# 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
 [1] Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181 [2] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [3] Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563 [4] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [5] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [6] Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703 [7] Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066 [8] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [9] Shubin Wang, Guowang Chen. Cauchy problem for the nonlinear Schrödinger-IMBq equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 203-214. doi: 10.3934/dcdsb.2006.6.203 [10] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 [11] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 [12] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [13] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389 [14] Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144 [15] Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253 [16] Nakao Hayashi, Pavel I. Naumkin, Patrick-Nicolas Pipolo. Smoothing effects for some derivative nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 685-695. doi: 10.3934/dcds.1999.5.685 [17] Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 [18] Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863 [19] Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 [20] Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131

2018 Impact Factor: 0.925