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 Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1601
Author(s):  
Zakieh Avazzadeh ◽  
Omid Nikan ◽  
José A. Tenreiro Machado
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Wave Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Ode System ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
De Vries

This paper investigates the solitary wave solutions of the generalized Rosenau–Korteweg-de Vries-regularized-long wave equation. This model is obtained by coupling the Rosenau–Korteweg-de Vries and Rosenau-regularized-long wave equations. The solution of the equation is approximated by a local meshless technique called radial basis function (RBF) and the finite-difference (FD) method. The association of the two techniques leads to a meshless algorithm that does not requires the linearization of the nonlinear terms. First, the partial differential equation is transformed into a system of ordinary differential equations (ODEs) using radial kernels. Then, the ODE system is solved by means of an ODE solver of higher-order. It is shown that the proposed method is stable. In order to illustrate the validity and the efficiency of the technique, five problems are tested and the results compared with those provided by other schemes.

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.

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