The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well ...The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well method,and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function.Meanwhile,blow-up of solutions in the unstable set is also obtained.展开更多
In this paper,the wave equation defined in a semi-infinite cylinder is considered,in which the nonlinear damping and source terms is included.By setting an arbitrary parameter greater than zero in the energy expressio...In this paper,the wave equation defined in a semi-infinite cylinder is considered,in which the nonlinear damping and source terms is included.By setting an arbitrary parameter greater than zero in the energy expression,the fast growth rate or decay rate of the solution with spatial variables is obtained by using energy analysis method and differential inequality technique.Secondly,we obtain the asymptotic behavior of the solution on the external domain of the sphere.In addition,in this paper we also give some useful remarks which show that our results can be extended to more models.展开更多
Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronizatio...Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.展开更多
In this paper,we consider two Cauchy systems of coupled two wave equations in the whole line R under one or two frictional dampings,where the coupling terms are either of order one with respect to the time variable or...In this paper,we consider two Cauchy systems of coupled two wave equations in the whole line R under one or two frictional dampings,where the coupling terms are either of order one with respect to the time variable or of order two with respect to the space variable.We prove some L^(2)(R)-norm decay estimates of solutions and their higher-order derivatives with respect to the space variable,where the decay rates depend on the number of the present frictional dampings,the regularity of the initial data,and some connections between the speeds of wave propagation of the two wave equations.Both our systems are considered under weaker conditions on the coefficients than the ones considered in the literature and they include the case where only one frictional damping is present,so they generate new difficulties and represent new situations that have not been studied earlier.展开更多
The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-eleme...The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.展开更多
In this paper the decay of global solutions to some nonlinear dissipative wave equations are discussed, which based on the method of prior estimate technique and a differenece inequality.
This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite tim...This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite time to the mixed problems with respect to Neumann boundary and Dirichlet boundary for various nonlinear conditions and initial value conditions which usually meet.展开更多
From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that ...From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.展开更多
The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = ...The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.展开更多
Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to n...Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.展开更多
The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions a...The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions are obtained by using multiplier techniques to establish identity ddtE（t）＋F（t）=0 and skillfully selecting f（t）,g（t）,h（t）when the initial data have a compact support.Using the similar method,the Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│＋t）-1 and a nonlinearity │u│p-1u is studied,similar solutions are obtained when the initial data have a compact support.展开更多
In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models...In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models. Approaches to develop the stable formulas which are of 2M-order accuracy in both time and space with Mbeing a positive integer for regular grids are discussed and illustrated by constructing the second order （M= 1） and the fourth order （M = 2） recursion formulas.展开更多
In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are...In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.展开更多
This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n ＋ ： u tt u ＋ u t ＋ divf （u） = 0, t 〉 0, x = （x...This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n ＋ ： u tt u ＋ u t ＋ divf （u） = 0, t 〉 0, x = （x 1 , x ′ ） ∈ R n ＋ ：= R ＋ × R n 1 , u（0, x） = u 0 （x） → u ＋ , as x 1 → ＋ ∞ , u t （0, x） = u 1 （x）, u（t, 0, x ′ ） = u b , x ′ = （x 2 , x 3 , ··· , x n ） ∈ R n 1 . （I） For the non-degenerate case f ′ 1 （u ＋ ） 〈 0, it was shown in [10] that the above initialboundary value problem （I） admits a unique global solution u（t, x） which converges to the corresponding planar stationary wave φ（x 1 ） uniformly in x 1 ∈ R ＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u（t, x） for t → ＋ ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ （u） 〈 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 （u b ） ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.展开更多
The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonli...The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the abovementioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.展开更多
The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained...The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.展开更多
In the present paper, we investigate the well-posedness of the global solutionfor the Cauchy problem of generalized long-short wave equations. Applying Kato＇s methodfor abstract quasi-linear evolution equations and a...In the present paper, we investigate the well-posedness of the global solutionfor the Cauchy problem of generalized long-short wave equations. Applying Kato＇s methodfor abstract quasi-linear evolution equations and a priori estimates of solution,we get theexistence of globally smooth solution.展开更多
The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is ...The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger （MNLS） equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.展开更多
基金This research was supported by the Natural Science Foundation of Zhejiang Province(Grant No.LY17A010009).
文摘The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied.The existence of global solutions for this problem is proved by using potential well method,and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function.Meanwhile,blow-up of solutions in the unstable set is also obtained.
基金Supported by Innovation Team Project of Humanities and Social Sciences in Colleges and Universities of Guangdong Province(Grant No.2020WCXTD008)Natural Sciences Key Projects of Universities in Guangdong Province(Grant No.2019KZDXM042)。
文摘In this paper,the wave equation defined in a semi-infinite cylinder is considered,in which the nonlinear damping and source terms is included.By setting an arbitrary parameter greater than zero in the energy expression,the fast growth rate or decay rate of the solution with spatial variables is obtained by using energy analysis method and differential inequality technique.Secondly,we obtain the asymptotic behavior of the solution on the external domain of the sphere.In addition,in this paper we also give some useful remarks which show that our results can be extended to more models.
文摘Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.
文摘In this paper,we consider two Cauchy systems of coupled two wave equations in the whole line R under one or two frictional dampings,where the coupling terms are either of order one with respect to the time variable or of order two with respect to the space variable.We prove some L^(2)(R)-norm decay estimates of solutions and their higher-order derivatives with respect to the space variable,where the decay rates depend on the number of the present frictional dampings,the regularity of the initial data,and some connections between the speeds of wave propagation of the two wave equations.Both our systems are considered under weaker conditions on the coefficients than the ones considered in the literature and they include the case where only one frictional damping is present,so they generate new difficulties and represent new situations that have not been studied earlier.
基金sponsored by the National Natural Science Foundation of China Research(Grant No.41274138)the Science Foundation of China University of Petroleum(Beijing)(No.KYJJ2012-05-02)
文摘The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.
文摘In this paper the decay of global solutions to some nonlinear dissipative wave equations are discussed, which based on the method of prior estimate technique and a differenece inequality.
文摘This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite time to the mixed problems with respect to Neumann boundary and Dirichlet boundary for various nonlinear conditions and initial value conditions which usually meet.
文摘From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10447007 and 10671156Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘The invariant sets and exact solutions of the （1 ＋ 2）-dimensional wave equations are discussed. It is shown that there exist a class of solutions to the equations which belong to the invariant set E0 = {u ： ux = vxF（u）,uy = vyF（u） }. This approach is also developed to solve （1 ＋ N）-dimensional wave equations.
文摘Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.
基金The National Natural Science Foundation of China(No.10771032)
文摘The Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│）-1 and a nonlinearity │u│p-1u is studied.The total energy decay estimates of the global solutions are obtained by using multiplier techniques to establish identity ddtE（t）＋F（t）=0 and skillfully selecting f（t）,g（t）,h（t）when the initial data have a compact support.Using the similar method,the Cauchy problem for the nonlinear wave equation with a critical potential type of damping coefficient（1＋│x│＋t）-1 and a nonlinearity │u│p-1u is studied,similar solutions are obtained when the initial data have a compact support.
基金National Basic Research Program of China Under Grant No. 2007CB714200National Natural Science Foundation of China Under Grant No. 90715038
文摘In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models. Approaches to develop the stable formulas which are of 2M-order accuracy in both time and space with Mbeing a positive integer for regular grids are discussed and illustrated by constructing the second order （M= 1） and the fourth order （M = 2） recursion formulas.
基金Project supported by the Anhui Key Laboratory of Information Materials and Devices (Anhui University),China
文摘In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.
基金The research of Fan Lili was supported by two grants from the National Natural Science Foundation of China (10871151 10925103)+1 种基金the research of Liu Hongxia was supported by National Natural Science Foundation of China (10871082)the research of Yin Hui was supported by National Natural Sciences Foundation of China (10901064)
文摘This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n ＋ ： u tt u ＋ u t ＋ divf （u） = 0, t 〉 0, x = （x 1 , x ′ ） ∈ R n ＋ ：= R ＋ × R n 1 , u（0, x） = u 0 （x） → u ＋ , as x 1 → ＋ ∞ , u t （0, x） = u 1 （x）, u（t, 0, x ′ ） = u b , x ′ = （x 2 , x 3 , ··· , x n ） ∈ R n 1 . （I） For the non-degenerate case f ′ 1 （u ＋ ） 〈 0, it was shown in [10] that the above initialboundary value problem （I） admits a unique global solution u（t, x） which converges to the corresponding planar stationary wave φ（x 1 ） uniformly in x 1 ∈ R ＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u（t, x） for t → ＋ ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ （u） 〈 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 （u b ） ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.
基金Supported by the National Natural Science Foundation of China(10371073)
文摘The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the abovementioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.
文摘The generalized conditional symmetry approach is applied to study the variable separation of the extended wave equations. Complete classification of those equations admitting functional separable solutions is obtained and exact separable solutions to some of the resulting equations are constructed.
文摘In the present paper, we investigate the well-posedness of the global solutionfor the Cauchy problem of generalized long-short wave equations. Applying Kato＇s methodfor abstract quasi-linear evolution equations and a priori estimates of solution,we get theexistence of globally smooth solution.
基金supported by the Central Public-Interest Scientific Institution Basal Research Fund of China(Grant No.TKS100108)
文摘The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger （MNLS） equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.
