scholarly journals On a Coupled System of Shallow Water Equations Admitting Travelling Wave Solutions

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
D. Burini ◽  
S. De Lillo ◽  
D. Skouteris
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Boussinesq Equations ◽  
Coupled System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Characteristic Parameters ◽  
Wave Solutions ◽  
Free Interfaces ◽  
Rigid Walls

We consider three inviscid, incompressible, irrotational fluids that are contained between the rigid wallsy=−h1andy=h+Hand that are separated by two free interfacesη1andη2. A generalized nonlocal spectral (NSP) formulation is developed, from which asymptotic reductions of stratified fluids are obtained, including coupled nonlinear generalized Boussinesq equations and(1+1)-dimensional shallow water equations. A numerical investigation of the(1+1)-dimensional case shows the existence of solitary wave solutions which have been investigated for different values of the characteristic parameters.

scholarly journals Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Ömer Faruk Gözükızıl ◽  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boussinesq Equation ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Higher Order ◽  
Sobolev Type ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions

By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

scholarly journals Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210307
Author(s):  
Demetrios T. Papageorgiou ◽  
Saleh Tanveer
Keyword(s):  
Global Existence ◽  
Evolution Equations ◽  
Coupled System ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Rigorous Results ◽  
Wave Solutions ◽  
Non Local ◽  
Two Fluid

This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

Exact travelling wave solutions for the generalized shallow water wave equation

2003 ◽  
Vol 17 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S.A. Elwakil ◽  
S.K. El-labany ◽  
M.A. Zahran ◽  
R. Sabry
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Shallow Water Wave ◽  
Wave Solutions

Integrability Test and Travelling-Wave Solutions of Higher-Order Shallow- Water Type Equations

2010 ◽  
Vol 65 (4) ◽  
pp. 353-356
Author(s):  
Mercedes Maldonado ◽  
María Celeste Molinero ◽  
Andrew Pickering ◽  
Julia Prada
Keyword(s):  
Shallow Water ◽  
Travelling Wave ◽  
Higher Order ◽  
Water Type ◽  
Travelling Wave Solutions ◽  
Compatibility Conditions ◽  
Wave Solutions

We apply the Weiss-Tabor-Carnevale (WTC) Painlev´e test to members of a sequence of higher-order shallow-water type equations. We obtain the result that the equations considered are non-integrable, although compatibility conditions at real resonances are satisfied. We also construct travelling-wave solutions for these and related equations.

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