On the basis of the integrable Kaup–Boussinesq version of the shallow-water equations, an analytical theory of undular bores is constructed. A complete classification for the problem of the decay of an initial discontinuity is made.

We investigate the response of a laboratory aquifer submitted to artificial rainfall, with an emphasis on the early stage of a rain event. In this almost two-dimensional experiment, the infiltrating rainwater forms a groundwater reservoir which exits the aquifer through one side. The resulting outflow resembles a typical stream hydrograph: the water discharge increases rapidly during rainfall a...

Abstract. A fully discrete a priori analysis of the finite element heterogenenous multiscale method (FE-HMM) introduced in [A. Abdulle, M. Grote, C. Stohrer, Multiscale Model. Simul. 2014] for the wave equation with highly oscillatory coefficients over long time is presented. A sharp a priori convergence rate for the numerical method is derived for long time intervals. The effective model over ...

We propose a modi…cation of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven ‡ows of viscous, incompressible ‡uids. This modi…cation is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations...

In a previous study [DD07b] we presented a novel visco-potential free surface flows formulation. The governing equations contain local and nonlocal dissipative terms. From physical point of view, local dissipation terms come from molecular viscosity but in practical computations, rather eddy viscosity should be used. On the other hand, nonlocal dissipative term represents a correction due to th...

Similarity transforms of the Boussinesq equation in a semi-in®nite medium are available when the boundary conditions are a power of time. The Boussinesq equation is reduced from a partial di€erential equation to a boundary-value problem. Chen et al. [Trans Porous Media 1995;18:15±36] use a hodograph method to derive an integral equation formulation of the new di€erential equation which they sol...

It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, ...

We study here instability problems of standing waves for the nonlinear Klein–Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave’s frequency or the wave’s speed of propagation and on the nonlinearity. © 2006 Elsevier Masson SAS. All rights...

We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999), 159-207]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for Bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic arguments, ...