Well-posedness for one-dimensional derivative nonlinear Schrödinger equations

Chengchun Hao

doi:10.3934/cpaa.2007.6.997

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2007, Volume 6, Issue 4: 997-1021. Doi: 10.3934/cpaa.2007.6.997

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Well-posedness for one-dimensional derivative nonlinear Schrödinger equations

Chengchun Hao^1,

1.
Institute of Mathematics, Academy of Mathematics & Systems Science, CAS, Beijing 100080

Received: January 2007

Revised: April 2007

Published: December 2007

Abstract / Introduction Related Papers Cited by

Abstract

In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{x x}+i\lambda |u|^k u_x=0$ with non-zero $\lambda\in \mathbb R$ and any real number $k\geq 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.

Keywords:

Derivative nonlinear Schrödinger equations,
Cauchy problem,
local well-posedness.

Mathematics Subject Classification: 35Q55, 35A07.

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