Shallow water waves: Conservation laws and symplectic geometry

1983 ◽  
Vol 95 (1) ◽  
pp. 27-28 ◽  
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Water Waves ◽  
Symplectic Geometry ◽  
Shallow Water Waves

scholarly journals The shallow water equations: conservation laws and symplectic geometry

1987 ◽  
Vol 10 (3) ◽  
pp. 557-562 ◽  
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Symplectic Form ◽  
Symplectic Geometry ◽  
Nonlinear Differential Equations ◽  
Shallow Water Waves ◽  
Hodograph Method ◽  
Sloping Bottom

We consider the system of nonlinear differential equations governing shallow water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Böcklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.

scholarly journals Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Emrullah Yaşar ◽  
Sait San ◽  
Yeşim Sağlam Özkan
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Function Method ◽  
Boussinesq Equation ◽  
Lie Point Symmetries ◽  
Shallow Water Waves ◽  
Non Linear ◽  
Ill Posed

AbstractIn this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

scholarly journals Conservation laws for shallow water waves on a sloping beach

1986 ◽  
Vol 9 (2) ◽  
pp. 387-396
Author(s):  
Yilmaz Akyildiz
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Water Waves ◽  
Simple Wave ◽  
Shallow Water Waves ◽  
Linear Partial Differential Equations ◽  
Wave Solutions ◽  
Hodograph Plane ◽  
Sloping Beach ◽  
Linear Nature

Shallow water waves are governed by a pair of non-linear partial differential equations. We transfer the associated homogeneous and non-homogeneous systems, (corresponding to constant and sloping depth, respectively), to the hodograph plane where we find all the non-simple wave solutions and construct infinitely many polynomial conservation laws. We also establish correspondence between conservation laws and hodograph solutions as well as Bäcklund transformations by using the linear nature of the problems on the hodogrpah plane.

scholarly journals Coupling Conditions for Water Waves at Forks

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 434 ◽  
Author(s):  
Jean-Guy Caputo ◽  
Denys Dutykh ◽  
Bernard Gleyse
Keyword(s):  
Conservation Laws ◽  
Numerical Solution ◽  
Shallow Water ◽  
Water Waves ◽  
Large Amplitude ◽  
Narrow Channel ◽  
Effective Model ◽  
Interface Conditions ◽  
Shallow Water Waves ◽  
Coupling Conditions

We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.

RESPONSE CHARACTERISTICS OF GRAVEL BEACH FROM THE VIEWPOINT OF SHALLOW WATER WAVES AND MOVEMENT OF GRAVELS

2020 ◽  
Vol 76 (2) ◽  
pp. I_565-I_570
Author(s):  
Shin-ichi AOKI ◽  
Tomoki HAMANO ◽  
Taishi NAKAYAMA ◽  
Eiichi OKETANI ◽  
Takahiro HIRAMATSU ◽  
...  
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Response Characteristics ◽  
Gravel Beach
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