In this paper, oscillatory properties of solutions of certain nonlinear hyperbolic partial differential equations are investigated and a series of sufficient conditions for oscillations of the equations are establishe...In this paper, oscillatory properties of solutions of certain nonlinear hyperbolic partial differential equations are investigated and a series of sufficient conditions for oscillations of the equations are established. The results fully indicate that the oscillations are caused by delay.展开更多
A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations...A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.展开更多
Making use of a new generalized ans?tze 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 ans?tze 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 extended tanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.展开更多
In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differen...In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.展开更多
In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of cl...In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.展开更多
Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-co...Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-complex special solutions which can be made up of hyperbolicfunctions or elliptic functions.展开更多
A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear tran...A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+ 1)-dimensional GBK system to a simple linear evolution equation. Solving this equation,we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.展开更多
In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its...In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.展开更多
A new class of second order accuracy semidiscrete difference schemes is presented for the two-dimensional nonlinear scalar hyperbolic conservation laws. It is based on flux splitting, piecewise linear cell-averaged re...A new class of second order accuracy semidiscrete difference schemes is presented for the two-dimensional nonlinear scalar hyperbolic conservation laws. It is based on flux splitting, piecewise linear cell-averaged reconstruction and upwind property in the spatial discretization. By using TVD Runge-Kutta time discretization method, the full discrete scheme is obtained and its MmB property is proved. The extension to the two-dimensionalnonlinear hyperbolic conservation law systems is straightforward by using component-wise manner. The main advantage is simple: no Riemann problem is solved, and so field-by-field decomposition is avoided and the complicated computation is reduced. Numerical results of two-dimensional Euler equations of compressible gas dynamics verify the accuracy and robustness of the method.展开更多
In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equati...In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.展开更多
A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value probl...A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.展开更多
This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = ...This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.展开更多
Many complete extremal surfaces of mixed type are constructed with explicit expressions. Itis proved that there exist complete extremal surfaces of mixed type which have a given numberof time-like spans and a given nu...Many complete extremal surfaces of mixed type are constructed with explicit expressions. Itis proved that there exist complete extremal surfaces of mixed type which have a given numberof time-like spans and a given number of annular ends.展开更多
The authors prove some global existence results for equations of Kirchhoff type, i.e., nonlinearstretched string with nonlocal terms, depending on a parameter. This general setting includes the known results on the Ki...The authors prove some global existence results for equations of Kirchhoff type, i.e., nonlinearstretched string with nonlocal terms, depending on a parameter. This general setting includes the known results on the Kirchhoff equation with small data. Moreover, the authors can also handle some cases of degeneracy, which escaped earlier methods.展开更多
The author considers the Klien-Gordon equations utt --△u+μu= f(u) (P >0, |f(u)| ≤ c|u|α+1 ). The necessary and sufficient condition of existence of global solutions is obtained for f(s)dsdx <d (d is the give...The author considers the Klien-Gordon equations utt --△u+μu= f(u) (P >0, |f(u)| ≤ c|u|α+1 ). The necessary and sufficient condition of existence of global solutions is obtained for f(s)dsdx <d (d is the given constant).展开更多
The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidski...The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidskiǐ inequalities.An elementary proof of the latter for hyperbolic polynomials is given.This proof follows an idea from H.Weinberger and is free from representation theory and Schubert calculus arguments,as well as from hyperbolic partial differential equations theory.展开更多
文摘In this paper, oscillatory properties of solutions of certain nonlinear hyperbolic partial differential equations are investigated and a series of sufficient conditions for oscillations of the equations are established. The results fully indicate that the oscillations are caused by delay.
基金supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province
文摘A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.
文摘Making use of a new generalized ans?tze 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 extended tanh-function method and other more sophisticated methods. More importantly, for some equations, we also obtain other new and more general solutions at the same time. The results include kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, periodic wave solutions, rational solutions, singular solutions and new formal solutions.
基金Supported by the Natural Science Foundation of China(10471086)Supported by the Science Research Foundation of Department of Education of Hunan Province(07C164)
文摘In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.
文摘In this paper we investigate the formation of singularities of hyperbolic systems.Employing the method of parametric coordinates and the existence of the solution of the blow-up system, we prove that the blow-up of classic solutions is due to the envelope of characteristics of the same family, analyze the geometric properties of the envelope of characteristics and estimate the blowup rates of the solution precisely.
文摘Through Pickering's and extended Painleve nonstandard truncated expansionmethod, this paper solves the phase-separating dynamics equation of diblock copolymer, and obtainsvarious exact solutions. We discuss non-complex special solutions which can be made up of hyperbolicfunctions or elliptic functions.
文摘A nonlinear transformation and some multi-solition solutions for the (2+1 )-dimensional generalized Broer-Kaup (GBK) system is first given by using the homogeneous balance method. Then starting from the nonlinear transformation, we reduce the (2+ 1)-dimensional GBK system to a simple linear evolution equation. Solving this equation,we can obtain some new explicit exact solutions of the original equations by means of the extended hyperbola function method.
基金The project partially supported by National Natural Science Foundation of China
文摘In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.
文摘A new class of second order accuracy semidiscrete difference schemes is presented for the two-dimensional nonlinear scalar hyperbolic conservation laws. It is based on flux splitting, piecewise linear cell-averaged reconstruction and upwind property in the spatial discretization. By using TVD Runge-Kutta time discretization method, the full discrete scheme is obtained and its MmB property is proved. The extension to the two-dimensionalnonlinear hyperbolic conservation law systems is straightforward by using component-wise manner. The main advantage is simple: no Riemann problem is solved, and so field-by-field decomposition is avoided and the complicated computation is reduced. Numerical results of two-dimensional Euler equations of compressible gas dynamics verify the accuracy and robustness of the method.
文摘In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.
基金The project is supported by the National Natural Science Foundation of China(10071048)
文摘A class of nonlocal singnlarly perturbed problems for the hyperbolic dif-ferential equation are considered. Under snitable conditions, we discuss the asymptoticbehavior of solution for the initial boundary value problems.
文摘This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.
文摘Many complete extremal surfaces of mixed type are constructed with explicit expressions. Itis proved that there exist complete extremal surfaces of mixed type which have a given numberof time-like spans and a given number of annular ends.
文摘The authors prove some global existence results for equations of Kirchhoff type, i.e., nonlinearstretched string with nonlocal terms, depending on a parameter. This general setting includes the known results on the Kirchhoff equation with small data. Moreover, the authors can also handle some cases of degeneracy, which escaped earlier methods.
文摘The author considers the Klien-Gordon equations utt --△u+μu= f(u) (P >0, |f(u)| ≤ c|u|α+1 ). The necessary and sufficient condition of existence of global solutions is obtained for f(s)dsdx <d (d is the given constant).
文摘The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidskiǐ inequalities.An elementary proof of the latter for hyperbolic polynomials is given.This proof follows an idea from H.Weinberger and is free from representation theory and Schubert calculus arguments,as well as from hyperbolic partial differential equations theory.
