Scattering of regularized-long-wave solitary waves

doi:10.1016/0167-2789(84)90014-9

Physica D: Nonlinear Phenomena

Volume 11, Issue 3, June 1984, Pages 324-336

https://doi.org/10.1016/0167-2789(84)90014-9 Get rights and content

Abstract

The Lagrangian density for the regularized-long-wave equation (also known as the BBM equation) is presented. Using the trial function technique, ordinary differential equations that describe the time dependence of the position of the peaks, amplitudes, and widths for the collision of two solitary waves are obtained. These equations are analyzed in the Born and “equal-width” approximations and compared with numerical results obtained by direct integration utilizing the split-step fast Fourier-transform method. The computations show that collisions are inelastic and that production of solitary waves may occur.

References (14)

J. Courtenay Lewis et al.
Phys. Lett.
(1979)
D.H. Peregrine
J. Fluid Mech.
(1966)
V. Petviashvili
Sov. J. Plasma Phys.
(1977)
J.D. Meiss et al.
Phys. Fluids
(1982)
T. Benjamin et al.
Phil. Trans. R. Soc. London
(1972)
M.D. Kruskal
K. Abdullov et al.
Phys. Lett.
(1976)
J.L. Bona et al.
Phys. Fluids
(1980)
P.J. Olver
Math. Proc. Camb. Phil. Soc.
(1979)

There are more references available in the full text version of this article.

Cited by (125)

Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics
2023, Alexandria Engineering Journal
In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35], [36], [37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32], [34], [35], [36], [37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality $α$ are illustrated by Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7. The fractional parameter $α$ masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number $α$ in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7. (4) The effect about free parameters $c, k$ can be seen from Fig. 8, Fig. 9, Fig. 10, Fig. 11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34], [35], [36], [37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Fig. 12, Fig. 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.
Novel analytical technique to find diversity of solitary wave solutions for Wazwaz-Benjamin-Bona Mahony equations of fractional order
2023, Results in Physics
The newly modified Benjamin-Bona-Mahony equations in three dimensions are examined in the current work through the use of conformable fractional derivatives to incorporate spatial and temporal fractional order derivatives. With the aid of the Wolfram Mathematica software, a variety of solutions, including hyperbolic and periodic function solutions, are created. The information was gathered to evaluate how well the proposed novel (G’/G²)-expansion method could compute exact solutions of the WBBM equations that could be applied to the implementation of the nonlinear water model in the ocean and coastal engineering. Table 1 shows a comparison between the recently discovered solutions and those that have previously been reported in the literature. There are some visual representations of the solutions.
Galilei-invariant and energy-preserving extensions of Benjamin–Bona–Mahony-type equations
2023, Partial Differential Equations in Applied Mathematics
The classical Benjamin–Bona–Mahony equation (BBM equation) models unidirectional propagation of long gravity surface waves of small amplitude. Unlike many other water wave models, it lacks the Galilean invariance, which is an essential property of physical systems. It is shown that by an addition of a higher asymptotic order nonlinear term, this deficiency can be corrected, giving rise to a new Galilei invariant Benjamin–Bona–Mahony equation (iBBM equation). Moreover, further additional higher-order terms can be chosen in a way that the augmented model preserves the energy conservation property along with Hamiltonian and Lagrangian structures. The resulting equation is referred to as energy-preserving Benjamin–Bona–Mahony equation (eBBM).
It is shown that both the classical BBM equation and the energy-preserving eBBM equations belong to a one-parameter ( $α$ ) family that shares essentially the same local and nonlocal symmetries, conservation laws, Hamiltonian, and Lagrangian structures, with the BBM and eBBM equations corresponding to parameter values $α = 0$ and $α = 1$ , respectively. Symmetry and conservation law classifications reveal a special case $α = 1 / 3$ , which is shown to correspond to a rescaled version of the celebrated integrable Camassa–Holm (CH) equation. Local symmetries and conservation laws are computed, and numerical solution behaviour is compared for the three BBM-type modes and the CH-equivalent eBBM $_{1 / 3}$ model.
Soliton solutions and fractional effects to the time-fractional modified equal width equation
2022, Alexandria Engineering Journal
The one-dimensional long water wave propagation in a nonlinear medium, including the dispersion process, is well simulated by the fractional-order modified equal-width (MEW) equation. This article establishes several recognized, standard, inclusive, and scores of typical exact wave solutions to the MEW equation using the double $(G^{'} / G, 1 / G)$ -expansion method. For specific parameter values, kink, periodic, periodic-singular, singular-kink, and other forms of solitons can be recovered from general solutions. The effect of the fractional parameter on wave forms has also been analyzed by depicting several graphs for different values of the fractional-order $α$ . In order to illustrate the potential characteristics, three- and two-dimensional combined plots using Maple have been drawn. It has been established that the introduced approach is a potential tool for extracting new exact solutions to various nonlinear evolution equations (NLEEs) arising in engineering, science, and applied mathematics.
New dynamical soliton propagation of fractional type couple modified equal-width and Boussinesq equations
2022, Alexandria Engineering Journal
The space–time fractional coupled modified equal-width equation and the coupled Boussinesq equation are a category of fractional partial differential equations, which might be crucial mathematical feathers in nonlinear optics, solid-state physics, vibrations in the nonlinear string, ion sound waves in plasma, hydro-magnetic waves in cold plasma and many more. To assemble such new exact solutions of the mentioned equations, the Sine-Gordon expansion (SGE) technique has been proposed with inside the sense of conformable derivative and the fractional order partial differential equation that is capable to change into an ordinary differential equation by using the traveling wave transform. In this article, the SGE technique has been employed to search the higher-dimensional fractional nonlinear evolution equations and hooked up consistent soliton solutions to the faster thought fractional nonlinear evolution equations through installing use of the prolonged higher-dimensional SGE technique. The compatibility of the extended SGE technique confirms through the scoring of soliton solutions. Moreover, we explored a couple of varieties of solutions over the maple calculations, including soliton, kink types, bell types, single soliton type, dark soliton, singular kink type, and anti-bell type solutions for distinct values of constants, which have been illustrated by the usage of 3D, list-point, contour analysis, and vector plotting. It is far incredible to understand that the feature of the solutions relies upon the selection of the parameters from the figures. This takes a look at an impactful position in studying higher-dimensional fractional nonlinear evolution equations through the prolonged SGE technique.
The space–time generalized finite difference scheme for solving the nonlinear equal-width equation in the long-time simulation
2022, Applied Mathematics Letters
In this research, the space–time (ST) coupled generalized finite difference method (GFDM) was extended to cooperate with the Newton–Raphson and time-marching methods for stably solving the nonlinear equal-width equation in long-time simulation. The equal-width equation is a time-dependent nonlinear partial differential equation and has a third-order derivation that is mixed space and time. In such a case, the action of fluid flow is over a long period, and the numerical scheme needs to rely on the state-steady condition or set up a specific time to stop the simulation. In the ST-GFDM, the temporal and spatial discretizations can be performed using the GFDM in an ST-domain. That makes the proposed scheme can easily deal with the mixed derivative. By applying the GFDM, which is advanced from the Taylor series expansion and the moving-least square method, the process of the numerical discretizations is only related to nearby nodes on the central node. Thus, the Jacobian matrix is sparse and can be solved by the Newton–Raphson method efficiently. Furthermore, the time-marching method is used to proceed with an ST-domain along the time axis. In this paper, two numerical examples are solved to verify the capability of the proposed ST-GFDM.

View all citing articles on Scopus

^∗: Also Department of Physics,

View full text

Scattering of regularized-long-wave solitary waves

Abstract

Phys. Lett.

J. Fluid Mech.

Sov. J. Plasma Phys.

Phys. Fluids

Phil. Trans. R. Soc. London

Phys. Lett.

Phys. Fluids

Math. Proc. Camb. Phil. Soc.