scholarly journals Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

2017 ◽  
Vol 20 (2) ◽  
pp. 631-653 ◽  
Author(s):  
Ryo Nemoto ◽  
Tatsuo Iguchi
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Initial Value

scholarly journals Spatially Quasi-Periodic Water Waves of Infinite Depth

2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Jon Wilkening ◽  
Xinyu Zhao
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Small Scale ◽  
Normal Velocity ◽  
Natural Setting ◽  
Two Dimensional ◽  
Periodic Functions ◽  
Initial Value

AbstractWe formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

scholarly journals Solvability of the initial value problem to a model system for water waves

2015 ◽  
Vol 38 (2) ◽  
pp. 470-491 ◽  
Author(s):  
Yuuta Murakami ◽  
Tatsuo Iguchi
Keyword(s):  
Initial Value Problem ◽  
Water Waves ◽  
Model System ◽  
Initial Value

An evaluation of a model equation for water waves

1981 ◽  
Vol 302 (1471) ◽  
pp. 457-510 ◽  
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Water Waves ◽  
Laboratory Experiments ◽  
Initial Conditions ◽  
Initial Value ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Spatial Periodicity ◽  
Long Channel

This study assesses a particular model for the unidirectional propagation of water waves, comparing its predictions with the results of a set of laboratory experiments. The equation to be tested is a one-dimensional representation of weakly nonlinear, dispersive waves in shallow water. A model for such flows was proposed by Korteweg & de Vries (1895) and this has provided the theoretical basis for a number of laboratory experiments. Some recent studies that have been made in the area are those of Zabusky & Galvin (1971), Hammack (1973) and Hammack & Segur (1974). In each case the theoretical model gave a good qualitative account of the experiments, but the quantitative comparisons were not very extensive. One of the purposes of this paper is to provide a more detailed quantitative assessment of a particular model than has been given to date. An important aspect of the formulation of the theoretical model is the specification of the initial conditions and the boundary conditions for the equation. Zabusky & Galvin (1971) considered an initial-value problem having spatial periodicity, whereas Hammack (1973) and Hammack & Segur (1974) considered an initial-value problem posed on the real line. In contrast, we shall consider an initial-value problem posed on the half line with boundary data specified at the origin. This problem was chosen to correspond with an experiment in which waves were generated at one end of a long channel, and obviates certain difficulties inherent in the other formulations.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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