A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclin...A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.展开更多
By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturb...By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturbation terms,namely we prove the existence of the weak solution.展开更多
New two-component vector breather solution of the modified Benjamin-Bona-Mahony(MBBM)equation is considered.Using the generalized perturbation reduction method,the MBBM equation is reduced to the coupled nonlinear Sch...New two-component vector breather solution of the modified Benjamin-Bona-Mahony(MBBM)equation is considered.Using the generalized perturbation reduction method,the MBBM equation is reduced to the coupled nonlinear Schr¨odinger equations for auxiliary functions.Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented.The two-component vector breather and single-component scalar breather of the MBBM equation is compared.展开更多
Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑...Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.展开更多
In this paper, we use perturbing families of gernerlized Lyapunov functions to discuss ' the relative stability of ordinary differential systems in terms of two measures and gain some criteria for this stability o...In this paper, we use perturbing families of gernerlized Lyapunov functions to discuss ' the relative stability of ordinary differential systems in terms of two measures and gain some criteria for this stability of asymptotic.'展开更多
基金Project supported by the National Natural Science Foundation of China (Nos. 10972240 and 11102045)the Natural Science Foundation of Guangdong Province of China (No. S20110400040)+2 种基金the Foundation of Guangdong Education Department of China (No. LYM10108)the Foundation of Guangzhou Education Bureau of China (No. 10A024)the Research Grant Council of Hong Kong of China (No. GRF-HKU-7173-09E)
文摘A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.
基金Supported by the Innovation Talents of Science and Technology of Henan University(2009-HASTIT-007)Supported by the Natural Science Program of Department of Education(2011A110006)
文摘By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturbation terms,namely we prove the existence of the weak solution.
文摘New two-component vector breather solution of the modified Benjamin-Bona-Mahony(MBBM)equation is considered.Using the generalized perturbation reduction method,the MBBM equation is reduced to the coupled nonlinear Schr¨odinger equations for auxiliary functions.Explicit analytical expressions for the profile and parameters of the vector breather oscillating with the sum and difference of the frequencies and wavenumbers are presented.The two-component vector breather and single-component scalar breather of the MBBM equation is compared.
基金Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).
文摘Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.
文摘In this paper, we use perturbing families of gernerlized Lyapunov functions to discuss ' the relative stability of ordinary differential systems in terms of two measures and gain some criteria for this stability of asymptotic.'
