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Influence of dissipation on solitary wave solution to generalized Boussinesq equation

Applied Mathematics and Mechanics(English Edition) 2023年第3期44卷 477-498页

作者： Weiguo ZHANG Siyu HONG Xingqian LING Wenxia LI College of Science University of Shanghai for Science and TechnologyShanghai 200093China

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations,and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves.The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail.The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied,and the critical values that can describe the magnitude of the dissipation effect are,respectively,found for the two cases of b_30 in the equation.The results show that,when the dissipation effect is significant(i.e.,r is greater than the critical value in a certain situation),the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution;when the dissipation effect is small(i.e.,r is smaller than the critical value in a certain situation),the traveling wave solution to the equation appears as the oscillation attenuation solution.By using the hypothesis undetermined method,all possible solitary wave solutions to the equation when there is no dissipation effect(i.e.,r=0)and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained;in particular,when the dissipation effect is small,an approximate solution of the oscillation attenuation solution can be achieved.This paper is further based on the idea of the homogenization principles.By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper,and by investigating the asymptotic behavior of the solution at infinity,the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained,which is an infinitesimal amount that decays exponentially.The influence of the dissipation coefficient on the ampli

关键词： generalized Boussinesq equation influence of dissipation qualitative analysis solitary wave solution oscillation attenuation solution error estimation

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Soliton solution of the generalized modified BBM equation and the generalized Boussinesq equation

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Journal of Ocean Engineering and Science 2017年第4期2卷 248-252页

作者： Ozkan Guner Faculty of Economics and Administrative Sciences Department of International TradeCankiri Karatekin UniversityCankiriTurkey

In this paper,we obtained the topological soliton solution of the(1+1)-dimensional generalized modified Benjamin-Bona-Mahony equation and shock wave solution of the generalized Boussinesq equation.We get that solutions by using solitary wave ansatz in terms of tanh^(p) functions.The velocity and the free parameters are the physical parameters in the soliton solutions.They can be obtained as functions of the dependent model coefficients.The domain restriction were also identified in the process.we hope that in nonlinear dynamical system these solutions will be explain some nonlinear physical phenomena.

关键词： Exact solution Topological soliton solution Shock wave solution generalized modified Benjamin-Bona-Mahony equation generalized Boussinesq equation

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Global Existence and Blow up for Damped generalized Boussinesq equation

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Acta Mathematicae Applicatae Sinica 2017年第1期33卷 251-262页

作者： Run-zhang XU Yong-bing LUO Ji-hong SHEN Shao-bin HUANG College of Science Harbin Engineering University Harbin 150001 China College of Automation Harbin Engineering University Harbin 150001 China College of Computer Science and Technology Harbin Engineering University Harbin 150001 China

We study the Cauchy problem of damped generalized Boussinesq equation utt - uxx ＋（uxx＋ f（u））xx - αuxxt = 0. First we give the local existence of weak solution and smooth solution. Then by using potential well method and convexity method we prove the global existence and finite time blow up of solution, then we obtain some sharp conditions for the well-posedness problem.

关键词： generalized Boussinesq equation damping Cauchy problem global existence blow up

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Some new exact solutions of Jacobian elliptic function about the generalized Boussinesq equation and boussinesq-Burgers equation

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Chinese Physics B 2008年第2期17卷 403-410页

作者：张亮张立凤李崇银 Institute of Meteorology PLA University of Science and TechnologyLaboratory of Applied Geophysics State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics Institute of Atmospheric PhysicsChinese Academy of Sciences

By using the modified mapping method, we find some new exact solutions of the generalized Boussinesq equation and the boussinesq-Burgers equation. The solutions obtained in this paper include Jacobian elliptic functio... 详细信息

关键词： generalized Boussinesq equation boussinesq-Burgers equation Jacobian elliptic function modified mapping method

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Multi-symplectic method for generalized Boussinesq equation

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Applied Mathematics and Mechanics(English Edition) 2008年第7期29卷 927-932页

作者：胡伟鹏邓子辰 School of Mechanics Civil Engineering and ArchitectureNorthwestern Polytechnical University State Key Laboratory of Structural Analysis of Industrial Equipment Dalian University of TechnologyDalian 116023Liaoning ProvinceP.R.China

The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.

关键词： generalized Boussinesq equation multi-symplectic method soliton solution conservation law

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