N-Soliton Solution of Two Dimensional Modified Boussinesq Equation

1988 ◽  
Vol 57 (9) ◽  
pp. 2936-2940 ◽  
Author(s):  
Michiaki Matsukawa ◽  
Shinsuke Watanabe
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Two Dimensional ◽  
Modified Boussinesq Equation

A Two-dimensional Boussinesq equation for water waves and some of its solutions

1996 ◽  
Vol 323 ◽  
pp. 65-78 ◽  
Author(s):  
R. S. Johnson
Keyword(s):  
Soliton Solution ◽  
Water Waves ◽  
Boussinesq Equation ◽  
Kdv Equation ◽  
Boussinesq Equations ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Completely Integrable ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

scholarly journals Existence and Uniqueness Results for Two-Dimensional Stochastic Linearised Boussinesq Equation

2020 ◽  
Vol 6 (1) ◽  
pp. 20
Author(s):  
Sofije Hoxha ◽  
Fejzi Kolaneci
Keyword(s):  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Boussinesq Equation ◽  
Initial Condition ◽  
Two Dimensional ◽  
Boussinesq Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Drainable Porosity

The water flow in saturated zones of the soil is described by two-dimensional Boussinesq equation. This paper is devoted to investigating the linearised stochastic Boussinesq problem in the presence of randomness in hydraulic conductivity, drainable porosity, recharge, evapotranspiration, initial condition and boundary condition. We use the Sabolev spaces and Galerkin method. Under some suitable assumptions, we prove the existence and uniqueness results, as well as, the continuous dependence on the data for the solution of linearised stochastic Boussinesq problem. Keywords: linearised stochastic Boussinesq equation, Galerkin method, existence and uniqueness results, and continuous dependence on the data.

scholarly journals Analysis of the effectiveness of the systems protecting against the impact of water damming in the river on the increase of groundwater level on the example of the Malczyce dam

2018 ◽  
Vol 23 ◽  
pp. 00011
Author(s):  
Arkadiusz Głogowski ◽  
Mieczysław Chalfen
Keyword(s):  
Boussinesq Equation ◽  
Chemical Pollution ◽  
Left Bank ◽  
Two Dimensional ◽  
Drainage Channel ◽  
The Finite Element Method ◽  
The Third ◽  
Stationary Version ◽  
The Impact ◽  
The Village

The aim of the article is to determine to what extent individual elements of the project protecting the village of Rzeczyca and adjacent areas against flooding after the planned damming up of water in the Odra on the Malczyce dam. The assessment of the impact of damming on the nearby towns was made using a mathematical model based on a two-dimensional and non-stationary version of the Boussinesq equation and the finite element method (FEM). In the simulations, the proprietary FIZ software was used for calculating water flow and chemical pollution in a porous medium. Four computer simulations were carried out, modelling the flow of groundwater in the left-bank Odra valley. The first simulation was run in pre-towering conditions, the second one included water damming without additional safeguards, the third one with a watertight membrane and the fourth one with a membrane and a drainage channel.

SOLITON INTERACTIONS FOR THE GENERALIZED (3+1)-DIMENSIONAL BOUSSINESQ EQUATION

2012 ◽  
Vol 26 (07) ◽  
pp. 1250062 ◽  
Author(s):  
XIAO-LING GAI ◽  
YI-TIAN GAO ◽  
XIN YU ◽  
ZHI-YUAN SUN
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Propagation Process ◽  
Soliton Propagation ◽  
Soliton Interactions ◽  
Pass Through ◽  
The Moment ◽  
The One

Generalized (3+1)-dimensional Boussinesq equation is investigated in this paper. Through the dependent variable transformation and symbolic computation, the one- and two-soliton solutions are obtained. With the one-soliton solution, the coefficient effects in the soliton propagation process are investigated. Through analyzing the two-soliton solution, two kinds of two-soliton interactions are presented: (i) Two solitons merge into a bigger one whose amplitude increases but does not exceed the sum of the two at the moment of the collision; (ii) Two solitons can pass through each other, and their shapes keep unchanged with a phase shift after the separation. In addition, two kinds of analytic solutions are discussed: (i) "Amplitudes" of the two analytic solutions immediately turn to negative (positive) infinity after the "collision"; (ii) Two analytic solutions are fused into a higher peak (valley) at the moment of "collision", whose "amplitudes" change to negative (positive) infinity after the separation.

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