Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations

Abstract

Communication: Restoring full size extensivity in internally contracted multireference coupled cluster theory J. Chem. Phys. 137, 131103 (2012) Analytical spatiotemporal soliton solutions to (3+1)-dimensional cubic-quintic nonlinear Schrödinger equation with distributed coefficients J. Math. Phys. 53, 103704 (2012) Time-domain determination of transmission in quantum nanostructures J. Appl. Phys. 112, 064325 (2012) The virial theorem for the smoothly and sharply, penetrably and impenetrably confined hydrogen atom J. Chem. Phys. 137, 114109 (2012) Communication: Phase space wavelets for solving Coulomb problemsIn this paper, we show that one dimension derivative nonlinear Schrödinger equation admits a whitney smooth family of small amplitude, real analytic quasiperiodic solutions with two Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an abstract infinite dimensional Kolmogorov-Arnold-Moser (KAM) theorem. C 2012 American Institute of Physics.Motivated by the classical Kolmogorov-Arnold-Moser (KAM) theory in finite dimensional Hamiltonian systems, to obtain quasi-periodic solutions, for an integer N > 1, one finds a parameter space O ⊂ R N and (symplectic) action-angle-normal coordinates I = (I 1 , . . . , θ = (θ 1 , . . . , θ N ), z = (z j ) j∈Z such that the Hamiltonian (1.3) can be transformed into a parametrized Hamiltonian normal formDenote d as the order of¯ j and δ as the order of the Hamiltonian vector field X P . We use d to measure the growth rate of the eigenvalue¯ j and δ to measure the unboundedness of X P . When δ ≤ 0, the vector field X P is called bounded perturbation, the existence of quasi-periodic solutions of such PDEs has been deeply and widely studied. There are plenty of papers dedicated to such these issues. See Refs. 2-6, 10, and 12 for example.When δ > 0, the vector field X P is called unbounded perturbation. In order to obtain the existence of quasi-periodic solutions for such PDEs, it is reasonable to assume δ ≤ d − 1, according to a well known example, which was referred in Klainerman 8 and Lax. 11 When 0 < δ < d − 1, the corresponding KAM theorem is due to Kuksin. 9 Kuksin's theorem is used to prove the existence of time quasi-periodic solutions of KdV equation subject to periodic boundary condition. See also For such unbounded perturbations, Bambusi-Graffi 1 gave another KAM theorem to prove that the time dependent linear Schrödinger equation has only pure point spectrum.When 0 < δ = d − 1, the nonlinearity of the PDE is the strongest. Liu-Yuan 17 gave a theorem, which generalized Kuksin's theorem from δ < d − 1 to δ ≤ d − 1. For the homological equations of variable coefficients(1.4) they developed some new estimate for the solution u covering not only the case δ < d − 1, but also the limit case δ = d − 1. Using the generalized Kuksin's theorem, Zhang-Gao-Yuan 19 established a KAM theorem for an infinite dimensional reversible system with unbounded perturbation and obtained many C ∞ (not real analytic) quasi-periodic solutions...

Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation

“…Since normal form obtained in [12] is independent of the angle variables θ, it is different from Kuksin's theorem [21] and Liu and Yuan's theorem [24]. Then, using an abstract KAM theorem with angle independent normal form, they obtain the real analytic quasiperiodic solutions for the derivative nonlinear Schrödinger equation (1.5) with only two Diophantine frequencies.…”

“…− i∂ ω u + λu + µ(θ)u = p(θ), |Im θ| < s, (1.3) Using the generalized Kuksin's theorem, Liu and Yuan [24] Then, Geng and Wu [12] consider the derivative nonlinear Schrödinger equation…”

Quasi-periodic solutions for a Schrödinger equation with a quintic nonlinear term depending on the time and space variables

“…Liang [11] considered the Schrödinger equation iu t -u xx + |u| 2p u = 0, p ∈ N and proved the existence of quasi-periodic solutions corresponding to two-dimensional invariant tori. Geng and Wu [12] showed that the one-dimensional derivative nonlinear Schrödinger equation…”

“…However, the models in [10][11][12][13] cannot explicitly contain the space variable. The authors in [10][11][12] applied the compact form condition, the generalized compact form condition, or the gauge invariant property.…”

“…The authors in [10][11][12] applied the compact form condition, the generalized compact form condition, or the gauge invariant property. Although these conditions simplify the normal forms and measure estimates, the equations they [10][11][12] considered cannot explicitly contain the space variable x. The x-dependent nonlinearity implies that the equation is variant in space translations and the momentum is not conserved.…”