- Previous Article
- DCDS Home
- This Issue
-
Next Article
Existence of nodal solutions for the sublinear Moore-Nehari differential equation
Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations
1. | Department of Mathematics and Statistics, Missouri University of Science & Technology, Rolla, MO, USA |
2. | Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan |
We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
References:
[1] |
J. E. Barab,
Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, , Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976. x+207 pp. |
[3] |
V. Bisognin, M. Sepúlveda and O. Vera,
On the nonexistence of asymptotically free solutions for a coupled nonlinear Schrödinger system, Appl. Numer. Math., 59 (2009), 2285-2302.
doi: 10.1016/j.apnum.2008.12.017. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323pp.
doi: 10.1090/cln/010. |
[5] |
Y. Cho and T. Ozawa,
On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.
doi: 10.1137/060653688. |
[6] |
S. Cuccagna and M. Maeda, On stability of small solitons of the 1–D NLS with a trapping delta potential, SIAM J. Math. Anal., 51 (2019), 4311–4331, arXiv: 1904.11869.
doi: 10.1137/19M1258402. |
[7] |
R. T. Glassey,
On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200.
doi: 10.1090/S0002-9947-1973-0330782-7. |
[8] |
R. T. Glassey,
Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations, Comm. Math. Phys., 53 (1977), 9-18.
doi: 10.1007/BF01609164. |
[9] |
J. Ginibre and G. Velo,
Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.
doi: 10.1007/BF02099195. |
[10] |
N. Hayashi, C. Li and P. Naumkin, Nonexistence of asymptotically free solutions to nonlinear Schrödinger systems, Electron. J. Differential Equations, 2012 (2012), 14 pp. |
[11] |
N. Hayashi, P. Naumkin and T. Niizato, Nonexistence of the usual scattering states for the generalized Ostrovsky-Hunter equation, J. Math. Phys., 55 (2014), 053502, 11pp.
doi: 10.1063/1.4874107. |
[12] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[13] |
S. Masaki and H. Miyazaki,
Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical inhomogeneous nonlinearity, Differential Integral Equations, 32 (2019), 121-138.
|
[14] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.
![]() |
[15] |
A. Shimomura,
Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.
|
[16] |
A. Shimomura and Y. Tsutsumi,
Nonexistence of scattering states for some quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 19 (2006), 1047-1060.
|
[17] |
W. A. Strauss, Nonlinear scattering theory, in Scattering Theory in Mathematical Physics, Reidel, Dordrecht, 9 (1974), 53–78.
doi: 10.1007/978-94-010-2147-0_3. |
[18] |
R. S. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
show all references
References:
[1] |
J. E. Barab,
Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[2] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, , Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976. x+207 pp. |
[3] |
V. Bisognin, M. Sepúlveda and O. Vera,
On the nonexistence of asymptotically free solutions for a coupled nonlinear Schrödinger system, Appl. Numer. Math., 59 (2009), 2285-2302.
doi: 10.1016/j.apnum.2008.12.017. |
[4] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323pp.
doi: 10.1090/cln/010. |
[5] |
Y. Cho and T. Ozawa,
On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.
doi: 10.1137/060653688. |
[6] |
S. Cuccagna and M. Maeda, On stability of small solitons of the 1–D NLS with a trapping delta potential, SIAM J. Math. Anal., 51 (2019), 4311–4331, arXiv: 1904.11869.
doi: 10.1137/19M1258402. |
[7] |
R. T. Glassey,
On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200.
doi: 10.1090/S0002-9947-1973-0330782-7. |
[8] |
R. T. Glassey,
Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations, Comm. Math. Phys., 53 (1977), 9-18.
doi: 10.1007/BF01609164. |
[9] |
J. Ginibre and G. Velo,
Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys., 144 (1992), 163-188.
doi: 10.1007/BF02099195. |
[10] |
N. Hayashi, C. Li and P. Naumkin, Nonexistence of asymptotically free solutions to nonlinear Schrödinger systems, Electron. J. Differential Equations, 2012 (2012), 14 pp. |
[11] |
N. Hayashi, P. Naumkin and T. Niizato, Nonexistence of the usual scattering states for the generalized Ostrovsky-Hunter equation, J. Math. Phys., 55 (2014), 053502, 11pp.
doi: 10.1063/1.4874107. |
[12] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[13] |
S. Masaki and H. Miyazaki,
Nonexistence of scattering and modified scattering states for some nonlinear Schrödinger equation with critical inhomogeneous nonlinearity, Differential Integral Equations, 32 (2019), 121-138.
|
[14] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.
![]() |
[15] |
A. Shimomura,
Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.
|
[16] |
A. Shimomura and Y. Tsutsumi,
Nonexistence of scattering states for some quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 19 (2006), 1047-1060.
|
[17] |
W. A. Strauss, Nonlinear scattering theory, in Scattering Theory in Mathematical Physics, Reidel, Dordrecht, 9 (1974), 53–78.
doi: 10.1007/978-94-010-2147-0_3. |
[18] |
R. S. Strichartz,
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[1] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[2] |
Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267 |
[3] |
Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021021 |
[4] |
Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 |
[5] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 |
[6] |
Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 |
[7] |
Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 |
[8] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[9] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
[10] |
Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 |
[11] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[12] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[13] |
Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024 |
[14] |
Jingni Guo, Junxiang Xu, Zhenggang He, Wei Liao. Research on cascading failure modes and attack strategies of multimodal transport network. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2020159 |
[15] |
Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452 |
[16] |
Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493 |
[17] |
Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 |
[18] |
Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031 |
[19] |
A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 |
[20] |
Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]