Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Bona,; Chen, Guihai; Saut,

doi:10.1007/s00332-002-0466-4

Cited by 437 publications

(594 citation statements)

References 36 publications

Supporting

Mentioning

584

Contrasting

Unclassified

Order By: Relevance

“…The derivation follows closely that of [5] for a single layer. Let us now consider waves whose typical amplitude, A, is small compared to the depth of the bottom layer h, and whose typical wavelength, ℓ, is large compared to the depth of the bottom layer 1 .…”

Section: Governing Equationsmentioning

confidence: 79%

A Boussinesq system for two-way propagation of interfacial waves

Nguyen

Dias

2008

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion. A Boussinesq system for two-way propagation of interfacial waves in a rigid lid configuration is derived. In most cases, the nonlinearity is quadratic. However, when the square of the depth ratio is close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. The propagation as well as the collision of solitary waves and/or fronts is studied numerically.

show abstract

Section: Governing Equationsmentioning

confidence: 79%

A Boussinesq system for two-way propagation of interfacial waves

Nguyen

Dias

2008

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…In [5], Bona, Chen and Saut studied in the one-layer case a three-parameter family of Boussinesq systems, which are all approximations to the full Euler equations at the same order (in the sense of consistency). The same structure applies also in the two-layer case, and we describe it quickly below.…”

Section: Remark 24mentioning

confidence: 99%

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Duchêne¹

2011

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83-103]. We study the wellposedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538-566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C.

show abstract

“…where λ and µ are real [8,10,11]. Particular choice of the parameters θ 2 = 1 3 , µ = 0 and arbitrary λ generates the CBS from the system (1) [9,10,12] as;…”

Section: Boussinesq Systemsmentioning

confidence: 99%

“…When the parameters are chosen as θ 2 = 2 3 , λ = 0, µ = 0, the Boussinesq system (1) reduces to the RBS of the form [11];…”

Section: Boussinesq Systemsmentioning

confidence: 99%

Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

2016

View full text Add to dashboard Cite

In the study, the collocation method based on exponential cubic B-spline functions is proposed to solve one dimensional Boussinesq systems numerically. Two initial boundary value problems for Regularized and Classical Boussinesq systems modeling motion of traveling waves are considered. The accuracy of the method is validated by measuring the error between the numerical and analytical solutions. The numerical solutions obtained by various values of free parameter p are compared with some solutions in literature.Keywords: Boussinesq sytems; solitary waves; exponential cubic B-spline; collocation. MSC2010: 35Q99;35C07;76B25;65D07;65M70. Boussinesq SystemsIn the light of the d' Alembert solution, describing two distinct waves moving in the opposite directions for the Cauchy problem for one dimensional wave equation, many physical problems modeled by linear and nonlinear partial differential equations have been solved in wave forms covering traveling waves, solitary waves, harmonic waves, etc. Waves are an important solution class for the model problems in physics, chemistry, and many fields of engineering. This solution type is widely constructed for different models in various media covering solids and fluids. Class of surface bell-shaped solitary waves, having long amplitude when compared with its width, is one of the most prominent classes of waves. In addition to linear equations, solitary wave solutions have also been studied for well known nonlinear equations and systems such as Korteweg-deVries(KdV), Schrödinger, Boussinesq, etc. Having analytical solutions only for some particular cases including additional restrictions and conditions, many numerical methods have been developed to obtain solutions for nonlinear problems. Dougalis et al.[1] studied the numerical behavior of solitary waves of a member of Boussinesq systems family (Bona-Smith system). In that study, long time stability and possible blow-up solutions were examined under small and large perturbations covering perturbation of amplitudes, system coefficients, etc. Numerical models were constructed with the Galerkin-finite element method on the space S h of smooth, periodic, cubic splines. Quintic B-spline collocation tecnique based on Crank-Nicolson formulation was set up for numerical solutions of Boussines type coupled-BBM system [2]. Two initial boundary value problems describing motion of single solitary wave and interaction of solitary waves were simulated by the proposed method. The numerical results were compared with the analytical ones. * Ozlem Ersoy,

show abstract

Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

Cited by 437 publications

References 36 publications

A Boussinesq system for two-way propagation of interfacial waves

A Boussinesq system for two-way propagation of interfacial waves

Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation

Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

Contact Info

Product

Resources

About