May  2010, 27(2): 389-439. doi: 10.3934/dcds.2010.27.389

The Cauchy problem for Schrödinger flows into Kähler manifolds

1. 

University of Chicago, United States

2. 

University of British Columbia, Canada

3. 

University of Washington, United States

4. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

5. 

Department of Mathematics, University of Washington, Seattle, Washington 98195–4350

Received  October 2009 Revised  February 2010 Published  February 2010

We prove local well-posedness of the Schrödinger flow from $\RR^n$ into a compact Kähler manifold $N$ with initial data in $H^{s+1}(\RR^n,N)$ for $s\geq[\frac{n}{2}]+4$.
Citation: Carlos Kenig, Tobias Lamm, Daniel Pollack, Gigliola Staffilani, Tatiana Toro. The Cauchy problem for Schrödinger flows into Kähler manifolds. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 389-439. doi: 10.3934/dcds.2010.27.389
[1]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431

[2]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131

[3]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[4]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039

[5]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233

[6]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[7]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102

[8]

Shubin Wang, Guowang Chen. Cauchy problem for the nonlinear Schrödinger-IMBq equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 203-214. doi: 10.3934/dcdsb.2006.6.203

[9]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028

[10]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure and Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867

[11]

Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

[12]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027

[13]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181

[14]

Editorial Office. Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3785-3785. doi: 10.3934/cpaa.2020167

[15]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

[16]

JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021221

[17]

Belkacem Said-Houari, Salim A. Messaoudi. General decay estimates for a Cauchy viscoelastic wave problem. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1541-1551. doi: 10.3934/cpaa.2014.13.1541

[18]

Hideo Takaoka. Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6351-6378. doi: 10.3934/dcds.2020283

[19]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[20]

Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (100)
  • HTML views (0)
  • Cited by (9)

[Back to Top]