The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out tha...The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.展开更多
In this paper, some misunderstam igs concerning the necessary conditions for resonance for ordinary differential equations with turning point have been corrected, and a recursive process for finding the sequence of ne...In this paper, some misunderstam igs concerning the necessary conditions for resonance for ordinary differential equations with turning point have been corrected, and a recursive process for finding the sequence of necessary conditions for resonance has beenoffered.展开更多
In this Paper,the stochastic layer of Duffing's equation hosed on the chosen resonance is investigated.A general method is provided for studying the stochastic layer near the assigned resonant orbit. Several appro...In this Paper,the stochastic layer of Duffing's equation hosed on the chosen resonance is investigated.A general method is provided for studying the stochastic layer near the assigned resonant orbit. Several approximate critical conditions are given for the theoretical predictions of that stochastic layer. For non-dissipative Duffing's equation, two critical conditions are obtained when its global stochastic layer occurs and vanishes for the given resonance. In addition, a limit critical condition has been presented when all Possible resonances exist risible in the stochastic layer.Our results are compared with the critical conditions resulting from both Chirikov overlap method and renormalization techniques. Finally, using the cirtical conditions, numerical simulations are Performed to check our theoretical predictions of the stochastic layer based on the given resonance.展开更多
Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung flui...Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equation.展开更多
The coupled equation method （CEM） has been applied to investigating the resonance structures for the ground state 1s^22s^ 2S of the neutral lithium from the first threshold up to 64.5 eV. Resonance structures of ato...The coupled equation method （CEM） has been applied to investigating the resonance structures for the ground state 1s^22s^ 2S of the neutral lithium from the first threshold up to 64.5 eV. Resonance structures of atomic lithium due to single excitations of the ls and 2s electrons are studied by infinite-order calculations in detail. The effect of spin-orbit splitting is also included for some of the low-lying ls2snp（↑↓） resonance, and the influence of the interference between 1s^2s^3 Snp .↓ and 1s2s^ 1 Snp ↑ states on the resonance structure has been confirmed theoretically. The results show that the presented technique can give the reasonable resonance structures very well in photoionization processes.展开更多
Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ ...Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.展开更多
基金Supported by the National Natural Science Foundation of China
文摘The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.
基金Projects Supported by the Science Fund of the Chinese Academy of Sciences
文摘In this paper, some misunderstam igs concerning the necessary conditions for resonance for ordinary differential equations with turning point have been corrected, and a recursive process for finding the sequence of necessary conditions for resonance has beenoffered.
文摘In this Paper,the stochastic layer of Duffing's equation hosed on the chosen resonance is investigated.A general method is provided for studying the stochastic layer near the assigned resonant orbit. Several approximate critical conditions are given for the theoretical predictions of that stochastic layer. For non-dissipative Duffing's equation, two critical conditions are obtained when its global stochastic layer occurs and vanishes for the given resonance. In addition, a limit critical condition has been presented when all Possible resonances exist risible in the stochastic layer.Our results are compared with the critical conditions resulting from both Chirikov overlap method and renormalization techniques. Finally, using the cirtical conditions, numerical simulations are Performed to check our theoretical predictions of the stochastic layer based on the given resonance.
文摘Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schr<span style="white-space:nowrap;">ö</span>dinger equation.
基金This work was supported by the Natural Science Foundation of Yantai Normal University under Grant No.22270301 and L20072804.
文摘The coupled equation method （CEM） has been applied to investigating the resonance structures for the ground state 1s^22s^ 2S of the neutral lithium from the first threshold up to 64.5 eV. Resonance structures of atomic lithium due to single excitations of the ls and 2s electrons are studied by infinite-order calculations in detail. The effect of spin-orbit splitting is also included for some of the low-lying ls2snp（↑↓） resonance, and the influence of the interference between 1s^2s^3 Snp .↓ and 1s2s^ 1 Snp ↑ states on the resonance structure has been confirmed theoretically. The results show that the presented technique can give the reasonable resonance structures very well in photoionization processes.
基金Supported by NSFC Innovation Grant(Grant No.10421101)
文摘Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.
