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

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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 and 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.
[2]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
[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.
[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.
[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.
[6]	D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.
[7]	M. Hamano, Global dynamics below the ground state for the quadratic Schödinger system in 5D, preprint, arXiv: 1805.12245.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[2]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
[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.
[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.
[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.
[6]	D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361.
[7]	M. Hamano, Global dynamics below the ground state for the quadratic Schödinger system in 5D, preprint, arXiv: 1805.12245.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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