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On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $
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Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five
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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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