In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the ...In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.展开更多
Research on seismic anisotropy and attenuation plays a significant role in exploration geophysics. To enhance the imaging quality for complicated structures, we develop several effective improvements for anisotropic a...Research on seismic anisotropy and attenuation plays a significant role in exploration geophysics. To enhance the imaging quality for complicated structures, we develop several effective improvements for anisotropic attenuation effects in reverse-time migration (Q-RTM) on surface and vertical seismic profiling (VSP) acquisition geometries. First, to suppress pseudo-shear wave artifact and numerical instability of the commonly used anisotropic pseudo-acoustic wave equations, an optimized pure P-wave dispersion relation is derived and the corresponding pure-mode wave equation is solved by combining the finite-difference and Possion methods. Second, a simplified anisotropic pure-mode visco-acoustic wave equation (PVAWE) based on standard linear solid model is established. Third, a time-dispersion correlation strategy is applied to improve the modeling accuracy. Fourth, we extend a target-oriented scheme to anisotropic attenuated modeling and imaging. Instead of the conventional wavefield modeling and RTM, the proposed approach can extract available wavefield information near the target regions and produce high imaging resolution for target structures. Last, both anisotropic surface and VSP Q-RTMs are executed by combining optimized PVAWE, time-dispersion correlation and target-oriented algorithm. Modeling examples demonstrate the advantages of our schemes. Moreover, our modified Q-compensated imaging workflow can be regarded as a supplement to the classical anisotropic RTM.展开更多
Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of t...Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.展开更多
The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the m...The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.展开更多
We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane ...We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.展开更多
A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly t...A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation.In view of this,based on the first-order CPML absorbing boundary condition,we propose a new CPML(NCPML)boundary which can be directly applied to the second-order wave equation.We first systematically extend the first-order CPML technique into second-order wave equations,neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain,avoiding the generation of convolution in the time domain.We then transform the technique back to the time domain through the inverse Fourier transform.Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time.A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation,and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary.展开更多
We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is...We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508).展开更多
An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variab...An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.展开更多
Kunio Hidano[4] has shown that the global and local C2-solutions for semilinear wave equations with spherical symmetry in three space dimensions. This paper studies the global and local C2-solutions for the semilinear...Kunio Hidano[4] has shown that the global and local C2-solutions for semilinear wave equations with spherical symmetry in three space dimensions. This paper studies the global and local C2-solutions for the semilinear wave equations without spherical symmetry in three space dimensions. A problem put forward by Hiroyuki Takamura[2] is partially answered.展开更多
This study develops an optimized finite difference iterative(OFDI) scheme for the two-dimensional(2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition(POD) method. It has...This study develops an optimized finite difference iterative(OFDI) scheme for the two-dimensional(2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition(POD) method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.展开更多
In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis o...In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.展开更多
With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.T...With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.展开更多
In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-...In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.展开更多
In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependen...In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.展开更多
This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the...This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.展开更多
This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity.Since the stress function is neither convex nor concave,the shock condition is degenerate.By introducing a degen...This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity.Since the stress function is neither convex nor concave,the shock condition is degenerate.By introducing a degenerate shock under the generalized shock condition,the global solutions are constructively obtained case by case.展开更多
This paper focuses on studying the symmetry of a practical wave equation on new lattices.It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogen...This paper focuses on studying the symmetry of a practical wave equation on new lattices.It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogeneous bar.The equation of motion of the bar can be changed into a discrete wave equation.With the new lattice equation,the translational and scaling invariant,not only is the infinitesimal transformation given,but the symmetry and Lie algebras are also calculated.We also give a new form of invariant called the ratio invariant,which can reduce the process of the computing invariant with the characteristic equation.展开更多
The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the ...The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the wave equation is determined by the viscoelastic media model.Equivalence relations exist between various frequency domain mathematical models and physical rheological models for the constant Q property.Considering two elastic moduli and three attenuation variables,24 kinds of wave equations based on diff erent generalized rheological models are divided into six classes in this study,and the 12 kinds of specifi c representation for the wave equations in the time domain are derived.On the basis of the equivalence relations between the generalized rheological models,the diff erence and equivalence relation between diff erent wave equations are proven and clarifi ed.Results show that the high-order generalized rheological model can accurately characterize the attenuation characteristics of seismic waves and has advantages in characterizing the dispersion characteristics in viscoelastic media.Lastly,the seismic refl ection characteristics caused by the diff erence of Q value are verifi ed by the forward modeling of the constant Q wave equation in this study,thereby providing a theoretical basis for the analysis and inversion of the formation Q value from refl ection seismic data.展开更多
This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of plana...This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefcients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.展开更多
In this paper we solve spin-weighted spheroidal wave equations through super-symmetric quantum mechanics with a different expression of the super-potential.We use the shape invariance property to compute the "exc...In this paper we solve spin-weighted spheroidal wave equations through super-symmetric quantum mechanics with a different expression of the super-potential.We use the shape invariance property to compute the "excited" eigenvalues and eigenfunctions.The results are beneficial to researchers for understanding the properties of the spin-weighted spheroidal wave more deeply,especially its integrability.展开更多
文摘In this paper,we establish global classical solutions of semilinear wave equations with small compact supported initial data posed on the product space R^(3)×T.The semilinear nonlinearity is assumed to be of the cubic form.The main ingredient here is the establishment of the L^(2)-L^(∞)decay estimates and the energy estimates for the linear problem,which are adapted to the wave equation on the product space.The proof is based on the Fourier mode decomposition of the solution with respect to the periodic direction,the scaling technique,and the combination of the decay estimates and the energy estimates.
基金supported by the National Key R&D Program of China(2021YFA0716902)National Natural Science Foundation of China(42004119,42174156)+1 种基金the Fundamental Research Funds for the Central Universities,CHD(300102261306)the National Engineering Research Center of Offshore Oil and Gas Exploration,No.6 Courtyard,Taiyanggong South Street,Chaoyang District,Beijing,100028.
文摘Research on seismic anisotropy and attenuation plays a significant role in exploration geophysics. To enhance the imaging quality for complicated structures, we develop several effective improvements for anisotropic attenuation effects in reverse-time migration (Q-RTM) on surface and vertical seismic profiling (VSP) acquisition geometries. First, to suppress pseudo-shear wave artifact and numerical instability of the commonly used anisotropic pseudo-acoustic wave equations, an optimized pure P-wave dispersion relation is derived and the corresponding pure-mode wave equation is solved by combining the finite-difference and Possion methods. Second, a simplified anisotropic pure-mode visco-acoustic wave equation (PVAWE) based on standard linear solid model is established. Third, a time-dispersion correlation strategy is applied to improve the modeling accuracy. Fourth, we extend a target-oriented scheme to anisotropic attenuated modeling and imaging. Instead of the conventional wavefield modeling and RTM, the proposed approach can extract available wavefield information near the target regions and produce high imaging resolution for target structures. Last, both anisotropic surface and VSP Q-RTMs are executed by combining optimized PVAWE, time-dispersion correlation and target-oriented algorithm. Modeling examples demonstrate the advantages of our schemes. Moreover, our modified Q-compensated imaging workflow can be regarded as a supplement to the classical anisotropic RTM.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11975156 and 12175148)。
文摘Two(3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion(CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new B?cklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.
基金supported by the National Natural Science Fundation of China (No. 11061021)the Science Research of Inner Mongolia Advanced Education (Nos. NJ10006, NJ10016, and NJZZ12011)the National Science Foundation of Inner Mongolia (Nos. 2011BS0102 and 2012MS0106)
文摘The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γh , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11361069 and 11775146).
文摘We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.
基金supported by the National Science and Technology Major Special Sub-project of China(No.2016ZX05024-001-008)the National Natural Science Foundation Joint Fund Prcject of China(No.U1562215).
文摘A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation.In view of this,based on the first-order CPML absorbing boundary condition,we propose a new CPML(NCPML)boundary which can be directly applied to the second-order wave equation.We first systematically extend the first-order CPML technique into second-order wave equations,neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain,avoiding the generation of convolution in the time domain.We then transform the technique back to the time domain through the inverse Fourier transform.Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time.A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation,and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary.
基金Supported by National Natural Science Foundation of China(11601122,11801145)。
文摘We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508).
基金Supported by the National Natural Science Fund(11061021)Supported by the Scientific Research Projection of Higher Schools of Inner Mongolia(NJZZ12011, NJ10006)+1 种基金Supported by the Program of Higher-level talents of Inner Mongolia University(125119)Supported by the Scientific Research Projection of Inner Mongolia University of Finance and Economics(KY1101)
文摘An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.
基金Supported by youth foundation of Sichuan province (1999-09)
文摘Kunio Hidano[4] has shown that the global and local C2-solutions for semilinear wave equations with spherical symmetry in three space dimensions. This paper studies the global and local C2-solutions for the semilinear wave equations without spherical symmetry in three space dimensions. A problem put forward by Hiroyuki Takamura[2] is partially answered.
基金Project supported by the National Natural Science Foundation of China(No.11671106)the Fundamental Research Funds for the Central Universities(No.2016MS33)
文摘This study develops an optimized finite difference iterative(OFDI) scheme for the two-dimensional(2D) viscoelastic wave equation. The OFDI scheme is obtained using a proper orthogonal decomposition(POD) method. It has sufficiently high accuracy with very few unknowns for the 2D viscoelastic wave equation. Existence, stability, and convergence of the OFDI solutions are analyzed. Numerical simulations verify efficiency and feasibility of the proposed scheme.
文摘In this article,we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions(n≥4).We use the Green’s function method.Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system.We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions.It is shown that the solution exhibits a generalized Huygens principle.
基金supported by the National Natural Science Foundation of China(grant number 11671081).
文摘With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.
基金Supported by the Natural Science Foundation of China(Grant No.11371175)Innovation Team Project in Colleges and Universities of Guangdong Province(Grant No.2020WCXTD008)+1 种基金Science Foundation of Huashang College Guangdong University of Finance&Economics(Grant No.2020HSDS01)Science Research Team Project in Guangzhou Huashang College(Grant No.2021HSKT01).
文摘In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.
文摘In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.
文摘This paper is devoted to the long time behavior of the solution to the initial boundary value problems for a class of the Kirchhoff wave equations with nonlinear strongly damped terms: . Firstly, in order to prove the smoothing effect of the solution, we make efficient use of the analytic property of the semigroup generated by the principal operator of the equation in the phase space. Then we obtain the regularity of the global attractor and construct the approximate inertial manifold of the equation. Finally, we prove that arbitrary trajectory of the Kirchhoff wave equations goes into a small neighbourhood of the approximate inertial manifold after large time.
基金Project supported by the National Natural Science Foundation of China(No.10971130)the Shanghai Leading Academic Dissipline Project(No.J50101)
文摘This paper studies the Riemann problem for a system of nonlinear degenerate wave equations in elasticity.Since the stress function is neither convex nor concave,the shock condition is degenerate.By introducing a degenerate shock under the generalized shock condition,the global solutions are constructively obtained case by case.
基金Project supported by the National Natural Science Foundation of China (Grant No.10672143)
文摘This paper focuses on studying the symmetry of a practical wave equation on new lattices.It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogeneous bar.The equation of motion of the bar can be changed into a discrete wave equation.With the new lattice equation,the translational and scaling invariant,not only is the infinitesimal transformation given,but the symmetry and Lie algebras are also calculated.We also give a new form of invariant called the ratio invariant,which can reduce the process of the computing invariant with the characteristic equation.
基金This work was supported by National Natural Science Foundation of China(No.41774137)111 project(No.B18055),and the Fundamental Research Funds for the Central Universities(No.19CX02002A).
文摘The constant Q property in viscoelastic media assumes that the quality factor Q does not change with frequency(i.e.,the Q value is independent of the frequency).For seismic waves propagating in viscoelastic media,the wave equation is determined by the viscoelastic media model.Equivalence relations exist between various frequency domain mathematical models and physical rheological models for the constant Q property.Considering two elastic moduli and three attenuation variables,24 kinds of wave equations based on diff erent generalized rheological models are divided into six classes in this study,and the 12 kinds of specifi c representation for the wave equations in the time domain are derived.On the basis of the equivalence relations between the generalized rheological models,the diff erence and equivalence relation between diff erent wave equations are proven and clarifi ed.Results show that the high-order generalized rheological model can accurately characterize the attenuation characteristics of seismic waves and has advantages in characterizing the dispersion characteristics in viscoelastic media.Lastly,the seismic refl ection characteristics caused by the diff erence of Q value are verifi ed by the forward modeling of the constant Q wave equation in this study,thereby providing a theoretical basis for the analysis and inversion of the formation Q value from refl ection seismic data.
基金Project supported by the National Natural Science Foundation of China(No.11071164)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ118)+1 种基金the Shanghai Leading Academic Discipline Project(No.XTKX2012)the Innovation Fund Project for Graduate Stu-dent of Shanghai(No.JWCXSL1201)
文摘This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefcients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.
基金supported by the National Natural Science Foundation of China (Grant No. 10875018)
文摘In this paper we solve spin-weighted spheroidal wave equations through super-symmetric quantum mechanics with a different expression of the super-potential.We use the shape invariance property to compute the "excited" eigenvalues and eigenfunctions.The results are beneficial to researchers for understanding the properties of the spin-weighted spheroidal wave more deeply,especially its integrability.
