[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We ...[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We also make the estimates on its fractal dimension and exponential attraction.展开更多
The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathem...The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.展开更多
In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed....In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.展开更多
In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together...In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.展开更多
In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local time- space region which is independent of the boundary condition and more essential than the global energy...In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local time- space region which is independent of the boundary condition and more essential than the global energy conservation law. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving （LEP） scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.展开更多
A new type of homoclinic arid heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreove...A new type of homoclinic arid heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreover, the homoclinic and heteroclinic structure with local oscillation and mechanicaL feature different from homoclinic and heterocliunic solutions are investigated. Result shows complexity of dynamics for complex nonlineaR evolution system. Moreover, the similarities and differences between homoclinic （heteroclinic） breather and homoclinic （heteroclinic） tube are exhibited. These results show that the diversity of the structures of homoclinic and heteroclinic solutions.展开更多
This paper considers the stability of the solitary waves for the generalized Zakharov system. By applying the abstract theory of Grillakis M. et al. and detailed spectral analysis, we obtain the stability of the solit...This paper considers the stability of the solitary waves for the generalized Zakharov system. By applying the abstract theory of Grillakis M. et al. and detailed spectral analysis, we obtain the stability of the solitary waves.展开更多
In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Lan...In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov展开更多
In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the glob...In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.展开更多
We consider the asymptotic behavior of solutions of an infinite lattice dynamical system of dissipative Zakharov equation. By introducing new weight inner product and norm in the space and establishing uniform estimat...We consider the asymptotic behavior of solutions of an infinite lattice dynamical system of dissipative Zakharov equation. By introducing new weight inner product and norm in the space and establishing uniform estimate on ＂Tail End＂ of solutions, we overcome some difficulties caused by the lack of Sobolev compact embedding under infinite lattice system, and prove the existence of the global attractor; then by using element decomposition and the covering property of a polyhedron in the finite-dimensional space, we obtain an upper bound for the Kolmogorov ε-entropy of the global attractor; finally, we present the upper semicontinuity of the global attractor.展开更多
A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference sche...A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference scheme and the error bounds of optimal order of the difference solutions are obtained in L^2 × H^1 × H^2 over a finite time interval （0, T]. Finally, the existence of a global attractor is proved for a discrete dynamical system associated with the fully discrete finite difference scheme.展开更多
文摘[1] has proved that the dissipative Zakharov system has an ε2-weak compact attractor. In this paper, we further show that the dissipative Langmuir waves in plasmas admit an inertial fractal set of (ε2,ε1)-type. We also make the estimates on its fractal dimension and exponential attraction.
文摘The nonlinear evolution equations have a wide range of applications,more precisely in physics,biology,chemistry and engineering fields.This domain serves as a point of interest to a large extent in the world’s mathematical community.Thus,this paper purveys an analytical study of a generalized extended(2+1)-dimensional quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering.The Lie group theory of differential equations is utilized to compute an optimal system of one dimension for the Lie algebra of the model.We further reduce the equation using the subalgebras obtained.Besides,more general solutions of the underlying equation are secured for some special cases of n with the use of extended Jacobi function expansion technique.Consequently,we secure new bounded and unbounded solutions of interest for the equation in various solitonic structures including bright,dark,periodic(cnoidal and snoidal),compact-type as well as singular solitons.The applications of cnoidal and snoidal waves of the model in oceanography and ocean engineering for the first time,are outlined with suitable diagrams.This can be of interest to oceanographers and ocean engineers for future analysis.Furthermore,to visualize the dynamics of the results found,we present the graphic display of each of the solutions.Conclusively,we construct conservation laws of the understudy equation via the application of Noether’s theorem.
基金supported by the Project for the National Natural Science Foundation of China(No.12261103).
文摘In this paper,we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system(QZS)with a dimensionless parameter 0<ε≤1,which is inversely proportional to the acoustic speed.In the subsonic limit regime,i.e.,when 0<ε?1,the solution of QZS propagates rapidly oscillatory initial layers in time,and this brings significant difficulties in devising numerical algorithm and establishing their error estimates,especially as 0<ε?1.The solvability,the mass and energy conservation laws of the scheme are also discussed.Based on the cut-off technique and energy method,we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data,respectively,which are uniform in both time and space forε∈(0,1]and optimal at the fourth order in space.Numerical results are reported to verify the error behavior.
基金supported in part by National Natural Science Foundation of China (GrantNos. 11001022 and 11071240)supported in part by National Natural Science Foundation of China(Grant Nos. 10801102,11171241 and 11071177)
文摘In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.
基金Project supported by the National Natural Science Foundation of China(Grant No.11771213)the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology(Grant No.2243141701090)
文摘In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local time- space region which is independent of the boundary condition and more essential than the global energy conservation law. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving （LEP） scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.
基金Supported by the Natural Science Foundation of China under Grant No.11061028
文摘A new type of homoclinic arid heteroclinic solutions, i.e. homoclinic and heteroclinic breather solutions, for Zakharov system are obtained using extended homoclinic test and two-soliton methods, respectively. Moreover, the homoclinic and heteroclinic structure with local oscillation and mechanicaL feature different from homoclinic and heterocliunic solutions are investigated. Result shows complexity of dynamics for complex nonlineaR evolution system. Moreover, the similarities and differences between homoclinic （heteroclinic） breather and homoclinic （heteroclinic） tube are exhibited. These results show that the diversity of the structures of homoclinic and heteroclinic solutions.
基金The research is supported by the Scientific Research Foundation of Yunnan Provincial Department and the Natural Science Foundation of Yunnan Province（2005A0026M）
文摘This paper considers the stability of the solitary waves for the generalized Zakharov system. By applying the abstract theory of Grillakis M. et al. and detailed spectral analysis, we obtain the stability of the solitary waves.
基金The research is supported by the Scientific Research Foundation of Yunnan Provincial Departmentthe Natural Science Foundation of Yunnan Province(No.2005A0026M).
文摘In the interaction of laser-plasma the system of Zakharov equation plays an important role.This system attracted many scientists' wide interest and attention.And the formation, evolution and interaction of the Langmuir solutions differ from solutions of the KDV equation. Here we consider the following generalized Zakharov
文摘In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.
基金supported by National Natural Science Foundation of People's Republic of China (10771139)Partly supported by A Project Supported by Scientific Research Fund of Hu'nan Provincial Education on Department (08A070 08A071)
文摘We consider the asymptotic behavior of solutions of an infinite lattice dynamical system of dissipative Zakharov equation. By introducing new weight inner product and norm in the space and establishing uniform estimate on ＂Tail End＂ of solutions, we overcome some difficulties caused by the lack of Sobolev compact embedding under infinite lattice system, and prove the existence of the global attractor; then by using element decomposition and the covering property of a polyhedron in the finite-dimensional space, we obtain an upper bound for the Kolmogorov ε-entropy of the global attractor; finally, we present the upper semicontinuity of the global attractor.
基金Supported by the National Natural Science Foundation of China(10371077)
文摘A fully discrete finite difference scheme for dissipative Zakharov equations is analyzed. On the basis of a series of the time-uniform priori estimates of the difference solutions, the stability of the difference scheme and the error bounds of optimal order of the difference solutions are obtained in L^2 × H^1 × H^2 over a finite time interval （0, T]. Finally, the existence of a global attractor is proved for a discrete dynamical system associated with the fully discrete finite difference scheme.
