Quasi-periodic solutions for the derivative nonlinear Schrödinger equation with finitely differentiable nonlinearities

Abstract

The authors are concerned with a class of derivative nonlinear Schrödinger equation $i{u_t} + {u_{xx}} + i \epsilon f\left( {u,\bar u,\omega t} \right){u_x} = 0,\left( {t,x} \right) \in \mathbb{R} \times \left[ {0,\pi } \right],$ subject to Dirichlet boundary condition, where the nonlinearity f(z ₁, z ₂, ϕ) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.

Keywords

Derivative NLS / KAM theory / Newton iterative scheme / Reduction theory / Quasi-periodic solutions / Smoothing techniques

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