American Institute of Mathematical Sciences

Advanced
November  2009, 8(6): 1803-1823. doi: 10.3934/cpaa.2009.8.1803

Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering

Alp Eden 1, and  Elİf Kuz 1,

1. 

Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey, Turkey

Received  October 2008 Revised  March 2009 Published  August 2009

We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type. We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.
Keywords: Davey-Stewartson equations, global existence, scattering, focusing solutions., Nonlinear Schrödinger equation.
Mathematics Subject Classification: Primary: 35G25, 35B40, 35Q5.
Citation: Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[1]

Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182
[2]

Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077
[3]

Jing Lu, Yifei Wu. Sharp threshold for scattering of a generalized Davey-Stewartson system in three dimension. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1641-1670. doi: 10.3934/cpaa.2015.14.1641
[4]

Zaihui Gan, Boling Guo, Jian Zhang. Sharp threshold of global existence for the generalized Davey-Stewartson system in $R^2$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 913-922. doi: 10.3934/cpaa.2009.8.913
[5]

Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286
[6]

Olivier Goubet, Manal Hussein. Global attractor for the Davey-Stewartson system on $\mathbb R^2$. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1555-1575. doi: 10.3934/cpaa.2009.8.1555
[7]

J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467
[8]

Caroline Obrecht, J.-C. Saut. Remarks on the full dispersion Davey-Stewartson systems. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1547-1561. doi: 10.3934/cpaa.2015.14.1547
[9]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[10]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
[11]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129
[12]

Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361
[13]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[14]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[15]

Guillaume Ferriere. The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6247-6274. doi: 10.3934/dcds.2020277
[16]

Rémi Carles. Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 385-398. doi: 10.3934/dcds.2005.13.385
[17]

Masahoto Ohta, Grozdena Todorova. Remarks on global existence and blowup for damped nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1313-1325. doi: 10.3934/dcds.2009.23.1313
[18]

Zaihui Gan, Boling Guo, Jian Zhang. Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1303-1312. doi: 10.3934/cpaa.2009.8.1303
[19]

Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
[20]

Daiwen Huang, Jingjun Zhang. Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1753-1773. doi: 10.3934/dcdss.2016073

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences