American Institute of Mathematical Sciences

December  2007, 6(4): 997-1021. doi: 10.3934/cpaa.2007.6.997

Well-posedness for one-dimensional derivative nonlinear Schrödinger equations

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

Received  January 2007 Revised  April 2007 Published  September 2007

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.
Citation: Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
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