This paper is concerned with the derivative nonlinear Schr?dinger equation with periodic boundary conditions.We obtain complete Birkhoff normal form of order six.As an application,the long time stability for solutions...This paper is concerned with the derivative nonlinear Schr?dinger equation with periodic boundary conditions.We obtain complete Birkhoff normal form of order six.As an application,the long time stability for solutions of small amplitude is proved.展开更多
In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic fu...In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 【 σ 【 1, and each given integer b 】 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.展开更多
In this paper, we consider the higher dimensional nonlinear beam equation:utt+△2u+σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t nea...In this paper, we consider the higher dimensional nonlinear beam equation:utt+△2u+σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.展开更多
In this paper, we consider the lattice SchrSdinger equations iqn（t） = tan π（na＋x）qn（t） ＋ε（qn＋1（t） ＋ qn-1（t）） ＋δVn（t）｜qn（t）｜2τ-2qn（t）,with a satisfying a certain Diophantine condition, x∈ ...In this paper, we consider the lattice SchrSdinger equations iqn（t） = tan π（na＋x）qn（t） ＋ε（qn＋1（t） ＋ qn-1（t）） ＋δVn（t）｜qn（t）｜2τ-2qn（t）,with a satisfying a certain Diophantine condition, x∈ R/Z, and t- = 1 or 2, where vn（t） is a spatial localized real bounded potential satisfying ｜vn（t）｜ Ce-plnl. We prove that the growth of H1 norm of the solution {qn（t）}n∈Z is at most logarithmic if the initial data {qn（0）}n∈Z ∈ H1 for e sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e.,展开更多
We consider Hamiltonian partial differential equations utt ＋ ｜δx｜u ＋ σu = f（u）, x ∈T, t ∈R, with periodic boundary conditions, where f（u） is a real-analytic function of the form f（u） = u5 ＋ σ（u5） nea...We consider Hamiltonian partial differential equations utt ＋ ｜δx｜u ＋ σu = f（u）, x ∈T, t ∈R, with periodic boundary conditions, where f（u） is a real-analytic function of the form f（u） = u5 ＋ σ（u5） near u = 0, σ ∈（0, 1） is a fixed constant, and T = R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ （0, 1）Q. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.展开更多
This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equati...This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].展开更多
基金Supported by NNSFC(Grant Nos.11671280,11822108)Fok Ying Tong Education Foundation(Grant No.161002)。
文摘This paper is concerned with the derivative nonlinear Schr?dinger equation with periodic boundary conditions.We obtain complete Birkhoff normal form of order six.As an application,the long time stability for solutions of small amplitude is proved.
基金Supported by Shandong Provincial Natural Science Foundation,China(ZR2012AM017)and(2011ZRA07006)partially supported by Jiangsu Planned Projects for Postdoctoral Research Funds(1302022C)+1 种基金China Postdoctoral Science Foundation funded project(2014M551583)Project supported by the National Natural Science Foundation of China(Grant NO.11401302)
基金supported by National Natural Science Foundation (Grant Nos.10531050, 10771098)National Basic Research Program of China (973 Projects) (Grant No. 2007CB814800)
文摘In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 【 σ 【 1, and each given integer b 】 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.
基金supported by National Natural Science Foundation of China (Grant Nos.10531050,10771098)the Major State Basic Research Development of China and the Natural Science Foundation of Jiangsu Province(Grant No.BK2007134)
文摘In this paper, we consider the higher dimensional nonlinear beam equation:utt+△2u+σu + f(u)=0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u)=u3+ h.o.t near u=0 and σ is a positive constant. It is proved that for any fixed σ>0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.
文摘In this paper, we consider the lattice SchrSdinger equations iqn（t） = tan π（na＋x）qn（t） ＋ε（qn＋1（t） ＋ qn-1（t）） ＋δVn（t）｜qn（t）｜2τ-2qn（t）,with a satisfying a certain Diophantine condition, x∈ R/Z, and t- = 1 or 2, where vn（t） is a spatial localized real bounded potential satisfying ｜vn（t）｜ Ce-plnl. We prove that the growth of H1 norm of the solution {qn（t）}n∈Z is at most logarithmic if the initial data {qn（0）}n∈Z ∈ H1 for e sufficiently small and a.e. x fixed. Furthermore, suppose that the linear equation has a time quasi-periodic potential, i.e.,
文摘We consider Hamiltonian partial differential equations utt ＋ ｜δx｜u ＋ σu = f（u）, x ∈T, t ∈R, with periodic boundary conditions, where f（u） is a real-analytic function of the form f（u） = u5 ＋ σ（u5） near u = 0, σ ∈（0, 1） is a fixed constant, and T = R/2πZ. A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with σ ∈ （0, 1）Q. The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.
基金Supported by NSFC(Grant Nos.11601487 and 11526189)
文摘This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation,iut—uxx+i(|u|4u)x=0,x eT.The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established.The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem.We mention that in the present paper the mean value of u does not need to be zero,but small enough,which is different from the assumption(1.7)in Geng-Wu[J.Math.Phys.、53,102702(2012)].
