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]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[2]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[3]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[4]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[5]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[6]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[7]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[8]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264
[9]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[10]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243
[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[12]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448
[14]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[15]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[16]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[17]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[18]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[19]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268
[20]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences