The initial value problem for the derivative nonlinear Schrödinger equation in the energy space. (English) Zbl 0787.35099
This paper is concerned with the Cauchy problem for the derivative nonlinear Schrödinger equation
\[
i \psi_ t+\psi_{xx}+2i \delta(| \psi |^ 2 \psi)_ x=0,\;(t,x) \in \mathbb{R} \times \mathbb{R},\;\psi(0,x)=\varphi(x),\;x \in \mathbb{R}
\]
where \(\delta=\pm 1/2\).
The author proves the following: Theorem. Assume that \(\varphi(x) \in H^ 1(\mathbb{R})\) and \(\| \varphi \|_{L^ 2}\) is sufficiently small. Then there exists a unique global solution \(\psi(t,x)\) such that \[ \psi(t,x) \in C(\mathbb{R};H^ 1(\mathbb{R})) \cap L^{1,2}_{\text{loc}} (\mathbb{R};H^{1,3}(\mathbb{R})). \] (Here he uses the usual notation of Sobolev spaces and \(H^{m,p}(\mathbb{R})\) denotes the \(L^ p\) Sobolev space with regularity of order \(m\).)
This equation was derived to study the propagation of circular polarized nonlinear Alfven waves in plasma. It is remarked that the usual \(L^ p- L^ q\) estimates are not directly applicable to this problem which causes the usage of special Sobolev spaces.
The author proves the following: Theorem. Assume that \(\varphi(x) \in H^ 1(\mathbb{R})\) and \(\| \varphi \|_{L^ 2}\) is sufficiently small. Then there exists a unique global solution \(\psi(t,x)\) such that \[ \psi(t,x) \in C(\mathbb{R};H^ 1(\mathbb{R})) \cap L^{1,2}_{\text{loc}} (\mathbb{R};H^{1,3}(\mathbb{R})). \] (Here he uses the usual notation of Sobolev spaces and \(H^{m,p}(\mathbb{R})\) denotes the \(L^ p\) Sobolev space with regularity of order \(m\).)
This equation was derived to study the propagation of circular polarized nonlinear Alfven waves in plasma. It is remarked that the usual \(L^ p- L^ q\) estimates are not directly applicable to this problem which causes the usage of special Sobolev spaces.
Reviewer: A.Tsutsumi (Sakai)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
Keywords:
existence; uniqueness; Cauchy problem; derivative nonlinear Schrödinger equation; global solution; Sobolev spaces; propagation of circular polarized nonlinear Alfven waves in plasmaReferences:
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