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March  2021, 41(3): 1507-1517. doi: 10.3934/dcds.2020328

Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations

Jason Murphy 1,, and  Kenji Nakanishi 2,

1. 

Department of Mathematics and Statistics, Missouri University of Science & Technology, Rolla, MO, USA
2. 

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

* Corresponding author: Jason Murphy

Received  July 2019 Revised  September 2019 Published  September 2020

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.

Keywords: nonlinear Schrödinger equations, long-range nonlinearities, failure of scattering.
Mathematics Subject Classification: Primary: 35Q55.
Citation: Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.   Google Scholar

[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.   Google Scholar
[15]

A. Shimomura, Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[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.   Google Scholar

[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅱ. Fourier Analysis, Self-adjointness, Academic Press, New York-London, 1975.   Google Scholar
[15]

A. Shimomura, Nonexistence of asymptotically free solutions for quadratic nonlinear Schrödinger equations in two space dimensions, Differential Integral Equations, 18 (2005), 325-335.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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