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Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media I: Derivation and linear theory
, 2002
"... Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surface of water in a canal. These syst ..."
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Cited by 111 (26 self)
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Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of longcrested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a fourparameter family of Boussinesq systems from the twodimensional Euler equations for freesurface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.
Long wave approximations for water waves
, 2005
"... Abstract: In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three spatial dimensions. These systems are symmetric and conservative. Rigorous convergence results are provided showing that solutions of the complete freesurface Euler equatio ..."
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Cited by 76 (7 self)
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Abstract: In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three spatial dimensions. These systems are symmetric and conservative. Rigorous convergence results are provided showing that solutions of the complete freesurface Euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large. Using this result as a tool, a rigorous justification of all the twodimensional, approximate systems recently put forward and analysed by Bona, Chen and Saut is obtained. In particular, this remark applies to the original system derived by Boussinesq. The estimates for the difference between the Euler variables and the system variables is better than that obtained in the twodimensional context by Schneider and Wayne who approximated with a decoupled pair of Korteweg de Vries equations. Indeed, the limitations inherent in approximating by a decoupled system are clarified in our analysis. Results are obtained both on an unbounded domain with solutions that evanesce at infinity as well as for solutions that are spatially periodic.
LaguerreGalerkin method for nonlinear partial differential equations on a semiinfinite interval
, 2000
"... A LaguerreGalerkin method is proposed and analyzed for the Burgers equation and BenjaminBonaMahony (BBM) equation on a semiinfinite interval. By reformulating these equations with suitable functional transforms, it is shown that the LaguerreGalerkin approximations are convergent on a semiinfini ..."
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Cited by 37 (18 self)
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A LaguerreGalerkin method is proposed and analyzed for the Burgers equation and BenjaminBonaMahony (BBM) equation on a semiinfinite interval. By reformulating these equations with suitable functional transforms, it is shown that the LaguerreGalerkin approximations are convergent on a semiinfinite interval withspectral accuracy. An efficient and accurate algorithm based on the LaguerreGalerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.
Comparison between threedimensional linear and nonlinear tsunami generation models, Theor. Comput. Fluid Dyn
"... The modeling of tsunami generation is an essential phase in understanding tsunamis. For tsunamis generated by underwater earthquakes, it involves the modeling of the sea bottom motion as well as the resulting motion of the water above it. A comparison between various models for threedimensional wat ..."
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Cited by 34 (18 self)
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The modeling of tsunami generation is an essential phase in understanding tsunamis. For tsunamis generated by underwater earthquakes, it involves the modeling of the sea bottom motion as well as the resulting motion of the water above it. A comparison between various models for threedimensional water motion, ranging from linear theory to fully nonlinear theory, is performed. It is found that for most events the linear theory is sufficient. However, in some cases, more sophisticated theories are needed. Moreover, it is shown that the passive approach in which the seafloor deformation is simply translated to the ocean surface is not always equivalent to the active approach in which the bottom motion is taken into account, even if the deformation is
A new dualPetrovGalerkin method for third and higher oddorder differential equations: Application to the KDV equation
 SIAM J. Numer. Anal
, 2003
"... Abstract. A new dualPetrov–Galerkin method is proposed, analyzed, and implemented for third and higher oddorder equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfyin ..."
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Cited by 25 (7 self)
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Abstract. A new dualPetrov–Galerkin method is proposed, analyzed, and implemented for third and higher oddorder equations using a spectral discretization. The key idea is to use trial functions satisfying the underlying boundary conditions of the differential equations and test functions satisfying the “dual ” boundary conditions. The method leads to linear systems which are sparse for problems with constant coefficients and well conditioned for problems with variable coefficients. Our theoretical analysis and numerical results indicate that the proposed method is extremely accurate and efficient and most suitable for the study ofcomplex dynamics ofhigher oddorder equations.
Conservation Laws With Vanishing Nonlinear Diffusion And Dispersion
, 1996
"... We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This ..."
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Cited by 18 (2 self)
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We study the limiting behavior of the solutions to a class of conservation laws with vanishing nonlinear diffusion and dispersion terms. We prove the convergence to the entropy solution of the first order problem under a condition on the relative size of the diffusion and the dispersion terms. This study is closely motivated by the pseudoviscosity approximation introduced by Von Neumann in the 50's.
Corrections to the KdV approximation for water waves
 SIAM J. Math. Anal
"... Abstract. In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consi ..."
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Cited by 11 (0 self)
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Abstract. In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are explicitly solvable and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. AMS classification: 76B15, 35Q51, 35Q53 1.
Forced Oscillations of a damped Korteweg–de Vries equation in a quarter plane
 Comm. Contemp. Math
"... Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in ..."
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Cited by 11 (5 self)
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Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Kortewegde Vries equation. Thus, consideration is given to the initialboundaryvalue problem
On the dual PetrovGalerkin formulation of the KDV equation in a finite interval
 Submitted, 2006. MANUSCRIPT HYPERBOLIC EQUATIONS 23
, 2007
"... Abstract. An abstract functional framework is developed for the dual PetrovGalerkin formulation of the initialboundaryvalue problems with a thirdorder spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval. 1. ..."
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Cited by 10 (3 self)
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Abstract. An abstract functional framework is developed for the dual PetrovGalerkin formulation of the initialboundaryvalue problems with a thirdorder spatial derivative. This framework is then applied to study the wellposedness and decay properties of the KdV equation in a finite interval. 1.
Nonhomogeneous Boundary Value Problems for the Kortewegde Vries and the Kortewegde VriesBurgers Equations in a Quarter Plane
, 2007
"... Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0, ..."
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Cited by 9 (3 self)
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Attention is given to the initialboundaryvalue problems (IBVPs) ut + ux + uux + uxxx = 0, for x, t ≥ 0, u(x, 0) = φ(x), u(0, t) = h(t) for the Kortewegde Vries (KdV) equation and ut + ux + uux − uxx + uxxx = 0, for x, t ≥ 0,