Single Solitary Wave Research Articles

Invariant preserving schemes for double dispersion equations

We propose and study two finite difference schemes (FDSs) for the double dispersion equations. The first FDS is symplectic, while the second one preserves the discrete momentum exactly. Both FDS conserve the discrete energy approximately with O(h^{2}+tau ^{2}) global error. The extensive numerical experiments agree well with the theoretical results for single solitary wave as well as for the interaction between two solitary waves.

Simulation on propagation characteristics of solitary waves in a one-dimensional charged granular chain

Solitary waves propagating in a one dimensional charged granular chain has been investigated numerically. Electrically long-range interaction is introduced into such a nonlinear system for the first time. The propagating characteristics of the waves due to physical parameters of materials are studied systematically. It is found that both single solitary wave and multi-solitary wave can be excited in the granular chain. Numerical results show that the amplitude of the solitary wave decays exponentially because of damping as time increases even if the Coulomb’s force is taken into account. Specifically, Young’s modulus and charge quantity have slightly effect on the amplitude attenuation rate. However, the damping coefficient and the density of particles have significant effect.

A numerical simulation of the generation and evolution of nonlinear internal waves across the Kara Strait

Nonlinear internal waves (NIWs) are ubiquitous around the Kara Sea, a part of the Arctic Ocean that is north of Siberia. Three hot spot sources for internal waves, one of which is the Kara Strait, have been identified based on Envisat ASAR. The generation and evolution of the NIWs through the interactions of the tide and topography across the strait is studied based on a nonhydrostatic numerical model. The model captures most wave characteristics shown by satellite data. A typical inter-packets distance on the Barents Sea side is about 25 km in summer, with a phase speed about 0.65 m/s. A northward background current may intensify the accumulation of energy during generation, but it has little influence on the other properties of the generated waves. The single internal solitary wave (ISW) structure is a special phenomenon that follows major wave trains, with a distance about 5–8 km. This wave is generated with the leading wave packets during the same tidal period. When a steady current toward the Kara Sea is included, the basic generation process is similar, but the waves toward the Kara Sea weaken and display an internal bore-like structure with smaller amplitude than in the control experiment. In winter, due to the growth of sea ice, stratification across the Kara Strait is mainly determined by the salinity, with an almost uniform temperature close to freezing. A pycnocline deepens near the middle of the water depth (Barents Sea side), and the NIWs process is not as important as the NIWs process in summer. There is no fission process during the simulation.

Numerical solution of regularised long ocean waves using periodised scaling functions

In this paper, a numerical technique for solving the regularised long wave equation (RLW) is presented using a wavelet Galerkin (WG) method in space and a fourth-order Runge–Kutta (RK) technique in time. We study the convergence analysis of the obtained numerical solutions and investigate the results for the motions of double and single solitary waves, undular bores and conservation properties of mass, energy and momentum in order to verify the applicability and performance of the proposed method. Simulation results are further compared with the known analytical solutions and some previous published numerical results. It is concluded that the present method remarkably improves the accuracy of the Galerkin-based methods for numerically solving a large class of nonlinear and weakly dispersive ocean waves.

An Efficient Approximation to Numerical Solutions for the Kawahara Equation Via Modified Cubic B-Spline Differential Quadrature Method

The main purpose of this work is to obtain the numerical solutions for the Kawahara equation via the Crank–Nicolson–Differential Quadrature Method based on modified cubic B-splines (MCBC-DQM). First, the Kawahara equation has been discretized using Crank–Nicolson scheme. Then, Rubin and Graves linearization technique has been utilized and differential quadrature method has been applied to obtain algebraic equation system. Four different test problems, namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, and wave generation, have been solved. Next, to be able to test the efficiency and accuracy of the newly applied method, the error norms $$ L _{2}$$ and $$ L _{\infty }$$ as well as the three lowest invariants $$ I _{1}$$ , $$ I _{2}$$ , and $$ I _{3}$$ have been computed. Besides those, the relative changes of invariants have been reported. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. The comparison of present results with earlier works showed that the newly method may be provide significant benefit in case of numerical solutions of the other nonlinear differential equations.

Galerkin finite element solution for Benjamin–Bona–Mahony–Burgers equation with cubic B-splines

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic B- spline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L2, L∞ and three invariants I1,I2 and I3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation.

A Quartic Subdomain Finite Element Method for the Modified KdV Equation

In this article, we have obtained numerical solutions of the modified Kortewegde Vries (MKdV) equation by a numerical technique attributed on subdomain finite element method using quartic B-splines. The proposed numerical algorithm is controlled by applying three test problems including single solitary wave, interaction of two and three solitary waves. To inspect the performance of the newly applied method, the error norms, L2 and L1, as well as the four lowest invariants, I1,I2; I3 and I4, have been computed. Linear stability analysis of the algorithm is also examined.

Numerical solutions of the generalized Rosenau–Kawahara-RLW equation arising in fluid mechanics via B-spline collocation method

Present study reports the solution of generalized Rosenau–Kawahara-RLW equation. It includes motion of single solitary wave, interaction of two solitary waves along with the calculated invariants and error norms. Gaussian and undular bore initial conditions are studied to show evolution of solitons. Developed train of solitons and conservation of invariants are shown via figures and tables in the respective sections. Various case studies are presented to demonstrate the efficiency of the proposed numerical scheme. Solutions so produced may be helpful for explaining various nonlinear physical phenomena in nonlinear dynamical systems.

A collocation algorithm based on quintic B-splines for the solitary wave simulation of the GRLW equation

In this article, a collocation algorithm based on quintic B-splines is proposed for the numerical solution of the non-linear generalized regularized long wave (GRLW) equation. Also, to analyse the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach is discussed on three test examples consisting of a single solitary wave, the collision of two solitary waves and the growth of an undular bore. The accuracy of the method is demonstrated by calculating the error in the L2 and L¥ norms and the conservative quantities I1, I2 and I3. The findings are compared with those of previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to different parameters.

Charge pumping in nanotube filled with electrolyte

Charge transport in nanotube filled with an incompressible viscous electrolyte is considered. The flow is induced by elastic soliton wave in the nanotube wall. The soliton appears as an axisymmetric local deformation of the wall boundary propagating along the length of a nanotube. The Stokes approximation is used for the flow. As for the charge transport, different factors (identical charges repulsion, diffusion, convection) are taken into account. Picture of the streamlines and charge distribution are described. We estimate a value of charge transported by single solitary wave.

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some of the published papers. As a conclusion, it can be said that the present approach is an effective and efficient one for solving the cmKdV equation and can also be used for numerical solutions of other problems.

Interaction of highly nonlinear solitary waves with rigid polyurethane foams

Experiments and simulations are performed to study the interaction of solitary waves in a granular crystal with a plastically compressible and rate-sensitive test medium in the form of rigid polyurethane (PU) foam. We study the effects of the foam's compressive strength and elastic modulus on the nonlinear wave dynamics near the interface and on the formation of reflected solitary waves. Experiments are conducted using a granular crystal composed of a vertical array of spherical steel particles, directly contacting the foamed polyurethane samples. Single solitary waves are generated in the chain through striker particle impact, and the temporal profiles of the incident and reflected solitary waves are recorded by placing an instrumented sensor particle in the chain. The measurements show that travel time and wave amplitude of the reflected solitary waves are highly sensitive to the elasto-plastic properties of the inspection medium, particularly when the intensity of the incident solitary waves is increased. The measurements were compared to the predictions obtained with a coupled discrete/finite element model and good agreement was reported. The predictions show that when the amplitude of the incident solitary wave increases, the response transitions from an elastic regime to a regime dominated by plastic compression.

Modeling of emerged semi-circular breakwater performance against solitary waves using SPH method

The semicircular breakwater (SBW) is a composite breakwater consisting of a hollow empty reinforced concrete caisson seated on the rubble mound breakwater. In this research, smoothed particle hydrodynamics (SPH) method was used to examine the run-up and overtopping of solitary waves on SBW. At first the SPH model was validated for regular wave–SBW interaction against experimental results. Comparisons showed that the present modified numerical model has a good agreement with experimental data. Parametric study was then performed on the effect of water depth and wave height of the single solitary waves and height of the rubble mound on wave run-up. Furthermore, double solitary waves were generated to study their interaction with SBW by the model. Run-up and reflection coefficient for double solitary wave–structure interaction was then investigated. The results show that in sum run-up in double solitary wave mode is smaller than that of single solitary wave. Moreover, the second solitary wave run-up is smaller than that of the first solitary wave in double solitary wave train. Overtopping discharge of double solitary wave is nearly constant for τ/T greater than 1.3, where τ/T is the ratio of time distance between wave crests of double wave to effective period of single solitary wave. Also overtopping discharge decreases with increasing in τ/T for its value smaller than 1.3. The results show that this model is a useful tool for simulating the interaction of solitary waves–structures with complicated geometry.

Solving Modified Regularized Long Wave Equation Using Collocation Method

In this paper, we suggest collocation method depending on Cubic trigonometric B-spline (CuTBS) approach based on finite difference scheme to solve the modified regularized long wave equation. The single solitary wave motion was studied using the proposed method; thus the accuracy and efficiency of the suggested method were computed from the L2, L∞ norms. Also, the von-Neumann method was used to study the linear stability analysis. The obtained results through the tested two problems exhibited that, the method is an effective numerical scheme to solve Modified Regularized Long Wave equation (MRLW).

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

AbstractThe large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.

A numerical technique based on collocation method for solving modified Kawahara equation

In this article, a numerical solution of the modified Kawahara equation is presented by septic B-spline collocation method. Applying the von-Neumann stability analysis, the present method is shown to be unconditionally stable. L2 and L∞ error norms and conserved quantities are given at selected times. The accuracy of the proposed method is checked by test problems including motion of the single solitary wave, interaction of solitary waves and evolution of solitons.

Solitary Wave Solutions and Rational Solutions for Modified Zakharov-Kuznetsov Equation with Initial Value Problem

The modified Zakharov-Kuznetsov equation with the initial value problem is studied numerically by means of homotopy perturbation method. The analytical approximate solutions of the modified Zakharov-Kuznetsov equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.

Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method

The mKdV equation with the initial value problem is studied numerically by means of the homotopy perturbation method. The analytical approximate solutions of the mKdV equation are obtained. Choosing the form of the initial value, the single solitary wave, two solitary waves and rational solutions are presented, some of which are shown by the plots.

OPERATOR SPLITTING FOR NUMERICAL SOLUTIONS OF THE RLW EQUATION

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.