Quasi-Periodic Solutions in a Nonlinear Schrödinger Equation

1991 Mathematics Subject Classification. Primary 37K55, 35B10, 35J10, 35Q40, 35Q55.In this paper, one-dimensional (1D) nonlinear Schrödinger equation [equation omitted] with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescribed integer N > 1, the equation admits a Whitney smooth family of small-amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkho ff normal form reduction and an improved KAM method.Partially supported by NSF grant DMS0204119

Quasi-Periodic Breathers in Hamiltonian Networks of Long-Range Coupling

1991 Mathematics Subject Classification. Primary 37K60, 37K55This work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.The first author is partially supported by NSFC grant 10771098, 973 projects of China and NSFJP grant BK2007134. The third author is partially supported by NSFC grant 10428101 and NSF grants DMS0204119, DMS0708331

A KAM Theorem for Hamiltonian Networks with Long Ranged Couplings

1991 Mathematics Subject Classification. Primary 37K60, 37K55.We consider Hamiltonian networks of long ranged and weakly coupled oscillators with variable frequencies.The second author is Partially supported by NSF grant DMS0204119

Invariant tori of full dimension for a nonlinear Schrödinger equation

AbstractIn this paper, we consider the one-dimensional nonlinear Schrödinger equationiut−uxx+mu+f(|u|2)u=0 with periodic boundary conditions or Dirichlet boundary conditions, where f is a real analytic function in some neighborhood of the origin satisfying f(0)=0, f′(0)≠0. We prove that for each given constant potential m, when the frequencies, as a function of the amplitudes, can be regarded as the independent parameters, the equation admits a Whitney smooth family of small-amplitude, time almost-periodic solutions with all frequencies. The proof is based on a Birkhoff normal form reduction and an improved version of the KAM theorem

Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation

AbstractIn this paper, one-dimensional (1D) nonlinear wave equation utt−uxx+mu+u3=0, subject to Dirichlet boundary conditions is considered. We show that for each given m>0, and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, which correspond to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. In particular, these 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

A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

AbstractIn this paper, one-dimensional (1D) nonlinear Schrödinger equationiut-uxx+mu+∂g(u,u¯)∂u¯=0,with Periodic Boundary Conditions is considered; m∉112Z is a real parameter and the nonlinearityg(u,u¯)=∑j,l,j+l⩾4ajluju¯l,ajl=alj∈R,a22≠0is a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system

Quasi-periodic solutions in a nonlinear Schrödinger equation

Abstract. In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx + mu + |u | 4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescibed interger N> 1, the equation admits a Whitney smooth family of small-amplitude, time quasi-periodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. Consider a nonlinear Schrödinger equatio