American Institute of Mathematical Sciences

Advanced
November  2009, 24(4): 1113-1127. doi: 10.3934/dcds.2009.24.1113

On scattering for NLS: From Euclidean to hyperbolic space

Valeria Banica 1, , Rémi Carles 2, and  Thomas Duyckaerts 3,

1. 

Département de Mathématiques, Univ. Evry, Bd. F. Mitterrand, 91025 Evry, France
2. 

Université Montpellier 2, Mathématiques, CC051, 34095 Montpellier, CNRS, UMR 5149, 34095 Montpellier
3. 

Département de Mathématiques, Univ. Cergy-Pontoise, CNRS UMR 8088, 2 avenue Adolphe Chauvin, BP 222, Pontoise, 95302 Cergy-Pontoise cedex, France

Received  March 2008 Revised  January 2009 Published  May 2009

We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space $\mathbb H^n$ in the radial case, for $n\ge 4$, and any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which sort of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.
Keywords: Nonlinear Schrödinger equation, scattering theory, hyperbolic space..
Mathematics Subject Classification: Primary: 35P25, 35Q55; Secondary: 35B40, 58J5.
Citation: Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113
[1]

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
[2]

Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122
[3]

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
[4]

Alexei Rybkin. On the boundary control approach to inverse spectral and scattering theory for Schrödinger operators. Inverse Problems & Imaging, 2009, 3 (1) : 139-149. doi: 10.3934/ipi.2009.3.139
[5]

Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183
[6]

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
[7]

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
[8]

Kimitoshi Tsutaya. Scattering theory for the wave equation of a Hartree type in three space dimensions. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2261-2281. doi: 10.3934/dcds.2014.34.2261
[9]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
[10]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[12]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284
[13]

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, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[14]

Jiaxi Huang, Youde Wang, Lifeng Zhao. Equivariant Schrödinger map flow on two dimensional hyperbolic space. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4379-4425. doi: 10.3934/dcds.2020184
[15]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[16]

Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323
[17]

Nan Lu. Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3533-3567. doi: 10.3934/dcds.2015.35.3533
[18]

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
[19]

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
[20]

Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy