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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 *** dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in *** 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 *** 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 *** 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 *** paper is further based on the idea of the homogenization *** 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 *** influ

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

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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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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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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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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 *** get that solutions by using solitary wave ansatz in terms of tanh^(p) *** velocity and the free parameters are the physical parameters in the soliton *** can be obtained as functions of the dependent model *** domain restriction were also identified in the *** 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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KAM tori for generalized boussinesq equation

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Journal of Southeast University(English Edition) 2015年第1期31卷 157-162页

作者：石艳玲陆雪竹东南大学数学系南京211189 盐城工学院基础部盐城224003

One-dimensional generalized boussinesq equation u tt-u xx＋（f（u）＋u xx）xx=*** periodic boundary condition is considered, where f（u） = u3. First, the above equation is written as a Hamiltonian system, and then by choosing the eigenfunctions of the linear operator as bases, the Hamiltonian system in the coordinates is expressed. Because of the intricate resonance between the tangential frequencies and normal frequencies, some quasi-periodic solutions with special structures are considered. Secondly, the regularity of the Hamiltonian vector field is verified and then the fourth-order terms are normalized. By the Birkhoff normal form, the non- degeneracy and non-resonance conditions are obtained. Applying the infinite dimensional Kolmogorov-Arnold-Moser （KAM） theorem, the existence of finite dimensional invariant tori for the equivalent Hamiltonian system is proved. Hence many small-amplitude quasi-periodic solutions for the above equation are obtained.

关键词： generalized boussinesq equation quasi-periodicsolution Hamiltonian system invariant toil

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Dynamics and Exact Solutions of (1 + 1)-Dimensional generalized boussinesq equation with Time-Space Dispersion Term

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Journal of Applied Mathematics and Physics 2024年第8期12卷 2723-2737页

作者： Dahe Feng Jibin Li Jianjun Jiao School of Mathematics and Statistics Guizhou University of Finance and Economics Guiyang China School of Mathematical Sciences Huaqiao University Quanzhou China

We study exact solutions to (1 + 1)-dimensional generalized boussinesq equation with time-space dispersion term by making use of improved sub-equation method, and analyse the dynamical behavior and exact solutions of the sub-equation after constructing the nonlinear transformation and constraint conditions. Accordingly, we obtain twenty families of exact solutions such as analytical and singular solitons and singular periodic waves. In addition, we discuss the impact of system parameters on wave propagation.

关键词： generalized boussinesq equation Improved Sub-equation Method Bifurcation Soliton Solution Periodic Solution

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A new model for algebraic Rossby solitary waves in rotation fluid and its solution

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Chinese Physics B 2015年第9期24卷 54-61页

作者：陈耀登杨红卫高玉芳尹宝树冯兴如 Key Laboratory of Meteorological Disaster of Ministry of Education Nanjing University of Information Science & Technology Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters Nanjing University of Information Science & Technology College of Mathematics and Systems Science Shandong University of Science and Technology Institute of Oceanology Chinese Academy of Sciences Key Laboratory of Ocean Circulation and Wave Chinese Academy of Sciences

A generalized boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and *** this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves,the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves,the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon.

关键词： generalized boussinesq equation algebraic Rossby solitary waves dissipation effect solitary waves fission

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Lienard equation and Exact Solutions for Some Soliton-Producing Nonlinearequations

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Communications in Theoretical Physics 2004年第6期41卷 849-858页

作者： ZHANGWei-Guo CHANGQian-Shun ZHANGQi-Ren DepartmentofBasicSciences UniversityofShanghaiforScienceandTechnologyShanghai200093~China InstituteofAppliedMathematics AcademyofMathematicsandSystemsSciencestheChineseAcademyofSciencesBeijing100080China

In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.

关键词： solitary wave Lienard equation compound KdV equation compound KdV-Burgers equation generalized boussinesq equation generalized KP equation Ginzburg-Landau equation

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A Modified Homogeneous Balance Method and Its Applications

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Communications in Theoretical Physics 2011年第8期56卷 223-227页

作者：刘春平 School of Mathematical Science Yangzhou University

A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance *** equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance *** boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour *** approach is also applicable to a large variety of nonlinear evolution equations.

关键词： homogeneous balance method bilinear equation generalized boussinesq equation KP equation mKdV equation

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