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August  2021, 41(8): 3817-3836. doi: 10.3934/dcds.2021018

Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five

Norman Noguera 1, and  Ademir Pastor 2,,

1. 

SM-UCR, Ciudad Universitaria Carlos Monge Alfaro, Departamento de Ciencias Naturales, Apdo: 111-4250, San Ramón, Alajuela, Costa Rica
2. 

IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, 13083-859, Campinas, São Paulo, Brazil

* Corresponding author

Received  August 2020 Published  August 2021 Early access  January 2021

Fund Project: The second author is partially supported by CNPq/Brazil grant 303762/2019-5 and FAPESP/Brazil grant 2019/02512-5

In this paper we study the scattering of radial solutions to a $ l $-component system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on the radial Sobolev embedding and a Morawetz estimate.

Keywords: Nonlinear Schrödinger system, quadratic-type interactions, scaterring, radial solutions, Morawetz estimate.
Mathematics Subject Classification: Primary:35Q55, 35B40;Secondary:35A01.
Citation: Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018
References:
[1]

A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Contin. Dyn. Syst., 39 (2019), 6643-6668.  doi: 10.3934/dcds.2019289.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[3]

M. Colin, L. Di Menza and J. C. Saut, Solitons in quadratic media, Nonlinearity, 29 (2016), 1000-1035.  doi: 10.1088/0951-7715/29/3/1000.  Google Scholar

[4]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.  doi: 10.1090/proc/13678.  Google Scholar

[5]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.  doi: 10.4310/MRL.2018.v25.n6.a5.  Google Scholar

[6]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.  doi: 10.1142/S0219891605000361.  Google Scholar

[7]

M. Hamano, Global dynamics below the ground state for the quadratic Schödinger system in 5D, preprint, arXiv: 1805.12245. Google Scholar

[8]

M. Hamano, T. Inui and K. Nishimura, Scattering for the quadratic nonlinear Schrödinger system in $ \mathbb{R}^5$ without mass-resonance condition, preprint, arXiv: 1903.05880. Google Scholar

[9]

N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.  Google Scholar

[10]

T. Inui, N. Kishimoto and K. Nishimura, Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst., 39 (2019), 6299-6353.  doi: 10.3934/dcds.2019275.  Google Scholar

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

Y. S. Kivshar, A. A. Sukhorukov, E. A. Ostrovskaya, T. J. Alexander, O. Bang, S. M. Saltiel, C. B. Clausen and P. L. Christiansen, Multi-component optical solitary waves, Physica A: Statistical Mechanics and its Applications, 288 (2000), 152-173.  doi: 10.1016/S0378-4371(00)00420-9.  Google Scholar

[13]

F. Meng and C. Xu, Scattering for mass-resonance nonlinear Schrödinger system in 5D, J. Differential Equations, 275 (2021), 837-857.  doi: 10.1016/j.jde.2020.11.005.  Google Scholar

[14]

N. Noguera and A. Pastor, Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six, preprint, arXiv: 2003.11103. Google Scholar

[15]

N. Noguera and A. Pastor, On the dynamics of a quadratic Schrödinger system in dimension $n = 5$, Dyn. Partial Differ. Equ., 17 (2020), 1. doi: 10.4310/DPDE.2020.v17.n1.a1.  Google Scholar

[16]

N. Noguera and A. Pastor, A system of Schrödinger equations with general quadratic-type nonlinearities, to appear in Commun. Contemp. Math., 2021. doi: 10.1142/S0219199720500236.  Google Scholar

[17]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[18]

A. Pastor, On three-wave interaction Schrödinger systems with quadratic nonlinearities: global well-posedness and standing waves, Commun. Pure Appl. Anal., 18 (2019), 2217-2242.  doi: 10.3934/cpaa.2019100.  Google Scholar

[19]

T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

[20]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/surv/081.  Google Scholar

[21]

H. Wang and Q. Yang, Scattering for the 5D quadratic NLS system without mass-resonance, J. Math. Phys., 60 (2019), 121508, 23 pp. doi: 10.1063/1.5119293.  Google Scholar

show all references

References:
[1]

A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Contin. Dyn. Syst., 39 (2019), 6643-6668.  doi: 10.3934/dcds.2019289.  Google Scholar

[2]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[3]

M. Colin, L. Di Menza and J. C. Saut, Solitons in quadratic media, Nonlinearity, 29 (2016), 1000-1035.  doi: 10.1088/0951-7715/29/3/1000.  Google Scholar

[4]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867.  doi: 10.1090/proc/13678.  Google Scholar

[5]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825.  doi: 10.4310/MRL.2018.v25.n6.a5.  Google Scholar

[6]

D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.  doi: 10.1142/S0219891605000361.  Google Scholar

[7]

M. Hamano, Global dynamics below the ground state for the quadratic Schödinger system in 5D, preprint, arXiv: 1805.12245. Google Scholar

[8]

M. Hamano, T. Inui and K. Nishimura, Scattering for the quadratic nonlinear Schrödinger system in $ \mathbb{R}^5$ without mass-resonance condition, preprint, arXiv: 1903.05880. Google Scholar

[9]

N. Hayashi, T. Ozawa and K. Tanaka, On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 661-690.  doi: 10.1016/j.anihpc.2012.10.007.  Google Scholar

[10]

T. Inui, N. Kishimoto and K. Nishimura, Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition, Discrete Contin. Dyn. Syst., 39 (2019), 6299-6353.  doi: 10.3934/dcds.2019275.  Google Scholar

[11]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[12]

Y. S. Kivshar, A. A. Sukhorukov, E. A. Ostrovskaya, T. J. Alexander, O. Bang, S. M. Saltiel, C. B. Clausen and P. L. Christiansen, Multi-component optical solitary waves, Physica A: Statistical Mechanics and its Applications, 288 (2000), 152-173.  doi: 10.1016/S0378-4371(00)00420-9.  Google Scholar

[13]

F. Meng and C. Xu, Scattering for mass-resonance nonlinear Schrödinger system in 5D, J. Differential Equations, 275 (2021), 837-857.  doi: 10.1016/j.jde.2020.11.005.  Google Scholar

[14]

N. Noguera and A. Pastor, Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six, preprint, arXiv: 2003.11103. Google Scholar

[15]

N. Noguera and A. Pastor, On the dynamics of a quadratic Schrödinger system in dimension $n = 5$, Dyn. Partial Differ. Equ., 17 (2020), 1. doi: 10.4310/DPDE.2020.v17.n1.a1.  Google Scholar

[16]

N. Noguera and A. Pastor, A system of Schrödinger equations with general quadratic-type nonlinearities, to appear in Commun. Contemp. Math., 2021. doi: 10.1142/S0219199720500236.  Google Scholar

[17]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[18]

A. Pastor, On three-wave interaction Schrödinger systems with quadratic nonlinearities: global well-posedness and standing waves, Commun. Pure Appl. Anal., 18 (2019), 2217-2242.  doi: 10.3934/cpaa.2019100.  Google Scholar

[19]

T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

[20]

M. E. Taylor, Tools for PDE, Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/surv/081.  Google Scholar

[21]

H. Wang and Q. Yang, Scattering for the 5D quadratic NLS system without mass-resonance, J. Math. Phys., 60 (2019), 121508, 23 pp. doi: 10.1063/1.5119293.  Google Scholar

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