Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations (Q2872308)

In this work, the authors consider the derivative nonlinear Schrödinger equation \(i u_{t}-u_{xx} -i(|u|^{4}u)_{x}=0\) in the class of functions with zero mean satisfying the periodic boundary condition \(u(t,x+2\pi) =u(t,x)\). By deriving a partial Birkhoff normal form of order six for the lattice Hamiltonian, and by using an abstract form of the Kolomgorov-Arnold-Moser (KAM) theorem, they establish a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions of the equation with two Diophantine frequencies.