The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as ...The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations is analyzed to shed light oi1 the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differentii equations and their corresponding exact solutions with generalized separated variables.展开更多
In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） wi...In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） with orders {k1, k2} （k1≥ k2） preserves the invariant subspace Wn1^1× Wn2^2 （n1 ≥ n2）, then n1 - n2 ≤ k2, n1 ≤2（k1 ＋ k2） ＋ 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq （q = 1, 2）. Several examples including the （1＋1）-dimensional diffusion system and Ito＇s type, Drinfel＇d-Sokolov-Wilson＇s type and Whitham-Broer-Kaup＇s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.展开更多
The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and condi- tional Lie=Bgcklund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonl...The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and condi- tional Lie=Bgcklund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher=order conditional Lie-B^icklund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical sys- tems.展开更多
By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibri...By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibrium points.Two types of periodic oscillation,such as 2T-focus/cycle periodic switching and 2T-focus/focus periodic switching,have been observed,the mechanism of which is presented through the switching relationship.The distribution of eigenvalues related to the equilibrium points determined by two subsystems is discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations.Furthermore,the invariant subspaces of the equilibrium point are investigated to reveal the mechanism of dynamical phenomena in the periodic switching.展开更多
基金supported by the State Administration of Foreign Experts Affairs of China,National Natural Science Foundation of China (Grant Nos. 10971136,10831003,61072147,11071159)Chunhui Plan of the Ministry of Education of China,Zhejiang Innovation Project (Grant No. T200905)the Natural Science Foundation of Shanghai and the Shanghai Leading Academic Discipline Project (Grant No.J50101)
文摘The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations is analyzed to shed light oi1 the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differentii equations and their corresponding exact solutions with generalized separated variables.
基金Project supported by the National Natural Science Foundation of China for Distinguished Young Scholars (No.10925104)the National Natural Science Foundation of China (No.11001240)+1 种基金the Doctoral Program Foundation of the Ministry of Education of China (No.20106101110008)the Zhejiang Provincial Natural Science Foundation of China (Nos.Y6090359,Y6090383)
文摘In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = （F1, F2） with orders {k1, k2} （k1≥ k2） preserves the invariant subspace Wn1^1× Wn2^2 （n1 ≥ n2）, then n1 - n2 ≤ k2, n1 ≤2（k1 ＋ k2） ＋ 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq （q = 1, 2）. Several examples including the （1＋1）-dimensional diffusion system and Ito＇s type, Drinfel＇d-Sokolov-Wilson＇s type and Whitham-Broer-Kaup＇s type equations are presented to illustrate the result. Furthermore, the estimate of dimension for m-component nonlinear systems is also given.
基金supported by National Natural Science Foundation of China for Distinguished Young Scholars(Grant No.10925104)the PhD Programs Foundation of Ministry of Education of China(Grant No.20106101110008)the United Funds of NSFC and Henan for Talent Training(Grant No.U1204104)
文摘The inhomogeneous nonlinear diffusion equation is studied by invariant subspace and condi- tional Lie=Bgcklund symmetry methods. It is shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher=order conditional Lie-B^icklund symmetries of the equations. As a consequence, a number of new solutions to the inhomogeneous nonlinear diffusion equations are constructed explicitly or reduced to solving finite-dimensional dynamical sys- tems.
基金supported by the National Natural Science Foundation of China (Grant Nos. 20976075 and 10972091)College Graduate Student Scientific Research Innovation Foundation of Jiangsu,China (Grant No. CXLX12-0619)
文摘By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibrium points.Two types of periodic oscillation,such as 2T-focus/cycle periodic switching and 2T-focus/focus periodic switching,have been observed,the mechanism of which is presented through the switching relationship.The distribution of eigenvalues related to the equilibrium points determined by two subsystems is discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations.Furthermore,the invariant subspaces of the equilibrium point are investigated to reveal the mechanism of dynamical phenomena in the periodic switching.
