scholarly journals Shallow water equations in Lagrangian coordinates: Symmetries, conservation laws and its preservation in difference models

2020 ◽  
Vol 89 ◽  
pp. 105343
Author(s):  
V.A. Dorodnitsyn ◽  
E.I. Kaptsov
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Lagrangian Coordinates

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

2016 ◽  
Vol 86 ◽  
pp. 185-195 ◽  
Author(s):  
Piyanuch Siriwat ◽  
Chompit Kaewmanee ◽  
Sergey V. Meleshko
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Lagrangian Coordinates

scholarly journals The shallow water equations in Lagrangian coordinates

2004 ◽  
Vol 200 (2) ◽  
pp. 654-669 ◽  
Author(s):  
J.L. Mead
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Lagrangian Coordinates

Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations

1991 ◽  
Vol 227 ◽  
pp. 393-406 ◽  
Author(s):  
Darryl D. Holm
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Hamiltonian Dynamics ◽  
Relative Equilibria ◽  
Hamiltonian Mechanics ◽  
Lagrangian Coordinates ◽  
Vortex Solutions ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

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 Two-Layer Viscous Shallow-Water Equations and Conservation Laws

2009 ◽  
Vol 3 (1) ◽  
pp. 373-384 ◽  
Author(s):  
Hiroshi KANAYAMA ◽  
Hiroshi DAN
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations

Discrete shallow water equations preserving symmetries and conservation laws

2021 ◽  
Vol 62 (8) ◽  
pp. 083508
Author(s):  
V. A. Dorodnitsyn ◽  
E. I. Kaptsov
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations

Nonlinear-dispersive shallow water equations on a rotating sphere and conservation laws

2014 ◽  
Vol 55 (3) ◽  
pp. 404-416 ◽  
Author(s):  
Z. I. Fedotova ◽  
G. S. Khakimzyanov
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Rotating Sphere

CONSERVATION LAWS FOR ONE-DIMENSIONAL SHALLOW WATER MODELS FOR ONE AND TWO-LAYER FLOWS

2006 ◽  
Vol 16 (01) ◽  
pp. 119-137 ◽  
Author(s):  
RICARDO BARROS
Keyword(s):  
Conservation Laws ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Shallow Water Model ◽  
Frobenius Problem ◽  
One Dimensional ◽  
Shallow Water Models ◽  
The One ◽  
Open Question

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

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