doi: 10.3934/dcds.2020328

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

* 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.

Citation: Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 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

[3]

V. BisogninM. 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

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. BisogninM. 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

[1]

Ricardo Weder, Dimitri Yafaev. Inverse scattering at a fixed energy for long-range potentials. Inverse Problems & Imaging, 2007, 1 (1) : 217-224. doi: 10.3934/ipi.2007.1.217

[2]

Juan Kalemkerian, Andrés Sosa. Long-range dependence in the volatility of returns in Uruguayan sovereign debt indices. Journal of Dynamics & Games, 2020, 7 (3) : 225-237. doi: 10.3934/jdg.2020016

[3]

Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555

[4]

Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027

[5]

Xing Cheng, Ze Li, Lifeng Zhao. Scattering of solutions to the nonlinear Schrödinger equations with regular potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2999-3023. doi: 10.3934/dcds.2017129

[6]

Van Duong Dinh. A unified approach for energy scattering for focusing nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6441-6471. doi: 10.3934/dcds.2020286

[7]

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

[8]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235

[9]

Bartosz Bieganowski, Jaros law Mederski. Nonlinear SchrÖdinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (1) : 143-161. doi: 10.3934/cpaa.2018009

[10]

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144

[11]

Juan Belmonte-Beitia, Víctor M. Pérez-García, Vadym Vekslerchik, Pedro J. Torres. Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 221-233. doi: 10.3934/dcdsb.2008.9.221

[12]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831

[13]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

[14]

Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020407

[15]

J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467

[16]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[17]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188

[18]

Alessio Pomponio, Simone Secchi. A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities. Communications on Pure & Applied Analysis, 2010, 9 (3) : 741-750. doi: 10.3934/cpaa.2010.9.741

[19]

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

[20]

João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621

2019 Impact Factor: 1.338

Article outline

[Back to Top]