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Study of exact analytical solutions and various wave profiles of a new extended(2+1)-dimensional boussinesq equation using symmetry analysis

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Journal of Ocean Engineering and Science 2022年第5期7卷 475-484页

作者： Sachin Kumar Setu Rani Department of Mathematics Faculty of Mathematical SciencesUniversity of DelhiDelhi110007India

This paper systematically investigates the exact solutions to an extended(2+1)-dimensional boussinesq equation,which arises in several physical applications,including the propagation of shallow-water waves,with the help of the Lie symmetry analysis method.We acquired the vector fields,commutation relations,optimal systems,two stages of reductions,and exact solutions to the given equation by taking advantage of the Lie group method.The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein.Furthermore,Lie symmetry analysis(LSA)is implemented for perceiving the symmetries of the boussinesq equation and then culminating the solitary wave solutions.The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles.These solutions show single soliton,multiple solitons,elastic behavior of combo soliton profiles,and stationary waves,as can be seen from the graphics.The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.

关键词： boussinesq equation Lie group method Exact invariant solutions Solitons Optimal system

A novel (G'/G)-expansion method and its application to the boussinesq equation

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Chinese Physics B 2014年第2期23卷 34-43页

作者： Md.Nur Alam Md.Ali Akbar Syed Tauseef Mohyud-Din Department of Mathematics Pabna University of Science and Technology Department of Applied Mathematics University of Rajshahi Department of Mathematics HITEC UniversityTaxila Cantt

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.

关键词：（G＇/G）-expansion method boussinesq equation solitary wave solutions auxiliary nonlinear ordinary differential equation

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended boussinesq equation in fluid mechanics

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Communications in Theoretical Physics 2021年第4期73卷 21-28页

作者： Gangwei Wang Abdul-Majid Wazwaz School of Mathematics and Statistics Hebei University of Economics and BusinessShijiazhuang050061China Department of Mathematics Saint Xavier UniversityChicagoIL 60655United States of America

In this work,we study a generalized double dispersion boussinesq equation that plays a significant role in fluid mechanics,scientific fields,and ocean engineering.This equation will be reduced to the Korteweg-de Vries equation via using the perturbation analysis.We derive the corresponding vectors,symmetry reduction and explicit solutions for this equation.We readily obtain Bäcklund transformation associated with truncated Painlevéexpansion.We also examine the related conservation laws of this equation via using the multiplier method.Moreover,we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.

关键词： boussinesq equation perturbation and symmetry analysis Bäcklund transformation conservation laws

A Generalization of F-Expansion Method and Its Application to (2+1)-Dimensional boussinesq equation

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Communications in Theoretical Physics 2007年第11期48卷 877-880页

作者： CHEN Jiang Lanzhou Institute of Physics,Lanzhou University,Lanzhou 730000,ChinaYANG Kong-Qing HE Hong-Sheng Institute of Applied Physics,Jimei University,Xiamen 361021,China Lanzhou Institute of Physics Lanzhou University Lanzhou 730000 China Institute of Applied Physics Jimei University Xiamen 361021 China

A new generalized F-expansion method is introduced and applied to the study of the (2+1)-dimensionalboussinesq equation.The further extension of the method is discussed at the end of this paper.

关键词： F-expansion method Jacobi elliptic function boussinesq equation solitary wave solution

PERTURBED PERIODIC SOLUTIONFOR boussinesq equation

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Acta Mathematica Scientia 2009年第3期29卷 705-722页

作者：江新华王振 Department of Mathematics and Information Science Beijing University of Chemical TechnologyBeijing 100029 China Wuhan Institute of Physics and Mathematics The Chinese Academy of SciencesWuhan 430071 China

We consider the solution of the good boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where... 详细信息

We consider the solution of the good boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.

关键词： boussinesq equation periodic solution Backlund transformation global existence uniform asymptotic expansion

Modified asymptotic Adomian decomposition method for solving boussinesq equation of groundwater flow

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Applied Mathematics and Mechanics(English Edition) 2014年第4期35卷 481-488页

作者：陈芳刘青泉 Key Laboratory for Mechanics in Fluid Solid Coupling Systems Institute of Mechanics Chinese Academy of Sciences School of Applied Science Beijing Information Science and Technology University

The Adomian decomposition method （ADM） is an approximate analytic method for solving nonlinear equations. Generally, an approximate solution can be ob- tained by using only a few terms. However, in applications, we need to use it flexibly according to the real problem. In this paper, based on the ADM, we give a modified asymptotic Adomian decomposition method and use it to solve the nonlinear boussinesq equation describing groundwater flows. The example shows effectiveness of the modified asymptotic Adomian decomposition method.

关键词： groundwater flow boussinesq equation Adomian decomposition asymp-totic Adomian decomposition

Blow-up of solution for a generalized boussinesq equation

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Applied Mathematics and Mechanics(English Edition) 2007年第11期28卷 1437-1443页

作者：王艳萍郭柏灵 Department of Mathematics and Physics Zhengzhou Institute of Aeronautical Industry Management Institute of Applied Physics and Computational Mathematics P.O.Box 8009Beijing 100088P.R.China Institute of Applied Physics and Computational Mathematics

This paper studies the initial boundary value problem for a generalized Boussinese equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreov... 详细信息

关键词： boussinesq equation initial boundary value problem local solution blow-up of the solution

Numerical Simulation of Abnormal Wave Height Change Induced by Excavated Waterway with boussinesq equation

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China Ocean Engineering 1998年第1期13卷 63-72页

作者： Zhang, YG Li, YC Shanghai Jiao Tong Univ Dept Mech Engn Shanghai 200030 Peoples R China Dalian Univ Technol Dept Civil Engn Dalian 116024 Peoples R China

A nonlinear numerical model has been set up by use of boussinesq equation with finite difference method, and has been applied to the simulation of the abnormal change of wave height induced by excavated waterway. Numerical results demonstrate that the abnormal change of wave height is due to the adding of the reflected wave height induced by excavated waterway to the incident wave height. Because the angle between the incident wave and the axis of the waterway is smaller than the critical angle, the reflected wave produced by the waterway may propagate to the breakwater and may be added with the incident wave, then the abnormal change of wave height before the breakwater may be caused. So the wave reflection caused by the change of water depth cannot be neglected.

关键词： boussinesq equation numerical simulation regular wave excavated waterway wave height

A boussinesq equation-Based Model for Nearshore Wave Breaking

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海洋工程：英文版 2004年第2期18卷 315-320页

作者：余建星张伟王广东杨树清 School of Construction

Based on the wave breaking model by Li and Wang (1999), this work is to apply Dally's analytical solution to the wave-height decay instead of the empirical and semi-empirical hypotheses of wave-height distribution within the wave breaking zone. This enhances the applicability of the model. Computational results of shoaling, location of wave breaking, wave-height decay after wave breaking, set-down and set-up for incident regular waves are shown to have good agreement with experimental and field data.

关键词： wave breaking numerical model boussinesq equation eddy viscosity