Abstract
This paper is concerned with a one dimensional derivative nonlinear Schrödinger equation with quasi-periodic forcing under periodic boundary conditions iut+uxx+ig(βt)(f(u2)u)x=0,x∈T≔R/2πZ, where g(βt) is real analytic and quasi-periodic on t with frequency vector β = (β1, β2, …, βm). f is real analytic in some neighborhood of the origin in ℂ, f(0) = 0 and f′(0) ≠ 0. We show that the above equation admits Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an abstract infinite dimensional Kolmogorov-Arnold-Moser theorem for unbounded perturbation vector fields and partial Birkhoff normal form.
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