Almost Global Solutions to Hamiltonian Derivative Nonlinear Schrödinger Equations on the Circle.

Consider a class of 1-d Hamiltonian derivative nonlinear Schrödinger equations i ψ t = ∂ xx ψ + V ∗ ψ + i ∂ x ( ∂ ψ ¯ F (x , ψ , ψ ¯) ) , x ∈ T , where V ∈ Θ m Θ m is defined in (1.5)]. The nonlinearity of these equations includes (ψ x , ψ ¯ x) and depends on space variable x periodically, which means that the nonlinearity is unbounded (see Definition 1.1) and the momentum set (see Definition 2.2) of the corresponding Hamiltonian function is unbounded. In this paper, we obtain that for any potential V outside a small measure subset of Θ m , if the initial value is smaller than R ≪ 1 in p-Sobolev norm, then the corresponding solution to this equation is also smaller than 2R during a time interval (- c R - r ∗ , c R - r ∗ ) (for any given positive r ∗ ). The main methods are constructing Birkhoff normal forms to unbounded Hamiltonian systems which have unbounded momentum sets and using the special symmetry of the Hamiltonian functions to control p-Sobolev norms of the solutions. [ABSTRACT FROM AUTHOR]