Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle. (English) Zbl 1386.35376
Summary: We consider the quadratic derivative nonlinear Schrödinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the Cole-Hopf transformation, we prove unconditional global well-posedness in \(L^2(\mathbb T)\), and more generally in certain Fourier-Lebesgue spaces \(\mathcal FL^{s,p}(\mathbb T)\), under the mean-zero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with non-small mean-zero initial data.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
quadratic derivative nonlinear Schrödinger equation; normal form; Cole-Hopf transform; Fourier-Lebesgue space; well-posedness; finite time blowupReferences:
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