American Institute of Mathematical Sciences

Advanced
July  2009, 25(2): 617-626. doi: 10.3934/dcds.2009.25.617

Analyticity of the nonlinear scattering operator

Benoît Pausader 1, and  Walter A. Strauss 1,

1. 

Department of Mathematics, Brown University, Providence, RI 02912, United States, United States

Received  March 2008 Revised  April 2009 Published  June 2009

We present a new and simpler proof that the nonlinear scattering operator $\S$ is analytic on energy space. We apply it in particular to a fourth-order nonlinear wave equation in Rn. In addition, we prove that $\S$ determines the scatterer uniquely and that for small powers there is no scattering.
Keywords: NLS, nonlinear beam equation, analytic operator., Nonlinear scattering, NLKG.
Mathematics Subject Classification: Primary: 74J25, 35P25; Secondary: 35Q7.
Citation: Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617
[1]

Pao-Liu Chow. Asymptotic solutions of a nonlinear stochastic beam equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 735-749. doi: 10.3934/dcdsb.2006.6.735
[2]

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
[3]

Yoshinori Morimoto, Chao-Jiang Xu. Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules. Kinetic & Related Models, 2020, 13 (5) : 951-978. doi: 10.3934/krm.2020033
[4]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[5]

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
[6]

Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki. Scattering and inverse scattering for nonlinear quantum walks. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3687-3703. doi: 10.3934/dcds.2018159
[7]

Yanan Li, Zhijian Yang, Fang Da. Robust attractors for a perturbed non-autonomous extensible beam equation with nonlinear nonlocal damping. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5975-6000. doi: 10.3934/dcds.2019261
[8]

K. T. Andrews, M. F. M'Bengue, Meir Shillor. Vibrations of a nonlinear dynamic beam between two stops. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 23-38. doi: 10.3934/dcdsb.2009.12.23
[9]

Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180
[10]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029
[11]

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
[12]

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
[13]

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
[14]

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
[15]

Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055
[16]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
[17]

Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197
[18]

A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052
[19]

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
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences