In this paper, an upper bound of fractal dimension of the compact kernel sections for the dissipative non-autonomous Klein-Gordon-Schr<span style="white-space:nowrap;">ö</span>dinger lat...In this paper, an upper bound of fractal dimension of the compact kernel sections for the dissipative non-autonomous Klein-Gordon-Schr<span style="white-space:nowrap;">ö</span>dinger lattice system is obtained, by applying a criterion for estimating fractal dimension of a family of compact subsets of a separable Hilbert space.展开更多
In this paper,we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schr odinger equation.The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system an...In this paper,we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schr odinger equation.The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system and discrete the system by using Fourier pseudospectral method in space and symplectic Euler method in time.After composing two different symplectic Euler methods for the ODEs resulted from semi-discretization in space,we get a new explicit scheme for the target equation which is of second order in space and spectral accuracy in time.The canonical Hamiltonian form of the resulted ODEs is presented and the new derived scheme is proved strictly to be symplectic.The new scheme is totally explicitwhereas symplectic scheme are generally implicit or semi-implicit.Linear stability analysis is carried and a necessary Courant-Friedrichs-Lewy condition is given.The numerical results are reported to test the accuracy and efficiency of the proposed method in long-term computing.展开更多
We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We ...We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^(s_1) and wave data in H^(s_2) × H^(s_2-1)for 3/4- α < s_1≤0 and-1/2 < s_2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 < s_2 < 1/2 by using the conservation law for the L^2 norm of u.展开更多
文摘In this paper, an upper bound of fractal dimension of the compact kernel sections for the dissipative non-autonomous Klein-Gordon-Schr<span style="white-space:nowrap;">ö</span>dinger lattice system is obtained, by applying a criterion for estimating fractal dimension of a family of compact subsets of a separable Hilbert space.
基金This work is supported by the Jiangsu Collaborative Innovation Center for Climate Change,the National Natural Science Foundation of China(Grant Nos.11271195 and 11271196)and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
文摘In this paper,we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schr odinger equation.The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system and discrete the system by using Fourier pseudospectral method in space and symplectic Euler method in time.After composing two different symplectic Euler methods for the ODEs resulted from semi-discretization in space,we get a new explicit scheme for the target equation which is of second order in space and spectral accuracy in time.The canonical Hamiltonian form of the resulted ODEs is presented and the new derived scheme is proved strictly to be symplectic.The new scheme is totally explicitwhereas symplectic scheme are generally implicit or semi-implicit.Linear stability analysis is carried and a necessary Courant-Friedrichs-Lewy condition is given.The numerical results are reported to test the accuracy and efficiency of the proposed method in long-term computing.
基金supported by National Natural Science Foundation of China (Grant No. 11201498)
文摘We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^(s_1) and wave data in H^(s_2) × H^(s_2-1)for 3/4- α < s_1≤0 and-1/2 < s_2 < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 < s_2 < 1/2 by using the conservation law for the L^2 norm of u.
基金This work is supported by National Natural Science Foundation of China under Grant(10771139)Inno-vation Program of Shanghai Municipal Education Commission under Grant(08ZZ70).
