On the Cauchy Problem for the Derivative Nonlinear Schrödinger Equation with Periodic Boundary Condition
نویسنده
چکیده
It is shown that the Cauchy problem associated to the derivative nonlinear Schrödinger equation ∂tu − i∂ xu = λ∂x(|u| u) is locally well-posed for initial data u(0) ∈ H(T), if s ≥ 1 2 and λ is real. The proof is based on an adaption of the gauge transformation to periodic functions and sharp multi-linear estimates for the gauge equivalent equation in Fourier restriction norm spaces. By the use of a conservation law, the problem is shown to be globally well-posed for s ≥ 1 and data which is small in L.
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تاریخ انتشار 2005