November  2017, 37(11): 5819-5841. doi: 10.3934/dcds.2017253

Energy transfer model for the derivative nonlinear Schrödinger equations on the torus

Department of Mathematics, Kobe University, 1-1, Rokkodai, Nada-ku, Kobe 657-8501, Japan

* Dedicated to Professor Yoshio Tsutsumi on his 60th birthday

Received  September 2016 Revised  June 2017 Published  July 2017

Fund Project: The author is supported by supported by JSPS KAKENHI Grant Number 10322794.

We consider the nonlinear derivative Schrödinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties of the particular solution consisting of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. The proof is based on the analysis of resonant dynamics via a finite dimensional ordinary differential system.

Citation: Hideo Takaoka. Energy transfer model for the derivative nonlinear Schrödinger equations on the torus. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5819-5841. doi: 10.3934/dcds.2017253
References:
[1]

H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.  Google Scholar

[2]

P. Clarkson and C. Cosgrove, Painlevé analysis of the non-linear Schrödinger family of equations, J. Phys. A: Math. Gen., 20 (1987), 2003-2024.  doi: 10.1088/0305-4470/20/8/020.  Google Scholar

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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

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J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.  Google Scholar

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[6]

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

[7]

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

[8]

A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

[9]

E. Haus and M. Procesi, KAM for beating solutions of the quintic NLS, Comm. Math. Phys., 354 (2017), 1101-1132.  doi: 10.1007/s00220-017-2925-7.  Google Scholar

[10]

N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[11]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[12]

S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not. , (2006), Art. ID 96763, 33 pp. doi: 10.1155/IMRN/2006/96763.  Google Scholar

[13]

C. MiaoY. Wu and G. Xu, Global well-posedness for Schrödinger equation with derivative in $H(\mathbb{R})^{\frac{1}{2}}$, J. Differential Equations, 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar

[14]

A. NahmodT. OhL. Rey-Bellet and G. Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., 14 (2012), 1275-1330.  doi: 10.4171/JEMS/333.  Google Scholar

[15]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163.  doi: 10.1512/iumj.1996.45.1962.  Google Scholar

[16]

H. Takaoka, Well-posedness for the one dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.   Google Scholar

[17]

H. Takaoka, A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below $H^{1/2}$, J. Differential Equations, 260 (2016), 818-859.  doi: 10.1016/j.jde.2015.09.011.  Google Scholar

[18]

Y. Y. S. Win, Global well-posedness of the derivative nonlinear Schrödinger equations on $\mathbf{T}$, Funkcial. Ekvac., 53 (2010), 51-88.  doi: 10.1619/fesi.53.51.  Google Scholar

show all references

References:
[1]

H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc., 353 (2001), 3649-3659.  doi: 10.1090/S0002-9947-01-02754-4.  Google Scholar

[2]

P. Clarkson and C. Cosgrove, Painlevé analysis of the non-linear Schrödinger family of equations, J. Phys. A: Math. Gen., 20 (1987), 2003-2024.  doi: 10.1088/0305-4470/20/8/020.  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, A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., 34 (2002), 64-86.  doi: 10.1137/S0036141001394541.  Google Scholar

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.  Google Scholar

[6]

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

[7]

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

[8]

A. Grünrock and S. Herr, Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data, SIAM J. Math. Anal., 39 (2008), 1890-1920.  doi: 10.1137/070689139.  Google Scholar

[9]

E. Haus and M. Procesi, KAM for beating solutions of the quintic NLS, Comm. Math. Phys., 354 (2017), 1101-1132.  doi: 10.1007/s00220-017-2925-7.  Google Scholar

[10]

N. Hayashi, The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20 (1993), 823-833.  doi: 10.1016/0362-546X(93)90071-Y.  Google Scholar

[11]

N. Hayashi and T. Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D, 55 (1992), 14-36.  doi: 10.1016/0167-2789(92)90185-P.  Google Scholar

[12]

S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not. , (2006), Art. ID 96763, 33 pp. doi: 10.1155/IMRN/2006/96763.  Google Scholar

[13]

C. MiaoY. Wu and G. Xu, Global well-posedness for Schrödinger equation with derivative in $H(\mathbb{R})^{\frac{1}{2}}$, J. Differential Equations, 251 (2011), 2164-2195.  doi: 10.1016/j.jde.2011.07.004.  Google Scholar

[14]

A. NahmodT. OhL. Rey-Bellet and G. Staffilani, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., 14 (2012), 1275-1330.  doi: 10.4171/JEMS/333.  Google Scholar

[15]

T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137-163.  doi: 10.1512/iumj.1996.45.1962.  Google Scholar

[16]

H. Takaoka, Well-posedness for the one dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations, 4 (1999), 561-580.   Google Scholar

[17]

H. Takaoka, A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below $H^{1/2}$, J. Differential Equations, 260 (2016), 818-859.  doi: 10.1016/j.jde.2015.09.011.  Google Scholar

[18]

Y. Y. S. Win, Global well-posedness of the derivative nonlinear Schrödinger equations on $\mathbf{T}$, Funkcial. Ekvac., 53 (2010), 51-88.  doi: 10.1619/fesi.53.51.  Google Scholar

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