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THE SMOOTH AND NONSMOOTH TRAVELLING waveSOLUTIONS IN A nonlinear wave equation

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Applied Mathematics and Mechanics(English Edition) 2001年第11期22卷 1333-1343页

作者： LI Shu-min(李庶民) Institute of Science Kunming University of Science and Technology Kunming 650093 P R China

The travelling wave solutions (TWS) in a class of P. D. E. is studied. The travelling wave equation of this P. D. E. is a planar cubic polynomial system in three-parameter space. The study for M became the topological classifications of bifurcations of phase portraits defined by the planar system. B using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(xi), and considering discontinuity of the right side of the equation of TWS when xi = x - ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.

关键词： nonlinear wave equation solitary travelling wave periodic travelling wave dissmoothness of wave

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

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Applied Mathematics and Mechanics(English Edition) 2014年第1期35卷 117-132页

作者：李想张卫国李正明 Ji-bin LI Business School University of Shanghai for Science and Technology College of Science University of Shanghai for Science and Technology

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

关键词： nonlinear wave equation bounded traveling wave solution shape analysis,approximate damped oscillatory solution error estimate

On the Numerical Solution to a nonlinear wave equation Associated with the First Painlev equation:an Operator-Splitting Approach

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Chinese Annals of Mathematics,Series B 2013年第2期34卷 237-254页

作者： Roland GLOWINSKI Annalisa QUAINI Department of Mathematics University of Houston4800 Calhoun RdHouston(TX) 77204USA Laboratoire Jacques-Louis Lions Universit Pierre et Marie Curie4Place Jussieu75252 Paris CEDEX 05France

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevd transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang＇s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows：（i） The first Painlevd ordinary differential equation, （ii） a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather ＂violent＂ phenomenon.

关键词： Painlevd equation nonlinear wave equation Blow-up solution Operator-SDlitting

Specific wave profiles and closed-form soliton solutions for generalized nonlinear wave equation in(3+1)-dimensions with gas bubbles in hydrodynamics and fluids

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Journal of Ocean Engineering and Science 2023年第1期8卷 91-102页

作者： Sachin Kumar Ihsanullah Hamid M.A.Abdou Department of Mathematics Faculty of Mathematical SciencesUniversity of DelhiDelhi 110007India Department of Mathematics College of SciencesUniversity of BishaP.O.Box 344Bisha 61922Saudi Arabia Physics Department Faculty of ScienceMansoura UniversityMansoura 35516Egypt

nonlinear evolution equations(NLEEs)are frequently employed to determine the fundamental principles of natural *** equations are studied extensively in nonlinear sciences,ocean physics,fluid dynamics,plasma physics,scientific applications,and marine *** generalized exponen-tial rational function(GERF)technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in(3+1)dimensions,which explains several more nonlinear phenomena in liquids,including gas bubbles.A large number of closed-form wave solutions are generated,including trigonometric function solutions,hyper-bolic trigonometric function solutions,and exponential rational functional *** the dynamics of distinct solitary waves,a variety of soliton solutions are obtained,including single soliton,multi-wave structure soliton,kink-type soliton,combo singular soliton,and singularity-form wave *** de-termined solutions have never previously been *** dynamical wave structures of some analyt-ical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free *** technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics,fluid dynamics,and other fields of nonlinear sciences.

关键词： Closed-form solutions Dynamical wave patterns Analytic solutions nonlinear wave equation GERF Method Solitary waves Solitons

Bifurcation and solitary waves of the nonlinear wave equation with quartic polynomial potential

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Chinese Physics B 2002年第6期11卷 547-552页

作者：化存才刘延柱 DepartmentofEngineeringMechanics ShanghaiJiaotongUniversityShanghai200030China

For the nonlinear wave equation with quartic polynomial potential, bifurcation and solitary waves are investigated. Based on the bifurcation and the energy integral of the two-dimensional dynamical system satisfied by the travelling waves, it is very interesting to find different sufficient and necessary conditions in terms of the bifurcation parameter for the existence and coexistence of bright, dark solitary waves and shock waves. The method of direct integration is developed to give all types of solitary wave solutions. Our method is simpler than other newly developed ones. Some results are similar to those obtained recently for the combined KdV-mKdV equation.

关键词： bifurcation solitary waves nonlinear wave equation

GLOBAL NONEXISTENCE OF THE SOLUTIONS FOR A nonlinear wave equation WITH THE Q-LAPLACIAN OPERATOR

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Journal of Partial Differential equations 2007年第1期20卷 71-79页

作者： Gao Hongjun Zhang Hui Department of Mathematics and Institute of Mathematics Nanjing Normal University Nanjing 210097 Jiangsu China

We study the global nonexistence of the solutions of the nonlinear q-Laplacian wave equation utt-△qu＋（-△）^αut=｜u｜^p-2u, where 0 〈 α ≤ 1, 2 ≤ q 〈 p. We obtain that the solution blows up in finite time if the initial energy is negative. Meanwhile, we also get the solution blows up in finite time with suitable positive initial energy under some conditions.

关键词： q-Laplacian operator nonlinear wave equation global nonexistence.

EXACT SOLUTIONS TO nonlinear wave equation

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Annals of Differential equations 2011年第1期27卷 13-18页

作者： Huabing Jia,Enrang Yan(Dept. of Math.,Baoji University of Arts and Sciences,Baoji 721013,Shaanxi)

In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Severa... 详细信息

关键词： invariant set exact solution nonclassical symmetry nonlinear wave equation

A LOCAL ATTRACTOR FOR A nonlinear wave equation ARISING IN ELASTIC waveGUIDE MODEL

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Annals of Differential equations 2014年第3期30卷 333-339页

作者： Zhihua Song School of Math. and Information Science North China University of Water Resources and Electric Power

This paper studies the asymptotic behavior of solutions to the initial boundary value problem for a nonlinear wave equation arising in an elastic waveguide model. It proves that under rather mild conditions on g the r... 详细信息

关键词： nonlinear wave equation local attractor longtime behavior of solution

The Existence and Non-Existence of Global Solutions for a nonlinear wave equation

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Journal of Mathematical Research and Exposition 2009年第1期29卷 164-168页

作者： WANG Yan Ping Department of Mathematics and Physics Zhengzhou Institute of Aeronautical Industry Management Henan 450015 China

This paper deals with the existence and uniqueness of the global solution of the initial boundary value problem of a class of wave *** the meantime,it gives the sufficient conditions of blow-up of the solution for the... 详细信息

关键词： nonlinear wave equation initial boundary value problem global solution blow-up of solution.