Nonlinear Dispersive Wave Model Research Articles

Lump Waves in a Spatial Symmetric Nonlinear Dispersive Wave Model in (2+1)-Dimensions

This paper aims to search for lump waves in a spatial symmetric (2+1)-dimensional dispersive wave model. Through an ansatz on positive quadratic functions, we conduct symbolic computations with Maple to generate lump waves for the proposed nonlinear model. A line of critical points of the lump waves is computed, whose two spatial coordinates travel at constant speeds. The corresponding maximum and minimum values are evaluated in terms of the wave numbers, and interestingly, all those extreme values do not change with time, either. The last section is the conclusion.

Fundamental calculus of the fractional derivative defined with Rabotnov exponential kernel and application to nonlinear dispersive wave model

Before going further with fractional derivative which is constructed by Rabotnov exponential kernel, there exist many questions that are not addressed. In this paper, we try to recapitulate all the fundamental calculus, which we can obtain with this new fractional operator. The problems in this paper are to determine the solutions of the fractional differential equations where the second members are constant functions, polynomial functions, exponential functions, trigonometric functions, or Mittag-Leffler functions. For all the fractional differential equations, the obtained solutions are represented graphically. The Laplace transform of the fractional derivative with Rabotnov exponential kernel is the primary tool in the investigations. Finally, we give the fundamental solution to the nonlinear time-fractional modified Degasperis–Procesi equation by considering the fractional operator with Rabotnov exponential kernel.

Mechanical Solving a Few Fractional Partial Differential Equations and Discussing the Effects of the Fractional Order

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

Numerical simulation of ship-borne waves using a 2DH post-Boussinesq model

This work presents results on the simulation of the generation and propagation of ship-borne waves, using an advanced nonlinear dispersive wave model based on the higher order Boussinesq-type equations. The model includes a single frequency dispersion term, expressed through convolution integrals; it is adapted to represent ship-borne waves by adopting the approach for wave generation by a moving local pressure disturbance, achieved through adding a respective pressure term in its governing equations. The model is tested against an analytical solution for the calculation of the angles of ship wakes and laboratory experiments of waves produced by a high-speed ship in a channel. Results compare well to data from both the aforementioned studies, confirming the model's capabilities and highlighting its accuracy in the representation of ship-borne waves; the model's applicability to practical coastal engineering applications is also investigated, through a set of numerical experiments of waves generated at sea and in a harbour.

Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel

The generalized equal width model is an important non-linear dispersive wave model which is naturally used to describe physical situations in a water channel. In this work, we implement the idea of the interpolation by radial basis function to obtain numerical solution of the non-linear time fractional generalized equal width model defined by Caputo sense. In this technique, firstly, a time discretization is accomplished via the finite difference approach and the non-linear term is linearized by a linearization method. Afterwards, with the help of the radial basis function approximation method is used to discretize the spatial derivative terms. The stability of the method is theoretically discussed using the von Neumann (Fourier series) method. Numerical results and comparisons are presented which illustrate the validity and accuracy of our proposed concepts.

Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel

The generalized equal width model is an important non-linear dispersive wave model which is naturally used to describe physical situations in a water channel. In this work, we implement the idea of the interpolation by radial basis function to obtain numerical solution of the non-linear time fractional generalized equal width model defined by Caputo sense. In this technique, firstly, a time discretization is accomplished via the finite difference approach and the non-linear term is linearized by a linearization method. Afterwards, with the help of the radial basis function approximation method is used to discretize the spatial derivative terms. The stability of the method is theoretically discussed using the von Neumann (Fourier series) method. Numerical results and comparisons are presented which illustrate the validity and accuracy of our proposed concepts.

Dynamics of surface and internal long waves generated by atmospheric pressure disturbances

Different from non-dispersive wave models such as the shallow water equations, MCC (Miyata, Choi, and Camassa) type strongly nonlinear dispersive wave model can describe not only the dynamics of surface waves, but also those of internal waves. The MCC model for a two-layer fluid system is applied to study dynamics of both surface waves and internal waves. The resonance mechanisms with atmospheric pressure disturbances are numerically studied not only for surface waves, so-called Proudman resonance, but also for internal waves. Baroclinic internal waves are generated when the speed of the atmospheric pressure disturbance is comparable to the critical linear internal wave speed. We compare the amplitudes of generated internal waves to the resonance theory for surface waves. Although most baroclinic internal wave phenomena can be explained by this linear theory, the relative magnitudes of amplitudes for forced and free waves do not agree with the theory. In addition, sequentially generated surface waves with resonance are observed when the radius of the atmospheric pressure disturbance is small. Finally, the dynamics of both surface and internal waves over non-uniform bottom topography are discussed in terms of Froude numbers to understand how they propagate and interact with each other when the Proudman resonance condition is satisfied while they travel.

A parametric study of tsunami fission generation in a bay geometry based on numerical simulation

ABSTRACTTsunami fission generates short waves at the front face of tsunami and these short waves may expand the inundation area, increase the hydrodynamic forces exerted on structures, and amplify the damage to coastal communities. It is therefore important to understand to what extent the tsunami fission generates such short wave components under variable conditions for development of better tsunami disaster mitigation strategies and plannings. This study carried out numerical experiments to describe under which conditions the tsunami fission occurs in a bay. The experiments indicated that the amplification of the nearshore tsunami height induced by the narrowing bay geometry and the bay-scale resonance enhance the tsunami fission. Based on the computations, a parametric diagram for the occurrence of tsunami fission was proposed in terms of the tsunami properties, the effect of bay geometry and the bay resonance. The developed diagram was applied to Sendai, Ryori, Yoshihama, and Toni Bays, which were affected by the 2011 Tohoku Tsunami. The proposed diagram consistently explained the occurrence of tsunami fission in Sendai Bay and in Ryori Bay as observed in the video footage. The diagram also suggested that the tsunami fission would have occurred in Yoshihama Bay. A high-resolution numerical computation was performed to simulate the tsunami deformation in the Yoshihama Bay and Toni Bay based on a nonlinear dispersive wave model. The result supported the validity of the diagram in describing the generation of tsunami fission in Yoshihama Bay.

Study on Locally Varying Inundation Characteristics Induced by Super Typhoon Haiyan. Part 2: Deformation of Storm Waves on the Beach with Fringing Reef Along the East Coast of Eastern Samar

This paper investigates characteristics of coastal damage caused by Super Typhoon Haiyan with special focuses on locally varying wave deformations along the coast of Eastern Samar fronted by shallow fringing reef. The paper first summarized the field survey results and indicated that the erosion depth of the beach had negative correlation with the width of fringing reef along the straight coastline while a drastic variation of the damage levels was observed around Matarinao Bay mouth. A phase-resolved nonlinear dispersive wave model and a linear mild slope equation model were respectively applied for computations of wave deformations on the long-straight coastline with fringing reef and on the coast around Matarinao Bay. Computed rapid decay of waves on the reef reasonably explained observed negative correlations between erosion depth and the reef width. Around Matarinao Bay, on the other hand, wave refraction and diffraction played a significant role to explain wave concentration on the reef just outside of the bay mouth and also significant wave attenuation inside the bay mouth. Observed wave deformation characteristics were not sensitive to incident wave angles and periods and were reasonably consistent with the observed alongshore variation of maximum water levels and corresponding damages along the coast.

Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model

The aim of this work is the quantification and prediction of rare events characterized by extreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional nonlinear model with deep-water waves dispersion relation, the Majda–McLaughlin–Tabak (MMT) model, in a dynamical regime that is characterized by a broadband spectrum and strong nonlinear energy transfers during the development of intermittent events with finite-lifetime. To understand the energy transfers that occur during the development of an extreme event we perform a spatially localized analysis of the energy distribution along different wavenumbers by means of the Gabor transform. A statistical analysis of the Gabor coefficients reveals (i) the low-dimensionality of the intermittent structures, (ii) the interplay between non-Gaussian statistical properties and nonlinear energy transfers between modes, as well as (iii) the critical scales (or critical Gabor coefficients) where a critical amount of energy can trigger the formation of an extreme event. We analyze the unstable character of these special localized modes directly through the system equation and show that these intermittent events are due to the interplay of the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wave breaking. These localized instabilities are triggered by random localizations of energy in space, created by the dispersive propagation of low-amplitude waves with random phase. Based on these properties, we design low-dimensional functionals of these Gabor coefficients that allow for the prediction of the extreme event well before the nonlinear interactions begin to occur.

Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.

Numerical simulations of the Boussinesq equation by He's variational iteration method

In this paper, we present a reliable algorithm to study the well-known model of nonlinear dispersive waves proposed by Boussinesq. We solve the Cauchy problem of Boussinesq equation using variational iteration method (VIM). The numerical results of this method are compared with the exact solution of an artificial model to show the efficiency of the method. The approximate solutions show that VIM is a powerful mathematical tool for solving nonlinear problems.

Four-wave resonant interactions in the classical quadratic Boussinesq equations

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

Finite element solution of the Boussinesq wave equation

In this work, we investigate a Boussinesq-type flow model for nonlinear dispersive waves by developing a computational model based on the finite element discretisation technique. Hermite interpolation functions were used to interpolate approximation elements. The system is modeled using a time dependent equation. Solution to the model is obtained, through a combination of two different schemes namely: a time approximation scheme (the Newmark Method) and the eigenvalue finite element method. Using this schemes, discrete solutions of the model at different time steps, were obtained. Graphical illustrations of solutions for the transient displacements at the center and right end of the rod are presented. The results obtained are very accurate and the model efficient. JONAMP Vol. 11 2007: pp. 223-238

Boundary-fitted numerical model for computing linear and nonlinear wave forces on bottom-mounted piles

A vertically-integrated nonlinear dispersive wave model in nonorthogonal curvilinear coordinate system is used for simulating linear and nonlinear wave forces on bottom mounted circular and elliptical piles. The wave model is in terms of the contravariant velocities hence the wall condition for arbitrarily shaped objects is satisfied accurately and easily. As basic test cases, computations of linear wave forces on vertical cylinders of circular and elliptical cross sections are compared with the available analytical results. Cnoidal waves incident on circular and elliptical cylinders are also simulated for different wave steepness ratios and wave shape parameter values to observe the effect of nonlinearity and wave form. Comparisons of linear and nonlinear force coefficients and relevant discussion are presented in closing.

Traveling waves in compressible elastic rods

We determine all weak traveling wave solutions of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods. Besides the previously known smooth, peaked, and cusped solutions, the equation is shown to admit compactons, stumpons, and fractal-like waves.

Boundary‐fitted non‐linear dispersive wave model for regions of arbitrary geometry

AbstractA vertically integrated non‐linear dispersive wave model is expressed in non‐orthogonal curvilinear co‐ordinate system for simulating shallow or deep water wave motions in regions of arbitrary geometry. Both dependent and independent variables are transformed so that an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Thus, the wall condition for enclosures surrounding a typical physical domain, such as a channel, port or harbor, is satisfied accurately and easily. The numerical scheme is based on staggered grid finite‐difference approximations, which result in implicit formulations for the momentum equations and semi‐explicit formulation for the continuity equation. Test cases of linear wave propagation in converging, diverging and circular channels are performed to check the reliability of model simulations against the analytical solutions. Cnoidal waves of different steepness values in a circular channel are also considered as examples to non‐linear wave propagation within curved walls. In closing, remarks concerning versatility and practical uses of the numerical model are made. Copyright © 2004 John Wiley & Sons, Ltd.

Improvement of Parabolic Nonlinear Dispersive Wave Model

Improvements to a previously published nonlinear parabolic wave model are developed and implemented. A second-order correction to a free-surface boundary condition used to develop the original model is formulated. The correction takes into account the complete second-order transformation between amplitudes of the velocity potential and those of the free-surface elevation. Additionally, wide-angle propagation terms are included in the model. It is shown that the model with the second-order correction retains the properties of third-order Stokes theory quite well in deep water. Comparisons of model behavior to data reveal that both nonlinearity and wide-angle propagation effects need to be included in the model for general wave transformation problems in shallow water. Skewness predictions are considerably improved by using both the second-order correction and by retaining a greater number of frequency components in the calculation. Asymmetry calculations are aided by incorporation of frequency-squared weighting for distribution of the dissipation function. Further improvement may entail a different form of the breaking model.

Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method

In this paper, we present a reliable algorithm to study the known model of nonlinear dispersive waves proposed by Boussinesq. The modified algorithm of Adomian decomposition method is used with an emphasis on the single soliton solution. New exact periodic solutions and polynomial solutions are obtained. The results of numerical examples are presented and only few terms are required to obtain accurate solutions.

Stability of a class of solitary waves in compressible elastic rods

We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.