Abstract
We establish existence and uniqueness of solutions to the Cauchy problem associated with a new one-dimensional weakly-nonlinear, weakly-dispersive system which arises as an asymptotical approximation of the full potential theory equations for modelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth, taking into account the effect of surface tension. Furthermore, numerical schemes of spectral type are introduced for approximating the evolution in time of solutions of this system and its travelling wave solutions, in both the periodic and nonperiodic case.
Highlights
In this paper we study the propagation of water waves on the surface of a shallow channel with variable depth, considering the effect of surface tension
Φ(x, y, t) denotes the potential velocity and η(x, t) the wave elevation measured with respect to the undisturbed free surface y = 0
We look for a solitary wave solution of (94) in the form
Summary
In this paper we study the propagation of water waves on the surface of a shallow channel with variable depth, considering the effect of surface tension. To describe this phenomenon we will derive a new water wave model from Euler’s equations (in dimensionless variables) for an inviscid, incompressible liquid bounded above by a free surface and bounded below by an impermeable bottom topography [1]: βφxx + φyy = 0 f or. Φ(x, y, t) denotes the potential velocity and η(x, t) the wave elevation measured with respect to the undisturbed free surface y = 0.
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