One- and two-soliton solutions to a new KdV evolution equation with nonlinear and nonlocal terms for the water wave problem

2015 ◽  
Vol 83 (4) ◽  
pp. 2461-2473 ◽  
Author(s):  
M. Fokou ◽  
T. C. Kofane ◽  
A. Mohamadou ◽  
E. Yomba
Keyword(s):  
Evolution Equation ◽  
Water Wave ◽  
Soliton Solutions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Nonlocal Terms

scholarly journals An alternative approach to study irrotational periodic gravity water waves

2021 ◽  
Vol 72 (4) ◽  
Author(s):  
Biswajit Basu ◽  
Calin I. Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Stagnation Points ◽  
Alternative Approach ◽  
New Formulation ◽  
Bounded Below ◽  
Surface Dynamic

AbstractWe are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

scholarly journals A remark on the two dimensional water wave problem with surface tension

2019 ◽  
Vol 266 (9) ◽  
pp. 5748-5771
Author(s):  
Shuanglin Shao ◽  
Hsi-Wei Shih
Keyword(s):  
Surface Tension ◽  
Water Wave ◽  
Two Dimensional ◽  
Wave Problem ◽  
Water Wave Problem

Unsteady non-linear waves in sloping channels

1958 ◽  
Vol 247 (1249) ◽  
pp. 186-198 ◽  
Keyword(s):  
Exact Solution ◽  
Water Wave ◽  
Legendre Function ◽  
Elliptic Integral ◽  
Vertical Wall ◽  
Auxiliary Function ◽  
Level Line ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Non Linear

It is shown in general that the exact solution to every non-degenerate unsteady water-wave problem in a straight channel inclined at arbitrary slope, governed by the non-linear hydraulic equations, can be obtained in terms of the complete elliptic integral of the second kind, E . By means of a non-Newtonian reference frame, every such wave problem for a sloping channel can be replaced by an associated problem for a horizontal channel. For the latter, the partial differential equations become reducible and thus permit hodograph inversion. The Riemann integration method for the resulting Euler-Poisson equation yields an auxiliary function for these hydraulic problems which is transformable into a Legendre function and then into the elliptic integral. In particular, the procedure is applied to obtain the exact solution for the water wave in a sloping channel produced by sudden release of the triangular wedge of water (the reservoir) initially at rest behind a vertical wall. The behaviour of the solution is exhibited for convenience in two level-line charts, and representative wave profiles and velocity distributions are presented.

A mathematical analysis of tsunami generation in shallow water due to seabed deformation

2011 ◽  
Vol 141 (3) ◽  
pp. 551-608 ◽  
Author(s):  
Tatsuo Iguchi
Keyword(s):  
Velocity Field ◽  
Shallow Water ◽  
Water Wave ◽  
Initial Velocity ◽  
Tsunami Generation ◽  
Wave Problem ◽  
Initial Displacement ◽  
Water Wave Problem ◽  
Submarine Earthquakes ◽  
Tsunami Model

In numerical computations of tsunamis due to submarine earthquakes, it is frequently assumed that the initial displacement of the water surface is equal to the permanent shift of the seabed and that the initial velocity field is equal to zero and the shallow-water equations are often used to simulate the propagation of tsunamis. We give a mathematically rigorous justification of this tsunami model starting from the full water-wave problem by comparing the solution of the full problem with that of the tsunami model. We also show that, in some cases, we have to impose a non-zero initial velocity field, which arises as a nonlinear effect.

Hamiltonian long-wave approximations to the water-wave problem

Wave Motion ◽  
1994 ◽  
Vol 19 (4) ◽  
pp. 367-389 ◽  
Author(s):  
Walter Craig ◽  
Mark D. Groves
Keyword(s):  
Water Wave ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Long Wave

Transmission of water waves through small apertures

1971 ◽  
Vol 49 (1) ◽  
pp. 65-74 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Transmission Coefficient ◽  
Asymptotic Expansions ◽  
Water Waves ◽  
Water Wave ◽  
Approximate Formula ◽  
Matched Asymptotic Expansions ◽  
Wave Problem ◽  
Water Wave Problem ◽  
Method Of Solution

A method of solution is proposed for flows through small apertures in otherwise impermeable barriers. This method, which is an application of the method of matched asymptotic expansions, is used to solve a specific water-wave problem, yielding an approximate formula for the transmission coefficient.

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