Solitary Waves Research Papers - Academia.edu (original) (raw)

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability...

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- Mathematics, Applied Mathematics, Physics, Nonlinear Waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter a ¼ a=h 0 and wavelength parameter b ¼ ðh 0 =lÞ 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number s ¼ C=qgh 2 0 , where C is the surface tension coefficient, q is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) b (ð1=3 À sÞ 6 1=3 and (ii) ð1=3 À sÞ ¼ OðbÞ. The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for s ¼ 1=3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.: S 0 0 2 0-7 2 2 5 (0 2) 0 0 1 8 0-5

In this paper, we are using the extended sech function method along with a type of cole-Hopf transformation to obtain the solutions for the nonlinear Korteweg-de Vries (KdV) equation. These types of solutions are represented as the hyperbolic function solutions including the solitary wave solution, shock wave solution and trigonometric function solution when the modulus m approaches to 1 and 0. Mathematica software is used in calculation and graphics.

Two fundamental one-dimensional (1D) models are proposed and applied to simulate the transient flows with the propagation of an interface in a water-filled duct. The proposed models are developed to simulate the unsteady open channel flows based on finite-volume method (FVM). The models presented herein are based on the continuity and momentum equations of free surface and pressurized flows and the momentum equation of an interface between both flows. However, the highly simplified marker and cell (HSMAC) method with pressure iteration procedures is applied to the pressurized flow region. The numerical simulations are performed under the hydraulic conditions of previous experiments, and then simulated results were compared with the experimental data. It is pointed out that the solitary wave solution is able to reproduce the air cavity profile. In contrast to the hydrostatic model, results of the Boussinesq model compare reasonably well to the experimental observations.

- by Hamid Bashiri and +1
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- Finite Volume Methods, Hydraulics, Free surface flows, Pipeline

A solitary wave is generated by impacting a dry chain of beads on one of its ends. Its speed depends on the speed v_0 of the striker and the details of the contact force. The time-of-flight (ToF) of the wave was measured as a function of v0, along with the effect of adding a fluid around the contact points. The ToF displays a complex dependence on the fluid's rheological properties not seen in previous works. A power-law dependence of the ToF on v_0 in both, dry and wet cases was found. It turned out that the Hertz plus viscoelastic interactions are not enough to account for our results. Two phenomenological models providing a unified and accurate account of our results were developed.

- by RAUL LABBE
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- Hydrodynamics (Physics), Solitary Waves, Granular Materials

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. Motivated by their work, we formally derive this equation from two-dimensional potential ¯ow equations governing the small amplitude long capillary-gravity waves on the surface of shallow water for Bond number very close to but less than 1/3. On the basis of far-®eld analyses and heuristic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-®eld. We review various analytical and numerical methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (®fth-order) KdV equation. Using these methods, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-®eld.

- by Ignacio Olivares
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- Engineering, Physics, Hydrodynamics (Physics), Mathematical Sciences

Internal solitary waves have been documented in several parts of the world. This paper intends to look at the effects of the variable topography and rotation on the evolution of the internal waves of depression. Here, the wave is considered to be propagating in a two-layer fluid system with the background topography is assumed to be rapidly and slowly varying. Therefore, the appropriate mathematical model to describe this situation is the variable-coefficient Ostrovsky equation. In particular, the study is interested in the transition of the internal solitary wave of depression when there is a polarity change under the influence of background rotation. The numerical results using the Pseudospectral method show that, over time, the internal solitary wave of elevation transforms into the internal solitary wave of depression as it propagates down a decreasing slope and changes its polarity. However, if the background rotation is considered, the internal solitary waves decompose and form a wave packet and its envelope amplitude decreases slowly due to the decreasing bottom surface. The numerical solutions show that the combination effect of variable topography and rotation when passing through the critical point affected the features and speed of the travelling solitary waves.

The propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed by using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed in Papoutsellis et al (Eur. J. Mech. B/Fluids, Vol. 72, 2018, pp. 199–224). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by nonlinear potential theory are expected to be nearly realistic.
The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20% to 45% and the force from 15% to 38%., depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4 – 1.7 times smaller.

- by Gerassimos A . Athanassoulis and +2
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- Free surface flows, Nonlinear Waves, Water Waves, Solitary Waves
- by Jaya Eddy
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- Civil Engineering, Geology, Coastal Engineering, Water

New experimental measurements of bed shear under solitary waves and solitary bores that
represent tsunamis are presented. The total bed shear stress was measured directly using a
shear cell apparatus. The solitary wave characteristics were measured using ultrasonic
wave gauges and free stream velocities were measured using an Acoustic Doppler
Velocimeter. The measurements were carried out in laminar and transitional flow regimes
(¡«104 < Re < ¡«105). This sort of data is sparsely available in literature. In the absence of
direct measurements, shear stress is indirectly estimated using velocity profiles or is
inferred using standard friction factors. However, this indirect method has its limitations,
e.g., under unsteady hydrodynamic conditions and relatively large roughness the
assumptions of both approaches are no longer valid. More than 168 experimental runs
comprising solitary waves and bores were carried out over a smooth flat bed with wave
height to water depth ratio varying between 0.12 and 0.69. Analytical modeling was
carried out to predict shear stresses using Fourier and convolution integration methods.
This paper presents comparison of the measured and predicted bed shear stress or skin
friction stress, together with estimates of traditional wave friction factors. Overall, the
models can predict the bed shear stress with a satisfactory degree of accuracy.

- by JAYA KUMAR S
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- Coastal Engineering, Tsunami, Solitary Waves, Bed Shear Stress

— Designing safe and economical seashore structures such as breakwaters is of great importance in the engineering of seashore structures. Besides, breakwaters have a wide variety of applications in commercial and recreational harbors, military activities and offshore operations. In the de-signing process of aforementioned structures, wave rose and wave transmission coefficient are important criteria in determining the height of these structures. Smoothed particle hydrodynamics method is also a powerful tool for studying the free-surface and simulation of waves and structures inter-action. Hence, in the present study, the performance of submerged breakwaters against solitary waves was examined. The main objective of this study was to investigate transmission coefficient of solitary waves in fixed submerged breakwaters. To study the standing waves formed in front of a trapezoid submerged breakwater, a two-dimensional numerical model, without Lagrangian mesh approach and using smoothed...

- by Mahmoud Reza Mollaeinia
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- Smoothed Particle Hydrodynamics, Solitary Waves

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

- by Saeed Booshi
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- Solitary Waves, Vertical Permeable Breakwater, Piston Type Wavemaker