Small-amplitude Gravity Waves Research Articles

Sensitivity and wave propagation analysis of the time-fractional (3+1)-dimensional shallow water waves model

AbstractThis study investigates the exact solutions of the time-fractional (3+1)-dimensional combined Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM) equation. The considered model describes the long surface gravity waves of small amplitude, which portrays the two-way propagation of waves. The modified generalized Kudryashov method and the exp(-$$\varphi (\xi $$ φ ( ξ ))-expansion methods are employed to resolve the aforementioned issue because of their effectiveness and simplicity. For generality, the time fractional version is studied; more advanced solutions that do not exist in the literature were obtained. As a result, a variety of the exact wave solutions of the conformable (3+1)-dimensional KdV-BBM equation are obtained. The dynamical behaviors of some obtained solutions are represented with the proper parameter values. The used methods yield noteworthy results in obtaining the analytical solutions of fractional differential equations under various conditions. Besides, the sensitivity of regarding dynamical system is assessed to show the numerical stability effects.

Lax pair, Darboux transformation, Weierstrass–Jacobi elliptic and generalized breathers along with soliton solutions for Benjamin–Bona–Mahony equation

This paper studies the Lax pair (LP) of the [Formula: see text]-dimensional Benjamin–Bona–Mahony equation (BBBE). Based on the LP, initial solution and Darboux transformation (DT), the analytic one-soliton solution will also be obtained for BBBE. This paper contains a bunch of soliton solutions together with bright, dark, periodic, kink, rational, Weierstrass elliptic and Jacobi elliptic solutions for governing model through the newly developed sub-ODE method. The BBBE has a wide range of applications in modeling long surface gravity waves of small amplitude. In addition, we will evaluate generalized breathers, Akhmediev breathers and standard rogue wave solutions for stated model via appropriate ansatz schemes.

A note on the control and stabilization of a higher-order water wave model

<p style='text-indent:20px;'>The paper deals with the internal controllability and stabilizability of a family of Boussinesq systems introduced by J. L. Bona, M. Chen and J.-C. Saut to describe two-way propagation of small amplitude gravity waves on the surface of water in a canal. By applying the moment method, we first obtain the exact controllability of the linearized system in suitable Hilbert spaces, which implies its exponential stabilization by suitable feedback. Then, by using a contraction mapping principle, we establish the local exact controllability and exponential stabilization for the original nonlinear system.</p>

Rapid exponential stabilization of a Boussinesq system of KdV–KdV Type

This paper studies the exponential stabilization of a Boussinesq system describing the two-way propagation of small amplitude gravity waves on the surface of an ideal fluid, the so-called Boussinesq system of the Korteweg–de Vries type. We use a Gramian-based method introduced by Urquiza to design our feedback control. By means of spectral analysis and Fourier expansion, we show that the solutions of the linearized system decay uniformly to zero when the feedback control is applied. The decay rate can be chosen as large as we want. The main novelty of our work is that we can exponentially stabilize this system of two coupled equations using only one scalar input.

Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current

This paper describes a numerical investigation of ripples generated on the front face of deep-water gravity waves progressing on a vertically sheared current with the linearly changing horizontal velocity distribution, namely parasitic capillary waves with a linear shear current. A method of fully nonlinear computation using conformal mapping of the flow domain onto the lower half of a complex plane enables us to obtain highly accurate solutions for this phenomenon with the wide range of parameters. Numerical examples demonstrated that, in the presence of a linear shear current, the curvature of surface of underlying gravity waves depends on the shear strength, the wave energy can be transferred from gravity waves to capillary waves and parasitic capillary waves can be generated even if the wave amplitude is very small. In addition, it is shown that an approximate model valid for small-amplitude gravity waves in a linear shear current can reasonably well reproduce the generation of parasitic capillary waves.

Stabilization of a Boussinesq system with localized damping

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen, and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damping term acts in one equation only. By means of spectral analysis and eigenvectors expansion of solutions, we prove that the energy associated to the model converges to zero as time tends to infinity. Also, we address the problem of unique continuation property for the corresponding conservative system.

A novel approach for stochastic solutions of wick-type stochastic time-fractional Benjamin–Bona–Mahony equation for modeling long surface gravity waves of small amplitude

Objectives: In the paper, two new reliable analytical methods have been devised for getting new exact analytical solutions of wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation. Moreover, the Hermite transform andinverse Hermite transform have been utilized for converting fractional stochastic differential equation to deterministic fractional partial differential equation and vice versa respectively. Here for reducing fractional partial differential equations (FPDE) to the ordinary differential equation (ODE), fractional complex transform has been utilized.Methods: The authors have used a newly proposed method and Kudryshov method for getting the solutions for wick-type stochastic time-fractional Benjamin-Bona-Mahony (BBM) equation.Results: By using two reliable methods, here, the authors find the new exact solutions for the governing equations.Conclusion: Two new approaches to find solutions of the aforementioned equation have been established. Also, the new exact solutions have been obtained for stochastic differential equation by using two methods.

On the analytical and numerical solutions of the Benjamin–Bona–Mahony equation

In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin–Bona–Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin–Bona–Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin–Bona–Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier–Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors $$L_{2}$$ and $$L_{\infty }$$ . We use the finite difference method in obtaining the numerical solutions of the Benjamin–Bona–Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9.

Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media

This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.

The new types of wave solutions of the Burger's equation and the Benjamin–Bona–Mahony equation

In this article, we suggest the two variable (G′/G, 1/G)-expansion method for extracting further general closed form wave solutions of two important nonlinear evolution equations (NLEEs) that model one-dimensional internal waves in deep water and the long surface gravity waves of small amplitude propagating uni-directionally. The method can be regarded as an extension of the(G′/G)-expansion method. The ansatz of this extension method to obtain the solution is based on homogeneous balance between the highest order dispersion terms and nonlinearity which is similar to the (G′/G) method whereas the auxiliary linear ordinary differential equation (LODE) and polynomial solution differs. We applied this method to find explicit form solutions to the Burger's and Benjamin–Bona–Mahony (BBM) equations to examine the effectiveness of the method and tested through mathematical computational software Maple. Some new exact travelling wave solutions in more general form of these two nonlinear equations are derived by this extended method. The method introduced here appears to be easier and faster comparatively by means of symbolic computation system.

Stabilization of a Boussinesq system with generalized damping

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin–Bona–Mahony with periodic boundary conditions. We study the stability properties of the resulting system when generalized damping operators are introduced in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, we show that the same property holds for the nonlinear system.

Blow-Up of Solution to Cauchy Problem for the Singularly Perturbed Sixth-Order Boussinesq-Type Equation

We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.

Numerical modeling study of the momentum deposition of small amplitude gravity waves in the thermosphere

Abstract. We study the momentum deposition in the thermosphere from the dissipation of small amplitude gravity waves (GWs) within a wave packet using a fully nonlinear two-dimensional compressible numerical model. The model solves the nonlinear propagation and dissipation of a GW packet from the stratosphere into the thermosphere with realistic molecular viscosity and thermal diffusivity for various Prandtl numbers. The numerical simulations are performed for GW packets with initial vertical wavelengths (λz) ranging from 5 to 50 km. We show that λz decreases in time as a GW packet dissipates in the thermosphere, in agreement with the ray trace results of Vadas and Fritts (2005) (VF05). We also find good agreement for the peak height of the momentum flux (zdiss) between our simulations and VF05 for GWs with initial λz ≤ 2π H in an isothermal, windless background, where H is the density scale height. We also confirm that zdiss increases with increasing Prandtl number. We include eddy diffusion in the model, and find that the momentum deposition occurs at lower altitudes and has two separate peaks for GW packets with small initial λz. We also simulate GW packets in a non-isothermal atmosphere. The net λz profile is a competition between its decrease from viscosity and its increase from the increasing background temperature. We find that the wave packet disperses more in the non-isothermal atmosphere, and causes changes to the momentum flux and λz spectra at both early and late times for GW packets with initial λz ≥ 10 km. These effects are caused by the increase in T in the thermosphere, and the decrease in T near the mesopause.

On the Particle Paths and the Stagnation Points in Small-Amplitude Deep-Water Waves

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.

Dispersion relation for water waves with non-constant vorticity

We derive the dispersion relation for linearized small-amplitude gravity waves for various choices of non-constant vorticity. To the best of our knowledge, this relation is only known explicitly in the case of constant vorticity. We provide a wide range of examples including polynomial, exponential, trigonometric and hyperbolic vorticity functions.

Control and stabilization of a family of Boussinesq systems

This paper studies the internal controllability and stabilizabilityof a family of Boussinesq systems recently proposed by J. L. Bona,M. Chen and J.-C. Saut to describe the two-way propagation of smallamplitude gravity waves on the surface of water in a canal. Thespace of the controllable data for the associated linear system isdetermined for all values of the four parameters. As an applicationof this newly established exact controllability, some simplefeedback controls are constructed such that the resultingclosed-loop systems are exponentially stable. When the parametersare all different from zero, the local exact controllability andstabilizability of the nonlinear system are also established.

Stabilization of a Boussinesq system of KdV–KdV type

A family of Boussinesq systems has recently been proposed by Bona, Chen, and Saut in [J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci. 12 (4) (2002) 283–318] to describe the two-way propagation of small-amplitude gravity waves on the surface of water in a canal. In this paper, we investigate the boundary stabilization of the Boussinesq system of KdV–KdV type posed on a bounded domain. We design a two-parameter family of feedback laws for which the solutions issuing from small data are globally defined and exponentially decreasing in the energy space.

Linear water waves with vorticity: Rotational features and particle paths

We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses.

Observations and numerical simulations of mountain waves in the presence of directional wind shear

Three research aircraft observed small-amplitude gravity waves over the south-western French Alps on 13 November 1999 during the Mesoscale Alpine Programme (MAP). Radiosonde ascents and Global Positioning System dropsondes deployed by MAP research aircraft indicate that the upstream flow was characterized by topographic blocking below 2 km and low-level directional wind shear associated with a synoptic-scale depression that extended above the Alpine crest. The in situ vertical velocity data from the three research aircraft, and backscatter from lidar onboard the low-level and upper-level aircraft, exhibit gravity-wave signatures in the lower troposphere that are characterized by relatively short horizontal wavelengths on a scale of 5 km or less. The in situ flight-level and backscatter data also suggest a rapid decrease in the wave amplitude with height, due to a directional critical layer that partially absorbs wave energy in the lower to middle troposphere in concert with a decrease in the Scorer parameter with height that traps the wave energy. Numerical simulation results, obtained from the non-hydrostatic Coupled Ocean–Atmosphere Mesoscale Prediction System model using a horizontal resolution of 556 m, indicate the presence of low-level wave breaking above the highest peaks, and confirm the presence of trapped waves and a directional critical level. Results based on linear theory and nonlinear model simulations suggest that the absorption of wave energy associated with the directional critical layer has an important impact on the wave characteristics including the momentum flux. Copyright © 2006 Royal Meteorological Society

On the reinitialization procedure in a narrow-band locally refined level set method for interfacial flows

A local level set algorithm for simulating interfacial flows described by the two-dimensional incompressible Navier–Stokes equations is presented. The governing equations are solved using a finite-difference discretization on a Cartesian grid and a second-order approximate projection method. The level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid, which is reconstructed every time step and refined using a simple uniform cell-splitting operation within the band. Instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band. The influence of different PDE-based reinitialization strategies on the accuracy of the results is investigated. The ability of the proposed method to accurately compute interfacial flows is discussed using different tests, namely the advection of a circle of fluid in two different time-reversed vortex flows, the advection of Zalesak's rotating disk, the propagation of small-amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water, and a classical test of Rayleigh–Taylor instability with and without surface tension effects. The interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method. Copyright © 2005 John Wiley & Sons, Ltd.