On the construction of almost periodic solutions for the derivative nonlinear Schrödinger equation

In this paper, we consider a derivative nonlinear Schrödinger equation $$\mathrm{i}{\mathrm{\partial }}_{t}u+{\mathrm{\partial }}_{xx}u-V\ast u+\mathrm{i}\mathrm{\mid }u{\mathrm{\mid }}^2{\mathrm{\partial }}_{x}u=0$$ on the torus $\mathbb{T}$, depending on some potential $V$. We prove that for `almost all’ potentials $V$, this equation admits an almost-periodic solution.