be used for further improvement of numerical modeling.

Numerical Simulations of the Interaction of Solitary Waves and Elastic Structures with a Fully Eulerian Method

Wave attenuation by elastic structures is a challenging issue when studying shoreline protection subjected to waves and erosion. For this purpose, a fully Eulerian method is then developed to solve the problem of fluid–elastic structure interactions for incompressible flows. The specificity of the proposed approach is to use a combination of a reinitialization technique for the interface and linear extrapolations for the deformations to prevent numerical instabilities for the computation of the elastic forces. First, we verify and validate our approach on several test cases. Next, we apply this method to illustrate the capacity of the model to study the wave damping phenomenon by a single or several elastic structures.

High-order unidirectional model with adjusted coefficients for large-amplitude long internal waves

To describe large amplitude internal solitary waves in a two-layer system, we consider the high-order unidirectional (HOU) model that extends the Korteweg–de Vries equation with high-order nonlinearity and leading-order nonlinear dispersion. While the original HOU model is valid only for weakly nonlinear waves, its coefficients depending on the depth and density ratios are adjusted such that the adjusted model can represent the main characteristics of large amplitude internal solitary waves, including effective wavelength, wave speed, and maximum wave amplitude. It is shown that the solitary wave solution of the adjusted HOU (aHOU) model agrees well with that of the strongly nonlinear Miyata–Choi–Camassa (MCC) model up to the maximum wave amplitude, which cannot be achieved by the original HOU model. To further validate the aHOU model, numerical solutions of the aHOU model are presented for the propagation and interaction of solitary waves and are shown to compare well with those of the MCC model. The aHOU model is further extended to the case of variable bottom and is solved numerically. In comparison with the MCC model for variable bottom, it is found that the aHOU model is a simple, but reliable theoretical model for large amplitude internal solitary waves, which would be useful for practical applications.

Compact structure-preserving algorithm with high accuracy extended to the improved Boussinesq equation

The improved Boussinesq equation is numerically studied using a higher-order compact finite difference technique. The aim is to achieve a mass and energy preserving scheme precisely on any time–space regions. The advantage of this scheme is that we can deal with a nonlinear partial differential equation with an implicit linear algorithm. Furthermore, the characteristics of the method are its simple steps and effective clearness. In addition, the convergence and stability analysis are then conducted to search a numerical solution whose the existence and uniqueness are guaranteed. The spatial accuracy is analyzed and found to be fourth order on a uniform grid. The numerical results are compared with established available data in literature for similar test cases, and the results are seen to be in good agreement. Besides, we perform relevant numerical simulations to illustrate the faithfulness of the present method by the evidences of the solitary wave interaction as well as the rapidly depressed solitary waves generation under sufficiently instantly decaying initial data.

Numerical investigation of an internal solitary wave interaction with horizontal cylinders

An internal solitary wave (ISW) interaction with different type horizontal cylinders are investigated. The forces induced by ISW on horizontal cylinders are presented. Distributions of the maximum dynamic pressures and flow fields around cylinders are described. Good agreements between numerical results and experimental data of forces on the isolated cylinder are achieved. For the extended cylinder, the vertical forces per unit length increase with the increase in the extended length. The horizontal forces on the extended cylinder change from increasing to decreasing after the centre-to-centre distance exceeding 3.5 times the diameter of the isolated cylinder. The peak and valley values of maximum dynamic pressures impose on the lower side of the leading edge and upper side of the trailing edge of the extended cylinder, respectively. The horizontal forces on each of the two tandem cylinders with different centre-to-centre distances are smaller than those on the isolated cylinder, indicating that the two cylinders play a role to protect each other. The comparison of forces between the two cylinders indicates that larger horizontal forces and smaller vertical forces are imposed on the rear cylinder. The maximum dynamic pressures around the rear cylinder are smaller than those around the front cylinder.

Interactions of solitary waves in integrable and nonintegrable lattices.

We study how the dynamics of solitary wave (SW) interactions in integrable systems is different from that in nonintegrable systems in the context of crossing of two identical SWs in the (integrable) Toda and the (non-integrable) Hertz systems. We show that the collision process in the Toda system is perfectly symmetric about the collision point, whereas in the Hertz system, the collision process involves more complex dynamics. The symmetry in the Toda system forbids the formation of secondary SWs (SSWs), while the absence of symmetry in the Hertz system allows the generation of SSWs. We next show why the experimentally observed by-products of SW-SW interactions, the SSWs, must form in the Hertz system. We present quantitative estimations of the amount of energy that transfers from the SW after collision to the SSWs using (i) dynamical simulations, (ii) a phenomenological approach using energy and momentum conservation, and (iii) using an analytical solution introduced earlier to describe the SW in the Hertz system. We show that all three approaches lead to reliable estimations of the energy in the SSWs.

The mixed interaction of localized, breather, exploding and solitary wave for the (3+1)-dimensional Kadomtsev–Petviashvili equation in fluid dynamics

The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation with weak nonlinearity, dispersion and perturbation can denote the development of the long waves and the surface waves in fluid dynamics. In this paper, the KP equation is illustrated with the symbolic computation. The mixed interaction solutions of local wave, solitary wave, breather wave, exploding wave and periodic wave for the equation are derived by the Hirota method. The effects of dispersion, nonlinearity and other parameters on the interactions are investigated. The solitary wave can be amplified via introducing the local wave. Adjusting the parameters can make the transmission of localized and breather wave more stable. Moreover, a new exploding and periodic wave is observed. It is useful for enriching the dynamic patterns of the wave solutions.

New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach

This manuscript investigates the fractional Phi-four equation by using q -homotopy analysis transform method ( q -HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and q -homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach’s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations.

Modeling porous coastal structures using a level set method based VRANS-solver on staggered grids

ABSTRACT Several engineering problems in the field of coastal and offshore engineering involve flow interaction with porous structures such as breakwaters, sediment screens, and scour protection devices. In this paper, the interaction of waves with porous coastal structures using an open-source computational fluid dynamics (CFD) model is presented. The fluid flow through porous media is modeled using the Volume-averaged Reynolds-averaged Navier-Stokes (VRANS) equations. Novel improvements to the numerical grid architecture and discretization schemes are made, with a staggered numerical grid for better pressure-velocity coupling and higher-order schemes for convection and time discretization. New interpolation schemes required for the VRANS equations on a staggered grid are implemented. The flow problem is solved as a two-phase problem and the free surface is captured with the level set function. The model is validated by comparing the numerical results to experimental data for different cases such as flow through crushed rock, solitary, and regular wave interaction with a porous abutment and wave interaction with a breakwater considering the three different porous layers. The numerical results are also seen to be highly grid-independent according to the grid convergence study and show a significantly better agreement to experimental data in comparison to current literature.

Experimental Investigation on Solitary Wave Interaction With Vertical Porous Barriers

Abstract Tsunami waves pose a threat to the coastal zone, and numerous studies have been carried out in the past to understand them. Solitary waves have been extensively used in research because they approximate certain important characteristics of tsunami waves. The present study focusses on the interaction and run-up of solitary waves on coastal protection structures in the form of thin, rigid vertical porous barriers with special attention given to the degree of energy dissipation. To understand the physics of energy dissipation, solitary wave interaction with a porous barrier has been studied from the viewpoint of energy balance. Based on this, a relationship for the wave energy dissipation has been developed. The experimental data show that the plate porosity that gives the optimal energy dissipation lies within the 10–20% range. From the experiments, the phase shift that the solitary wave undergoes upon interaction with the porous barrier models has also been recorded. In addition, a formula is proposed for maximum wave run-up on the porous barrier, which should be useful in the planning, design, construction, and maintenance of coastal protection structures.

Solitary wave, M-lump and localized interaction solutions to the (4+1)-dimensional Fokas equation

At present, many researchers pay close attention to the solutions of nonlinear partial differential equations. Because these solutions can explain physical phenomena well. In this paper, soliton solutions, fissionable wave solutions, M-lump solutions and interaction solutions of the (4+1)-dimensional Fokas equation are explicitly generated. Taking advantage of the Hirota bilinear method, N-soliton solutions are constructed. Then, on bases of the obtained solutions, fissionable wave solutions are ascertained by appropriate selections of parameters. Applying the long wave limit method and restricting the conjugation conditions on the related solitons, M-lump solutions are derived. Afterwards, the interaction solutions which describe interactions of a lump and a soliton, a lump and two solitons, a lump and fissionable solitary waves are gained. Moreover, some graphs are given out to demonstrate the dynamic properties of the explicit analytical localized wave solutions.

Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method

An efficient and accurate Crank–Nicolson meshless spectral radial point interpolation (CN-MSRPI) method is proposed for the numerical solution of nonlinear Rosenau–KdV equation. The proposed method uses meshless shape functions, owing Kronecker delta property, for approximation of spatial operator. Crank–Nicolson difference scheme is used for temporal operator approximation. Single solitary wave motion, interaction of double and triple solitary waves as well as generation of train of solitary waves from initial data are numerically simulated. Error analysis is made via computation of discrete $$L_{\infty }$$, $$L_{2}$$ and $$L_{\mathrm{rms}}$$ error norms. Efficiency of the proposed numerical scheme is assessed via variation of number of nodes N and time step-size $$\tau $$. Two invariant quantities correspond to mass and energy are computed using the proposed method for further validation. Stability of the proposed method is discussed and verified computationally. Comparison of obtained results made with exact and existing results in the literature revealed the proposed CN-MSRPI method superiority.

Quantum effects in bi-dust plasmas

In this paper a theoretical approach has been carried out using a fluid model on a bi-dust plasma which consist of two dusts negative charged different particles in an electron–ion plasma. In such system, this fluid model describe a dense strong coupled heavy dust grain together with a weak coupled light dusty particle. The bulk is immersed in a system of electrons and ions dominated by a Thomas–Fermi density distribution, where interaction and propagation of solitary dust acoustic solitary waves will be analyzed. This investigation will be carried out for arbitrary amplitude of solitary dust acoustic waves. These are originated through the presence of both dust grains in the plasma system. Therefore, it is used a set of fluid equation which determine general expressions for the concentrations of different dust particles. The onset of the Sagdeev method is then employed in order to look for influence of the light component on the quantum system. In this way there is an increasing or decreasing in the Sagdeev potential. This is a quantum effect phenomena.

Solitary wave solutions, lump solutions and interactional solutions to the (2+1)-dimensional potential Kadomstev–Petviashvili equation

In this paper, the [Formula: see text]-solitary wave solution to the (2[Formula: see text]+[Formula: see text]1)-dimensional potential Kadomstev–Petviashvili (PKP) equation is obtained with the Hirota bilinear method. Via the limit technique of long wave, the [Formula: see text]-lump solution can be derived from resulting [Formula: see text]-solitary wave solution. In addition, interactional solutions consisting of lumps and solitary waves for the PKP equation are obtained, which can describe elastic interactions of lumps and solitary waves. These results are illustrated by graphics of several sample examples.

A fully nonlinear and weakly dispersive water wave model for simulating the propagation, interaction, and transformation of solitary waves

This paper presents the development of a theoretical model of fully nonlinear and weakly dispersive (FNWD) waves and numerical techniques for simulating the propagation, interaction, and transformation of solitary waves. Using the standard expansion method and without the limit of small nonlinear parameter defined as the ratio of the wave height versus water depth, a set of model equations describing the FNWD waves in a domain of moderately varying bottom topography are formulated. Exact solitary wave solutions satisfying the FNWD equations are also derived. Numerically, a time-accurate and stabilized finite-element code to solve the governing equations is developed for wave simulations. The solitary wave solutions of FNWD, weakly nonlinear and weakly dispersive (WNWD), and Laplace equations based models in terms of wave profile and phase speed are compared to examine their related features and differences. Investigations on the overtaking collision of two unidirectional solitary waves of different amplitudes, i.e., α1 and α2 where α1 > α2, are carried out using both the FNWD and WNWD water wave models. Selected cases by running the FNWD and WNWD models are performed to identify the critical values of α1/α2 for forming a flattened merging wave peak, which is the condition used to determine if the stronger wave is to pass through the weaker one or both waves are to remain separated during the encountering process. It is interesting to note the critical values of α1/α2 obtained from the FNWD and WNWD models are found to be different and greater than the value of 3 proposed by Wu through the theoretical analysis of the Korteweg-de Vries (KdV) equations. Finally, the phenomena of wave splitting and nonlinear focusing of a solitary wave propagating over a three-dimensional semicircular shoal are simulated. The results obtained from both the FNWD and WNWD models showing the fission process of separating a main solitary wave into multiple waves of decreasing amplitudes are presented, compared, and discussed.

Numerical study of solitary wave interaction with a vegetated platform

For shore protection under extreme wave climates, a simple platform, vegetation stems, and vegetation roots were used to assemble a new type of vegetated platform breakwater. To study the interaction of a solitary wave with it, a 2-D numerical model was conducted using a macroscopic approach based on the OlaFlow solver. The numerical model was well-validated using the experimental data in the literature. The simulated results indicate that the simple platform plays a significant role, and the vegetation plays a supporting role in wave damping. The vegetation roots tend to perform better than stems in reducing wave transmission. However, the roots always contribute to diminishing the wave elevation oscillation and sub-peak in the undulating tails. In addition, the velocity distributions of the green water and underflow above and below the platform are investigated to determine the individual drag coefficients for vegetation stems and roots, respectively. In detail, the effects of the relative incident solitary wave height, relative stem height and root height, vegetation density, and relative platform width on the hydrodynamic coefficients of vegetated platforms are discussed. Finally, empirical equations are proposed for predicting the wave transmission and reflection for a solitary wave propagating over the simple platform and the vegetated platform.

Wave generation through the interaction of a mode-2 internal solitary wave and a broad, isolated ridge

An investigation of a mode-2 internal solitary wave passing over a broad ridge finds that it is gradually adjusted into multiple waves of different form. The leading wave remains intact but is reduced in amplitude and energy. Incident amplitude, speed, and ridge height determine the strength of the resultant waves.

Numerical experiments for long nonlinear internal waves via Gardner equation with dual-power law nonlinearity

This paper studies Gardner equation, which represents long nonlinear internal waves. The collocation method based on B-splines is applied to the equation. The stability of the proposed numerical scheme is analyzed by using von Neumann theory. To observe some physical properties of long nonlinear internal waves, three test problems which contain the propagation of solitary waves, the interaction of solitary waves and evolution of solitons are considered. Also, the effect of nonlinearity on physical problems is investigated. In order to see this effect clearly, the same parameters are used during the computation for different degrees of nonlinearity in each problem.