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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

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