A reformulation and applications of interfacial fluids with a free surface

2009 ◽  
Vol 631 ◽  
pp. 375-396 ◽  
Author(s):  
T. S. HAUT ◽  
M. J. ABLOWITZ
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Solitary Waves ◽  
Wave Equations ◽  
Computational Studies ◽  
Free Interface ◽  
Long Wave ◽  
Ideal Fluids ◽  
Non Local ◽  
Two Fluid

A non-local formulation, depending on a free spectral parameter, is presented governing two ideal fluids separated by a free interface and bounded above either by a free surface or by a rigid lid. This formulation is shown to be related to the Dirichlet–Neumann operators associated with the two-fluid equations. As an application, long wave equations are obtained; these include generalizations of the Benney–Luke and intermediate long wave equations, as well as their higher order perturbations. Computational studies reveal that both equations possess lump-type solutions, which indicate the possible existence of fully localized solitary waves in interfacial fluids with sufficient surface tension.

SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS

2008 ◽  
Vol 06 (04) ◽  
pp. 323-348 ◽  
Author(s):  
M. J. ABLOWITZ ◽  
T. S. HAUT
Keyword(s):  
Wave Equation ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Fluid System ◽  
Free Interface ◽  
Long Wave ◽  
Petviashvili Equation ◽  
Versus Amplitude ◽  
Two Fluid

A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.

Hamiltonian Formulation and Long Wave Models for Internal Waves

2007 ◽  
Author(s):  
Walter Craig ◽  
Philippe Guyenne ◽  
Henrik Kalisch
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Wave Equations ◽  
Hamiltonian Formulation ◽  
Solitary Wave Solutions ◽  
Free Interface ◽  
Wave Solutions ◽  
Long Wave ◽  
Wave Models ◽  
Upper Boundary

We derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free interface coupled with a free surface, this latter situation occurring more commonly in experiment and in nature. Based on the linearized equations, we highlight the discrepancies between the cases of rigid lid and free surface upper boundary conditions, which in some circumstances can be significant. We also derive systems of nonlinear dispersive long wave equations in the large amplitude regime, and compute solitary wave solutions of these equations.

scholarly journals Solving the Modified Regularized Long Wave Equations via Higher Degree B-Spline Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Pshtiwan Othman Mohammed ◽  
Manar A. Alqudah ◽  
Y. S. Hamed ◽  
Artion Kashuri ◽  
Khadijah M. Abualnaja
Keyword(s):  
Solitary Waves ◽  
Wave Motion ◽  
Wave Equations ◽  
The Other ◽  
Collocation Methods ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Current Article ◽  
The Stability

The current article considers the sextic B-spline collocation methods (SBCM1 and SBCM2) to approximate the solution of the modified regularized long wave ( MRLW ) equation. In view of this, we will study the solitary wave motion and interaction of higher (two and three) solitary waves. Also, the modified Maxwellian initial condition into solitary waves is studied. Moreover, the stability analysis of the methods has been discussed, and these will be unconditionally stable. Moreover, we have calculated the numerical conserved laws and error norms L 2 and L ∞ to demonstrate the efficiency and accuracy of the method. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods. The results show that our proposed methods are more accurate than the other methods.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

The Solution of the Regularized Long Wave Equation Using the Fourier Leap-Frog Method

2010 ◽  
Vol 65 (4) ◽  
pp. 268-276 ◽  
Author(s):  
Hany N. Hassan ◽  
Hassan K. Saleh
Keyword(s):  
Fourier Transform ◽  
Solitary Waves ◽  
Wave Equations ◽  
Space Dimension ◽  
Nonlinear Wave Equations ◽  
Long Wave ◽  
Rlw Equation ◽  
Regularized Long Wave Equation ◽  
Waves Interaction ◽  
Space Dependence

An efficient numerical method is developed for solving nonlinear wave equations by studying the propagation and stability properties of solitary waves (solitons) of the regularized long wave (RLW) equation in one space dimension using a combination of leap frog for time dependence and a pseudospectral (Fourier transform) treatment of the space dependence. Our schemes follow very accurately these solutions, which are given by simple closed formulas. Studying the interaction of two such solitons and three solitary waves interaction for the RLW equation. Our implementation employs the fast Fourier transform (FFT) algorithm.

Notiz: A Symbolic Computation-Based Method and Two Nonlinear Evolution Equations for Water Waves

1997 ◽  
Vol 52 (3) ◽  
pp. 295-296
Author(s):  
Yi-Tian Gao ◽  
Bo Tian
Keyword(s):  
Symbolic Computation ◽  
Solitary Waves ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Long Wave ◽  
Shallow Water Wave Equations

Abstract A symbolic-computation-based method, which has been newly proposed, is considered for a (2+1)-dimensional generalization of shallow water wave equations and a coupled set of the (2 +1)-dimensional integrable dispersive long wave equations. New sets of soliton-like solutions are constructed, along with solitary waves.

scholarly journals Orbital Stability of Solitary Waves for Generalized Symmetric Regularized-Long-Wave Equations with Two Nonlinear Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Weiguo Zhang ◽  
Xu Chen ◽  
Zhengming Li ◽  
Haiyan Zhang
Keyword(s):  
Explicit Expression ◽  
Solitary Waves ◽  
Wave Speed ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Orbital Stability ◽  
Long Wave ◽  
Stability And Instability ◽  
Stability Of Solitary Waves

This paper investigates the orbital stability of solitary waves for the generalized symmetric regularized-long-wave equations with two nonlinear terms and analyzes the influence of the interaction between two nonlinear terms on the orbital stability. SinceJis not onto, Grillakis-Shatah-Strauss theory cannot be applied on the system directly. We overcome this difficulty and obtain the general conclusion on orbital stability of solitary waves in this paper. Then, according to two exact solitary waves of the equations, we deduce the explicit expression of discriminationd′′(c)and give several sufficient conditions which can be used to judge the orbital stability and instability for the two solitary waves. Furthermore, we analyze the influence of the interaction between two nonlinear terms of the equations on the wave speed interval which makes the solitary waves stable.

scholarly journals A note on solitary waves with variable surface tension in water of infinite depth

2006 ◽  
Vol 48 (2) ◽  
pp. 225-235 ◽  
Author(s):  
E. Özuğurlu ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Solitary Waves ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Infinite Depth ◽  
Free Surface Boundary Condition ◽  
Variable Surface ◽  
Dimensional Gravity ◽  
Surface Boundary Condition

AbstractTwo-dimensional gravity-capillary solitary waves propagating at the surface of a fluid of infinite depth are considered. The effects of gravity and of variable surface tension are included in the free-surface boundary condition. The numerical results extend the constant surface tension results of Vanden-Broeck and Dias to situations where the surface tension varies along the free surface.

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