In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation i...In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L2, L∞ and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.展开更多
A Fourier spectral scheme is proposed for solving the periodic problem of nonlinear Klein-Gordon equation. Its stability and convergence are investigated. Numerical results are also presented.
A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and d...A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and demonstrate the e?ciency of this approach.展开更多
In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve ...In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.展开更多
By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gord...By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.展开更多
In terms of the solutions of the generalized Riccati equation,a new algebraic method,which contains theterms of radical expression of functions f(ξ),is constructed to explore the new exact solutions for nonlinear evo...In terms of the solutions of the generalized Riccati equation,a new algebraic method,which contains theterms of radical expression of functions f(ξ),is constructed to explore the new exact solutions for nonlinear evolutionequations.Being concise and straightforward,the method is applied to nonlinear Klein Gordon equation,and some newexact solutions of the system are obtained.The method is of important significance in exploring exact solutions for othernonlinear evolution equations.展开更多
We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Co...We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.展开更多
In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existe...In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.展开更多
The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid...The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.展开更多
The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference oper...The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.展开更多
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti...In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.展开更多
The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple ba...The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.展开更多
In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infin...In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.展开更多
New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's pa...New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's parameters and travelling wave transformation parameters. Some figures for a specific kind of solution are also presented.展开更多
A Legendre pseudospectral scheme is proposed for solving initial-boundary value problem of nonlinear Klein-Gordon equation. The numerical solution keeps the discrete conservation. Its stability and convergence are inv...A Legendre pseudospectral scheme is proposed for solving initial-boundary value problem of nonlinear Klein-Gordon equation. The numerical solution keeps the discrete conservation. Its stability and convergence are investigated. Numerical results are also presented, which show the high accuracy. The technique in the theoretical analysis provides a framework for Legendre pseudospectral approximation of nonlinear multi-dimensional problems.展开更多
This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p + q 〈 4/N. By using the variational calculus and scaling argumen...This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p + q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.展开更多
A generalized nonlinear disturbed Klein-Gordon equation is studied. Using the homotopic mapping method, the corresponding homotopic mapping is constructed. A suitable initial approximation is selected, and an arbitrar...A generalized nonlinear disturbed Klein-Gordon equation is studied. Using the homotopic mapping method, the corresponding homotopic mapping is constructed. A suitable initial approximation is selected, and an arbitrary-order approximate solution to the soliton is calculated: A weakly disturbed equation is also studied.展开更多
An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-aga...An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-agation in a bar supported on a spring foundation.The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure.The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation.For each term,a corresponding substructure composed of dampers and masses is constructed.Finally,the equivalent mechan-ical structure is obtained by parallelly connecting these substructures.The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements.Numerical examples show that with just a few extra degrees of freedom,the proposed approach effec-tively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.展开更多
The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently at...The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently attained in[1]for the one-dimensional case.More precisely,as the coefficient a(x)in the nonlinearity is sign-changing,the modified term 2(Δ(|u|2))u is still helpful for obtaining multiple positive solutions in a high dimension,even if a sign condition like∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold.展开更多
We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(R...We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.展开更多
文摘In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The L2, L∞ and Root-Mean-Square (RMS) errors in the solutions show the efficiency of the method computationally.
文摘A Fourier spectral scheme is proposed for solving the periodic problem of nonlinear Klein-Gordon equation. Its stability and convergence are investigated. Numerical results are also presented.
基金This work is supported in part by NSF of China, N.10471095, SF of Shanghai N.04JC14062, The Fund of ChineseEducation Ministry N.20040270002, The Shanghai Leading Academic Discipline Project N. T0401, The Funds forE-institutes of Universities N.E03004 and The special Funds for Major Specialities and N.04DB15 of ShanghaiEducation Commission.
文摘A Legendre rational spectral method is proposed for the nonlinear Klein-Gordon equation on the whole line. Its stability and convergence are proved. Numerical results coincides well with the theoretical analysis and demonstrate the e?ciency of this approach.
基金supported in part by NSF of China N.10871131The Science and Technology Commission of Shanghai Municipality,Grant N.075105118+1 种基金Shanghai Leading Academic Discipline Project N.T0401Fund for E-institute of Shanghai Universities N.E03004.
文摘In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.
文摘By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.
基金the Science and Technology Foundation of Guizhou Province of China under Grant No.20072009
文摘In terms of the solutions of the generalized Riccati equation,a new algebraic method,which contains theterms of radical expression of functions f(ξ),is constructed to explore the new exact solutions for nonlinear evolutionequations.Being concise and straightforward,the method is applied to nonlinear Klein Gordon equation,and some newexact solutions of the system are obtained.The method is of important significance in exploring exact solutions for othernonlinear evolution equations.
基金supported by the NSFC(12261044)the STP of Education Department of Jiangxi Province of China(GJJ210302)。
文摘We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.
基金supported by the National Natural Science Foundation of China(12071491,12001113)。
文摘In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.
文摘The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.
文摘The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L<sub>2</sub>-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.
文摘In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.
文摘The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.
文摘In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.
文摘New exact solutions, expressed in terms of the Jacobi elliptic functions, to the nonlinear Klein-Gordon equation are obtained by using a modified mapping method. The solutions include the conditions for equation's parameters and travelling wave transformation parameters. Some figures for a specific kind of solution are also presented.
文摘A Legendre pseudospectral scheme is proposed for solving initial-boundary value problem of nonlinear Klein-Gordon equation. The numerical solution keeps the discrete conservation. Its stability and convergence are investigated. Numerical results are also presented, which show the high accuracy. The technique in the theoretical analysis provides a framework for Legendre pseudospectral approximation of nonlinear multi-dimensional problems.
基金Supported by the National Natural Science Foundation of China (No. 10771151, 10801102)Sichuan Youth Sciences and Technology Foundation(No. 07ZQ026-009)China Postdoctoral Science Foundation Funded Project
文摘This paper deals with the standing waves for a class of coupled nonlinear Klein-Gordon equations with space dimension N ≥ 3, 0 〈 p, q 〈 2/N-2 and p + q 〈 4/N. By using the variational calculus and scaling argument, we establish the existence of standing waves with ground state, discuss the behavior of standing waves as a function of the frequency ω and give the sufficient conditions of the stability of the standing waves with the least energy for the equations under study.
基金Project supported by the National Natural Science Foundation of China (No. 40876010)the Main Direction Program of the Knowledge Innovation Project of Chinese Academy of Sciences (No. KZCX2YW-Q03-08)+2 种基金the Research and Development Special Fund for Public Welfare Industry (Meteorology)(No. GYHY200806010)the LASG State Key Laboratory Special Fundthe Foundation of E-Institute of Shanghai Municipal Education Commission (No. E03004)
文摘A generalized nonlinear disturbed Klein-Gordon equation is studied. Using the homotopic mapping method, the corresponding homotopic mapping is constructed. A suitable initial approximation is selected, and an arbitrary-order approximate solution to the soliton is calculated: A weakly disturbed equation is also studied.
基金supported by the National Natural Science Foundation of China(Grant Nos.11832001 and 11702046).
文摘An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-agation in a bar supported on a spring foundation.The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure.The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation.For each term,a corresponding substructure composed of dampers and masses is constructed.Finally,the equivalent mechan-ical structure is obtained by parallelly connecting these substructures.The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements.Numerical examples show that with just a few extra degrees of freedom,the proposed approach effec-tively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.
基金Partially the NSFC(11671085)the Science Foundation of Fujian Province(2020J01160)。
文摘The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4<p<2·2*shall be investigated in a high dimension.The conclusions extend the relative results recently attained in[1]for the one-dimensional case.More precisely,as the coefficient a(x)in the nonlinearity is sign-changing,the modified term 2(Δ(|u|2))u is still helpful for obtaining multiple positive solutions in a high dimension,even if a sign condition like∫_(R)N a(x)e_(1)^(p)dx<0(also named“a necessary condition”see[2,3])does not hold.
基金supported by the National Key Research and Development Program of China (Grant Nos. 2021YFC2201803 and 2020YFC2200104)。
文摘We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schr ¨odinger equation(GNLSE): one is the conservation quantity error adaptive step-control method(RK4IP-CQE), and the other is the local error adaptive step-control method(RK4IP-LEM). The methods are developed in the vector form of fourthorder Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.