In this paper, we shall show that the Hamiltonian structure can be defined for any nonlinear evolution equations which describe surfaces of a constant negative curvature, so that the densities of conservation laws can...In this paper, we shall show that the Hamiltonian structure can be defined for any nonlinear evolution equations which describe surfaces of a constant negative curvature, so that the densities of conservation laws can be considered as corresponding Hamiltonians. This paper obtains the soliton solution and conserved quantities of a new fifth-order nonlinear evolution equation by the aid of inverse scattering method.展开更多
This paper studies a generalized nonlinear evolution equation. Using the homotopic mapping method, it constructs a corresponding homotopic mapping transform. Selecting a suitable initial approximation and using homoto...This paper studies a generalized nonlinear evolution equation. Using the homotopic mapping method, it constructs a corresponding homotopic mapping transform. Selecting a suitable initial approximation and using homotopic mapping, it obtains an approximate solution with an arbitrary degree of accuracy for the solitary wave. From the approximate solution obtained by using the homotopic mapping method, it possesses a good accuracy.展开更多
The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parab...The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations (PSE). Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results.展开更多
A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the pa...A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.展开更多
This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx+(ζθ)x+aθxx,ζt=-θ...This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx+(ζθ)x+aθxx,ζt=-θx+βζxx;with initial data and end states (ζθ)(x,0)=(ζ0,θ0)(x)→(ζ±,θ±)as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.展开更多
To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are pr...To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.展开更多
In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local exist...In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.展开更多
For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numer...For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.展开更多
A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while t...A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1_st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example, namely, the two_dimensional Navier_Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.展开更多
In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution ...In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.展开更多
Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equati...Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations.As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extendedtanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain othernew and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profilesolitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.展开更多
With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a res...With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.展开更多
In this paper, we establish sufficient conditions for the controllability of nonlinear neutral evolution equations with nonlocal conditions. The result is obtained by using Krasnoselski-Schaefer type fixed point theorem.
This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measur...This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.展开更多
In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new typ...In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.展开更多
The electron movement based on the multi-photon nonlinear Compton scattering with the extra-intense stationary laser field is discussed by using KMR (Kroll-Morton-Rosenbluth) theory.We find that there exists only an e...The electron movement based on the multi-photon nonlinear Compton scattering with the extra-intense stationary laser field is discussed by using KMR (Kroll-Morton-Rosenbluth) theory.We find that there exists only an evolution from periodicity to non-periodicity of the un-captured electron phase orbits after the energy exchange between the electron beam and laser fields.With the increase of the absorbed photon number n by an electron, this evolution will be more and more faster, while it is rapidly decreased with the enhancement of the collision non-flexibility ξ of the electrons and photons; When the electrons are captured by the laser fields, the evolution is finished, the electrons will stably transport,and the photons dont give up the energy to these electrons.展开更多
The impacts of initial perturbations on the computational stability of nonlinear evolution equations for non-conservative difference schemes and non-periodic boundary conditions are studied through theoretical analysi...The impacts of initial perturbations on the computational stability of nonlinear evolution equations for non-conservative difference schemes and non-periodic boundary conditions are studied through theoretical analysis and numerical experiments for the case of onedimensional equations.The sensitivity of the difference scheme to initial values is further analyzed.The results show that the computational stability primarily depends on the form of the initial values if the difference scheme and boundary conditions are determined.Thus,the computational stability is sensitive to the initial perturbations.展开更多
文摘In this paper, we shall show that the Hamiltonian structure can be defined for any nonlinear evolution equations which describe surfaces of a constant negative curvature, so that the densities of conservation laws can be considered as corresponding Hamiltonians. This paper obtains the soliton solution and conserved quantities of a new fifth-order nonlinear evolution equation by the aid of inverse scattering method.
基金supported by the National Natural Science Foundation of China(Grant Nos 40676016 and 40876010)the Knowledge Innovation Project of Chinese Academy of Sciences(Grant No KZCX2-YW-Q03-08)+1 种基金LASG State Key Laboratory Special fundE-Institutes of Shanghai Municipal Education Commission of China(Grant No E03004)
文摘This paper studies a generalized nonlinear evolution equation. Using the homotopic mapping method, it constructs a corresponding homotopic mapping transform. Selecting a suitable initial approximation and using homotopic mapping, it obtains an approximate solution with an arbitrary degree of accuracy for the solitary wave. From the approximate solution obtained by using the homotopic mapping method, it possesses a good accuracy.
文摘The process of evolution, especially that of nonlinear evolution, of C-type instability of laminar-turbulent flow transition in nonparallel boundary layers are studied by means of a newly developed method called parabolic stability equations (PSE). Initial conditions, which are very important for the nonlinear problem, are investigated by computing initial solution of the harmonic waves, modifying the mean-flow-distortion, and giving initial value of TS wave and its subharmonic waves at initial station by solving linear PSE. A numerical method with high-order accuracy are developed in the text, the key normalization conditions in the PSE are satisfied, and nonlinear PSE are solved efficiently and implemented stably by the spatial marching. It has been shown that the computed process of nonlinear evolution of C-type instability in Blasius boundary layer is in good agreement with the experimental results.
基金Supported by the Postdoctoral Science Foundation of ChinaChinese Basic Research Plan"MathematicsMechanization and A Platform
文摘A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.
基金supported by the Natural Science Foundation of China(11001095)the Ph.D.specialized grant of the Ministry of Education of China(20100144110001)+2 种基金the Special Fund for Basic Scientific Research of Central Colleges(CCNU12C01001)supported by the Fundamental Research Funds for the Central Universities(2015IA009)the Natural Science Foundation of China(61573012)
文摘This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx+(ζθ)x+aθxx,ζt=-θx+βζxx;with initial data and end states (ζθ)(x,0)=(ζ0,θ0)(x)→(ζ±,θ±)as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.
基金supported by the National Natural Science Foundation of China(Grant No.10862003)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZZ07031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2010MS0111)
文摘To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.
基金supported by a grant from the National High Technology Researchand and Development Program of China (863 Program) (2009AA044501)by NSFC (10776035+2 种基金10771085)by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationby the 985 program of Jilin University
文摘In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.
基金supported by the project"Global Changefor Regional Response"of the Important Study Project of the National Natural Science Foundation of China (Grant No.902110041)the Key Innovation Project of the Chinese Academy of Sciences (KZCX3-SW-213).
文摘For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.
文摘A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0_th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1_st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example, namely, the two_dimensional Navier_Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11675084 and 11435005)the Fund from the Educational Commission of Zhejiang Province,China(Grant No.Y201737177)+1 种基金Ningbo Natural Science Foundation(Grant No.2015A610159)the K C Wong Magna Fund in Ningbo University
文摘In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.
文摘Making use of a new generalized ansatze and a proper transformation, we generalized the extended tanh-function method. Applying the generalized method with the aid of Maple, we consider some nonlinear evolution equations.As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extendedtanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain othernew and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profilesolitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.
基金The project supported by the National Key Basic Research Development Project Program under Grant No.G1998030600
the Foundation of Liaoning Normal University
文摘With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.
文摘In this paper, we establish sufficient conditions for the controllability of nonlinear neutral evolution equations with nonlocal conditions. The result is obtained by using Krasnoselski-Schaefer type fixed point theorem.
基金supported by the National Natural Science Foundation of China(11471230,11671282)。
文摘This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.
文摘The electron movement based on the multi-photon nonlinear Compton scattering with the extra-intense stationary laser field is discussed by using KMR (Kroll-Morton-Rosenbluth) theory.We find that there exists only an evolution from periodicity to non-periodicity of the un-captured electron phase orbits after the energy exchange between the electron beam and laser fields.With the increase of the absorbed photon number n by an electron, this evolution will be more and more faster, while it is rapidly decreased with the enhancement of the collision non-flexibility ξ of the electrons and photons; When the electrons are captured by the laser fields, the evolution is finished, the electrons will stably transport,and the photons dont give up the energy to these electrons.
基金supported by the"Strategic Priority Research Program-Climate Change:Carbon Budget and Relevant Issues"of the Chinese Academy of Sciences (Grant No.XDA01020304)
文摘The impacts of initial perturbations on the computational stability of nonlinear evolution equations for non-conservative difference schemes and non-periodic boundary conditions are studied through theoretical analysis and numerical experiments for the case of onedimensional equations.The sensitivity of the difference scheme to initial values is further analyzed.The results show that the computational stability primarily depends on the form of the initial values if the difference scheme and boundary conditions are determined.Thus,the computational stability is sensitive to the initial perturbations.