For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coeff...For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CC-EIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.展开更多
Shallow moment models are extensions of the hyperbolic shallow water equations.They admit variations in the vertical profile of the horizontal velocity.This paper introduces a non-hydrostatic pressure to this framewor...Shallow moment models are extensions of the hyperbolic shallow water equations.They admit variations in the vertical profile of the horizontal velocity.This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables.The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction.Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface.The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given.A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.展开更多
Phosphorus(P)leaching in alkaline soils,exacerbated by excessive fertilizer application,represents a significant pathway for P loss.While soil pore structure and texture critically regulate P transport,mechanisms gove...Phosphorus(P)leaching in alkaline soils,exacerbated by excessive fertilizer application,represents a significant pathway for P loss.While soil pore structure and texture critically regulate P transport,mechanisms governing P loss in texturally diverse alkaline soils remain unclear.This study investigated P leaching dynamics and transport parameters across four alkaline soil textures(silty clay,clay loam,loam,sandy loam)using a one-dimensional convective-diffusion equation(CDE)based on column experiments.Results indicated that phosphorus leaching kinetics were predominantly governed by diffusion transport,evidenced by low Peclet numbers(P_(e))(ranged from 0.02 to 0.31)across varying textures and initial P concentrations(C_(0)).Comparative analysis of transport parameters revealed significant textural effects on dispersion coefficient(D),retardation factor(R),pore water velocity(V),P_(e),and diffusion coefficient(λ)(F>523.42,p<0.001).Among these,only D,P_(e) andλexhibited substantial differences in response to variations in C_(0)(F>89.47,p<0.001).Saturated hydraulic conductivity(K_(s))(R^(2)=62.9%,p<0.01)and total pore area(A)(R^(2)=12.4%,p<0.01)emerged as primary regulators of P leaching.Enhanced clay content increased total pore area while reducing average pore diameter,concurrently decreasing pore water velocity and saturated infiltration rates.These textural modifications amplified diffusive P transport within soil matrices.The findings provide mechanistic insights into texturedependent P mobility in alkaline environments,informing targeted strategies for agricultural phosphorus management.展开更多
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the...The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.展开更多
Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-par...Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.展开更多
New exact solutions expressed by the Jacobi elliptic functions are obtained to the （2＋1）-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave sol...New exact solutions expressed by the Jacobi elliptic functions are obtained to the （2＋1）-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave solutions and triangular periodic wave solutions are obtained as well.展开更多
In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double d...In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.展开更多
The dispersion equation of the Scholte wave was reviewed using the homogeneous elastic half-space covered by a liquid layer,and the range of the Scholte wave propagation velocity was examined using the dispersion equa...The dispersion equation of the Scholte wave was reviewed using the homogeneous elastic half-space covered by a liquid layer,and the range of the Scholte wave propagation velocity was examined using the dispersion equation.The displacement expressions of the Scholte waves in liquid and solid were derived.Additionally,the mode of motion of Scholte waves in liquid and solid and their variation with depth was studied.The following results were obtained:The dispersion equation shows that the propagation velocity of the fundamental Scholte wave was greater than the P-wave in liquid and less than that of the Scholte wave in homogeneous elastic half-space.In contrast,the velocity of higher-order Scholte waves was greater than that of P waves in liquid and S-waves in solid.Only the fundamental Scholte wave has no cutoff frequency.The Scholte wave at the liquid surface moved only vertically,while the particles inside the liquid medium moved elliptically.The amplitude variation with depth in the solid medium caused the particle motion to change from a retrograde ellipse to a prograde ellipse.The above results imply the study of Scholte waves in the ocean and oceanic crust and help estimate ocean depths.展开更多
This paper studies the propagation of horizontally polarized shear waves in an internal magnetoelastic monoclinic stratum with irregularity in lower interface. The stratum is sandwiched between two magnetoelastic mono...This paper studies the propagation of horizontally polarized shear waves in an internal magnetoelastic monoclinic stratum with irregularity in lower interface. The stratum is sandwiched between two magnetoelastic monoclinic semi-infinite media. Dispersion equation is obtained in a closed form. In the absence of magnetic field and irregularity of the medium, the dispersion equation agrees with the equation of classical case in three layered media. The effects of magnetic field and size of irregularity on the phase velocity are depicted by means of graphs.展开更多
This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentia...This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentially and periodically along the depth. The displacements of the wave are found in the individual medium followed by a dispersion equation using a suitable analytic approach and a boundary condition. The prominent effect of inhomogeneity contained in the media, the rigid boundary plane, and the initial stress on the phase and group velocities is shown graphically.展开更多
The dispersion curves of real-valued modes in a fluid-filled borehole are widely used in acoustic well logging.The accurate dispersion curves are the precondition of theoretical analysis and inversion process.Generall...The dispersion curves of real-valued modes in a fluid-filled borehole are widely used in acoustic well logging.The accurate dispersion curves are the precondition of theoretical analysis and inversion process.Generally,these curves can be obtained by solving the conventional dispersion equation for isotropic formations and most vertically transverse isotropy(VTI)formations.However,if the real-valued solutions exist when the radial wavenumbers for the formation quasi-P and quasi-S equals to each other,the existed methods based on the conventional dispersion equation could lead to incorrect results for some VTI formations.Few studies have focused on the influence of these real-valued solutions on dispersion curve extraction.To remove these real-valued solutions,we have proposed a modified dispersion equation and its corresponding solving process.When solving the dispersion equation,the Scholte wave velocity of VTI formation at high frequency is used as the initial guess.The two synthetic examples including fast and slow VTI formations validate that these real-valued solutions do not contribute to the wavefield,and the new dispersion curve extraction method is suitable for all kinds of VTI formations.Consequently,the method can provide reliable dispersion curves for both theoretical analysis and anisotropic parameters inversion in VTI formations.展开更多
The plasma temperature （or the kinetic pressure） anisotropy is an intrinsic characteristic of a collisionless magnetized plasma. In this paper, based on the two-fluid model, a dispersion equation of low-frequency ...The plasma temperature （or the kinetic pressure） anisotropy is an intrinsic characteristic of a collisionless magnetized plasma. In this paper, based on the two-fluid model, a dispersion equation of low-frequency （ω〈〈ωci, ωci the ion gyrofrequency） waves, including the plasma temperature anisotropy effect, is presented. We investigate the properties of low-frequency waves when the parallel temperature exceeds the perpendicular temperature, and especially their dependence on the propagation angle, pressure anisotropy, and energy closures. The results show that both the instable Alfven and slow modes are purely growing. The growth rate of the Alfven wave is not affected by the propagation angle or energy closures, while that of the slow wave depends sensitively on the propagation angle and energy closures as well as pressure anisotropy. The fast wave is always stable. We also show how to elaborate the symbolic calculation of the dispersion equation performed using Mathematica Notebook.展开更多
The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite ...The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.展开更多
Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact s...Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the （2＋1）-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.展开更多
In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichart...In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal展开更多
Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）-...Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.展开更多
With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transfor...With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.展开更多
We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we mak...We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we make some remarks.展开更多
We present a modified method to solve the surface plasmons （SPs） of semi-infinite metal/dielectric superlattices and predicted new SP modes in physics. We find that four dispersion-equation sets and all possible SP ...We present a modified method to solve the surface plasmons （SPs） of semi-infinite metal/dielectric superlattices and predicted new SP modes in physics. We find that four dispersion-equation sets and all possible SP modes are determined by them. Our analysis and numerical calculations indicate that besides the SP mode obtained in the original theory, the other two SP modes are predicted, which have either a positive group velocity or a negative group velocity. We also point out the possible defect in the previous theoretical method in accordance to the linear algebra principle.展开更多
This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2+1)-dimensional dispersive long-wave equations . Starting from the homogeneous ba...This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2+1)-dimensional dispersive long-wave equations . Starting from the homogeneous balance method, we find that the richness of the localized coherent structures of the model is caused by the entrance of two variable-separated arbitrary functions. For some special selections of the arbitrary functions, it is shown that the localized structures of the model may be dromions, lumps, breathers, instantons and ring solitons.展开更多
基金supported in part by the National Key R&D Program of China No.2023YFA1009003 and the NSFC grant 12031001Chi-Wang Shu research is supported in part by the NSF grants DMS-2010107 and DMS-2309249.
文摘For the high-order diffusion and dispersion equations, the general practice of the explicit-implicit-null (EIN) method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation, and then apply the standard implicit-explicit method to the equivalent equation. We call this approach the constant-coefficient EIN method in this paper and hereafter denote it by “CC-EIN”. To reduce the error in the CC-EIN method, the variable-coefficient explicit-implicit-null (VC-EIN) method, which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient, is proposed and studied in this paper. Coupled with the local discontinuous Galerkin (LDG) spatial discretization, the VC-EIN method is shown to be unconditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear equations. In addition, although the computational cost slightly increases, the VC-EIN method can obtain more accurate results than the CC-EIN method, if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.
基金supported by the Research Training Group Energy,Entropy,and Dissipative Dynamics(EDDy),Project No.320021702/RTG2326,of the German Research Foundation(DFG).
文摘Shallow moment models are extensions of the hyperbolic shallow water equations.They admit variations in the vertical profile of the horizontal velocity.This paper introduces a non-hydrostatic pressure to this framework and shows the systematic derivation of dimensionally reduced dispersive equation systems which still hold information on the vertical profiles of the flow variables.The derivation from a set of balance laws is based on a splitting of the pressure followed by a same-degree polynomial expansion of the velocity and pressure fields in a vertical direction.Dimensional reduction is done via Galerkin projections with weak enforcement of the boundary conditions at the bottom and at the free surface.The resulting equation systems of order zero and one are presented in linear and nonlinear forms for Legendre basis functions and an analysis of dispersive properties is given.A numerical experiment shows convergence towards the resolved reference model in the linear stationary case and demonstrates the reconstruction of vertical profiles.
基金supported by the National Natural Science Foundation of China(Nos.42077067,42277329)the Projects of Talents Recruitment of GDUPT(No.XJ2005000301)。
文摘Phosphorus(P)leaching in alkaline soils,exacerbated by excessive fertilizer application,represents a significant pathway for P loss.While soil pore structure and texture critically regulate P transport,mechanisms governing P loss in texturally diverse alkaline soils remain unclear.This study investigated P leaching dynamics and transport parameters across four alkaline soil textures(silty clay,clay loam,loam,sandy loam)using a one-dimensional convective-diffusion equation(CDE)based on column experiments.Results indicated that phosphorus leaching kinetics were predominantly governed by diffusion transport,evidenced by low Peclet numbers(P_(e))(ranged from 0.02 to 0.31)across varying textures and initial P concentrations(C_(0)).Comparative analysis of transport parameters revealed significant textural effects on dispersion coefficient(D),retardation factor(R),pore water velocity(V),P_(e),and diffusion coefficient(λ)(F>523.42,p<0.001).Among these,only D,P_(e) andλexhibited substantial differences in response to variations in C_(0)(F>89.47,p<0.001).Saturated hydraulic conductivity(K_(s))(R^(2)=62.9%,p<0.01)and total pore area(A)(R^(2)=12.4%,p<0.01)emerged as primary regulators of P leaching.Enhanced clay content increased total pore area while reducing average pore diameter,concurrently decreasing pore water velocity and saturated infiltration rates.These textural modifications amplified diffusive P transport within soil matrices.The findings provide mechanistic insights into texturedependent P mobility in alkaline environments,informing targeted strategies for agricultural phosphorus management.
文摘The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
文摘Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004 zx16
文摘New exact solutions expressed by the Jacobi elliptic functions are obtained to the （2＋1）-dimensional dispersive long-wave equations by using the modified F-expansion method. In the limit case, new solitary wave solutions and triangular periodic wave solutions are obtained as well.
文摘In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
基金supported by the National Natural Science Fondation of China(Nos.42174074,41674055,41704053)the Earthquake Science Spark Program of Hebei Province(No.DZ20200827053)+1 种基金Fundamental Research Funds for the Central Universities(No.ZY20215117)the Hebei Key Laboratory of Earthquake Dynamics(No.FZ212105).
文摘The dispersion equation of the Scholte wave was reviewed using the homogeneous elastic half-space covered by a liquid layer,and the range of the Scholte wave propagation velocity was examined using the dispersion equation.The displacement expressions of the Scholte waves in liquid and solid were derived.Additionally,the mode of motion of Scholte waves in liquid and solid and their variation with depth was studied.The following results were obtained:The dispersion equation shows that the propagation velocity of the fundamental Scholte wave was greater than the P-wave in liquid and less than that of the Scholte wave in homogeneous elastic half-space.In contrast,the velocity of higher-order Scholte waves was greater than that of P waves in liquid and S-waves in solid.Only the fundamental Scholte wave has no cutoff frequency.The Scholte wave at the liquid surface moved only vertically,while the particles inside the liquid medium moved elliptically.The amplitude variation with depth in the solid medium caused the particle motion to change from a retrograde ellipse to a prograde ellipse.The above results imply the study of Scholte waves in the ocean and oceanic crust and help estimate ocean depths.
基金supported by the Department of Science and Technology of New Delhi(No.SR/S4/MS:436/07)
文摘This paper studies the propagation of horizontally polarized shear waves in an internal magnetoelastic monoclinic stratum with irregularity in lower interface. The stratum is sandwiched between two magnetoelastic monoclinic semi-infinite media. Dispersion equation is obtained in a closed form. In the absence of magnetic field and irregularity of the medium, the dispersion equation agrees with the equation of classical case in three layered media. The effects of magnetic field and size of irregularity on the phase velocity are depicted by means of graphs.
基金supported by the National Natural Science Foundation of China(No.11471087)the China Postdoctoral Science Foundation(No.2013M540270)+2 种基金the Heilongjiang Postdoctoral Foundation(No.LBH-Z13056)the Support Plan for the Young College Academic Backbone of Heilongjiang Province(No.1252G020)the Fundamental Research Funds for the Central Universities
文摘This paper studies dispersion of a G-type earthquake wave under the influence of a suppressed rigid boundary. Inside the Earth, the density and rigidity of the crustal layer and the mantle of the Earth vary exponentially and periodically along the depth. The displacements of the wave are found in the individual medium followed by a dispersion equation using a suitable analytic approach and a boundary condition. The prominent effect of inhomogeneity contained in the media, the rigid boundary plane, and the initial stress on the phase and group velocities is shown graphically.
基金financial support provided by the National Natural Science Foundation of China(Grant No.42104127 and 42004117)the Fundamental Research Funds for the Central Universities,China University of Geosciences(Wuhan)(Grant No.162301192696).
文摘The dispersion curves of real-valued modes in a fluid-filled borehole are widely used in acoustic well logging.The accurate dispersion curves are the precondition of theoretical analysis and inversion process.Generally,these curves can be obtained by solving the conventional dispersion equation for isotropic formations and most vertically transverse isotropy(VTI)formations.However,if the real-valued solutions exist when the radial wavenumbers for the formation quasi-P and quasi-S equals to each other,the existed methods based on the conventional dispersion equation could lead to incorrect results for some VTI formations.Few studies have focused on the influence of these real-valued solutions on dispersion curve extraction.To remove these real-valued solutions,we have proposed a modified dispersion equation and its corresponding solving process.When solving the dispersion equation,the Scholte wave velocity of VTI formation at high frequency is used as the initial guess.The two synthetic examples including fast and slow VTI formations validate that these real-valued solutions do not contribute to the wavefield,and the new dispersion curve extraction method is suitable for all kinds of VTI formations.Consequently,the method can provide reliable dispersion curves for both theoretical analysis and anisotropic parameters inversion in VTI formations.
基金supported by National Natural Science Foundation of China(Nos.10973043,41074107)Ministry of Science and Technology of China(No.2011CB811402)
文摘The plasma temperature （or the kinetic pressure） anisotropy is an intrinsic characteristic of a collisionless magnetized plasma. In this paper, based on the two-fluid model, a dispersion equation of low-frequency （ω〈〈ωci, ωci the ion gyrofrequency） waves, including the plasma temperature anisotropy effect, is presented. We investigate the properties of low-frequency waves when the parallel temperature exceeds the perpendicular temperature, and especially their dependence on the propagation angle, pressure anisotropy, and energy closures. The results show that both the instable Alfven and slow modes are purely growing. The growth rate of the Alfven wave is not affected by the propagation angle or energy closures, while that of the slow wave depends sensitively on the propagation angle and energy closures as well as pressure anisotropy. The fast wave is always stable. We also show how to elaborate the symbolic calculation of the dispersion equation performed using Mathematica Notebook.
基金This work was supported in part by the National Science Foundation under grant DMS-1620288。
文摘The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.
基金The project supported by the China Postdoctoral Science Foundation under Grant No. 2004036086, K.C. Wong Education Foundation, Hong Kong, and partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000 . The authors are grateful to professor Gao Xiao-Shan for his enthusiastic guidance and help.
文摘Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the （2＋1）-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.
基金Supported by the National Natural Science Foundation of China the Doctoral Foundation of NEM of China
文摘In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz＇s inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal
基金supported by the Natural Science Foundation of Shandong Province of China under Grant Nos.Q2005A01
文摘Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.
基金supported by the Scientific Research Foundation of Beijing Information Science and Technology UniversityScientific Creative Platform Foundation of Beijing Municipal Commission of Education
文摘With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.
文摘We study the WKB dispersion equation in non-uniform optical waveguide.There are three methods given in this paper:(1) method of Airy function;(2)method of connection formula;and(3) method of phase shift.At last we make some remarks.
基金supported by the National Natural Science Foundation of China(Grant No.11074061)
文摘We present a modified method to solve the surface plasmons （SPs） of semi-infinite metal/dielectric superlattices and predicted new SP modes in physics. We find that four dispersion-equation sets and all possible SP modes are determined by them. Our analysis and numerical calculations indicate that besides the SP mode obtained in the original theory, the other two SP modes are predicted, which have either a positive group velocity or a negative group velocity. We also point out the possible defect in the previous theoretical method in accordance to the linear algebra principle.
文摘This article is concerned with the extended homogeneous balance method for studying the abundant localized solution structures in the (2+1)-dimensional dispersive long-wave equations . Starting from the homogeneous balance method, we find that the richness of the localized coherent structures of the model is caused by the entrance of two variable-separated arbitrary functions. For some special selections of the arbitrary functions, it is shown that the localized structures of the model may be dromions, lumps, breathers, instantons and ring solitons.
