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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. |
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