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Global well-posedness for Schrödinger equations with derivative. (English) Zbl 1002.35113
Summary: We prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term $$\align & i\partial_t u+\partial^2_x u=i\lambda\partial_x(|u|^2u),\\ & u(x,0)=u_0(x),\qquad x\in\bbfR,\ t\in\bbfR,\endalign$$ is globally well-posed in $H^{s}$ for $s>2/3$, for small $L^{2}$ data. The result follows from an application of the “I-method”. This method allows us to define a modification of the energy norm $H^{1}$ that is “almost conserved” and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>2/3.$

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35A05General existence and uniqueness theorems (PDE) (MSC2000)
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