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November  2020, 40(11): 6351-6378. doi: 10.3934/dcds.2020283

Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations

Department of Mathematics, Kobe University, Kobe, 657-8501, Japan

Received  December 2019 Revised  June 2020 Published  July 2020

Fund Project: This work was supported by JSPS KAKENHI Grant Number 18H01129

We study the dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrödinger equation. Grébert and Thomann [9] proved that there exist solutions with initial data built on four Fourier modes, that confirm the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar energy exchange in long-period waves.

Citation: Hideo Takaoka. Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6351-6378. doi: 10.3934/dcds.2020283
References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅰ: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.  Google Scholar

[2]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[4]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.  doi: 10.1007/s00222-010-0242-2.  Google Scholar

[5]

B. G. Dodson, Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension for low regularity data, Int. Math. Res. Not. IMRN, 2012 (2012), 870-893.  doi: 10.1093/imrn/rnr037.  Google Scholar

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.  Google Scholar

[7]

E. FaouP. Germain and Z. Hani, The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., 29 (2016), 915-982.  doi: 10.1090/jams/845.  Google Scholar

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[9]

B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 455-477.  doi: 10.1016/j.anihpc.2012.01.005.  Google Scholar

[10] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[11]

M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[12]

H. Takaoka, Energy transfer model for the derivative nonlinear Schrödinger equations on the torus, Discrete Contin. Dyn. Syst., 37 (2017), 5819-5841.  doi: 10.3934/dcds.2017253.  Google Scholar

[13]

Y. Tsutsumi, $\mathrm{L}^{2}$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

show all references

References:
[1]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part Ⅰ: Schrödinger equation, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.  Google Scholar

[2]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.  Google Scholar

[4]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math., 181 (2010), 39-113.  doi: 10.1007/s00222-010-0242-2.  Google Scholar

[5]

B. G. Dodson, Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension for low regularity data, Int. Math. Res. Not. IMRN, 2012 (2012), 870-893.  doi: 10.1093/imrn/rnr037.  Google Scholar

[6]

B. Dodson, Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schrödinger equation when $d = 1$, Amer. J. Math., 138 (2016), 531-569.  doi: 10.1353/ajm.2016.0016.  Google Scholar

[7]

E. FaouP. Germain and Z. Hani, The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation, J. Amer. Math. Soc., 29 (2016), 915-982.  doi: 10.1090/jams/845.  Google Scholar

[8]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Functional Analysis, 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.  Google Scholar

[9]

B. Grébert and L. Thomann, Resonant dynamics for the quintic nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 455-477.  doi: 10.1016/j.anihpc.2012.01.005.  Google Scholar

[10] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, The Clarendon Press, Oxford University Press, New York, 1979.   Google Scholar
[11]

M. B. Nathanson, Additive Number Theory. Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-3845-2.  Google Scholar

[12]

H. Takaoka, Energy transfer model for the derivative nonlinear Schrödinger equations on the torus, Discrete Contin. Dyn. Syst., 37 (2017), 5819-5841.  doi: 10.3934/dcds.2017253.  Google Scholar

[13]

Y. Tsutsumi, $\mathrm{L}^{2}$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

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