doi: 10.3934/dcds.2022064
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Invariant tori for the derivative nonlinear Schrödinger equation with nonlinear term depending on spatial variable

1. 

College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

2. 

School of Mathematics, Shandong University, Jinan 250100, China

*Corresponding author: Shimin Wang

Received  September 2021 Revised  January 2022 Early access May 2022

We focus on a class of derivative nonlinear Schrödinger equation with reversible nonlinear term depending on spatial variable
$ x $
:
$ \begin{equation*} \mathrm{i} u_t+u_{xx}-\bar{u}u_{x}^2 + F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = 0, \quad x\in \mathbb{T}: = \mathbb{R}/2\pi\mathbb{Z}, \end{equation*} $
where the nonlinear term
$ F $
is an analytic function of order at least five in
$ u, \bar{u}, u_{x}, \bar{u}_{x} $
and satisfies
$ \begin{equation*} F(x, u, \bar{u}, u_{x}, \bar{u}_{x}) = \overline{F(x, \bar{u}, u, \bar{u}_{x}, u_{x})}. \end{equation*} $
Moreover, we also assume that
$ F $
satisfies the homogeneous condition (6) to overcome the degeneracy. We prove the existence of small amplitude, smooth quasi-periodic solutions for the above equation via establishing an abstract infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with unbounded perturbation.
Citation: Zhaowei Lou, Jianguo Si, Shimin Wang. Invariant tori for the derivative nonlinear Schrödinger equation with nonlinear term depending on spatial variable. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022064
References:
[1]

P. BaldiM. BertiE. Haus and R. Montalto, KAM for gravity water waves in finite depth, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 215-236.  doi: 10.4171/RLM/802.

[2]

P. BaldiM. BertiE. Haus and R. Montalto, Time quasi-periodic gravity water waves in finite depth, Invent. Math., 214 (2018), 739-911.  doi: 10.1007/s00222-018-0812-2.

[3]

P. BaldiM. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.  doi: 10.1007/s00208-013-1001-7.

[4]

P. BaldiM. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[5]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301–373. doi: 10.24033/asens.2190.

[6]

M. BertiL. Biasco and M. Procesi, KAM for reversible derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955.  doi: 10.1007/s00205-014-0726-0.

[7]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.  doi: 10.2307/121001.

[8]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.  doi: 10.1007/s002200050824.

[9]

M. Daniel and E. Gutkin, The dynamics of generalized Heisenberg ferromagnetic spin chain, Chaos, 5 (1995), 439-442.  doi: 10.1063/1.166114.

[10]

L. EliassonB. Grébert and S. Kuksin, KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.  doi: 10.1007/s00039-016-0390-7.

[11]

L. Eliasson and S. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.  doi: 10.4007/annals.2010.172.371.

[12]

R. FeolaF. Giuliani and M. Procesi, Reducible KAM tori for the Degasperis-Procesi equation, Comm. Math. Phys., 377 (2020), 1681-1759.  doi: 10.1007/s00220-020-03788-z.

[13]

M. Gao and J. Liu, Invariant Cantor manifolds of quasi-periodic solutions for the derivative nonlinear Schrödinger equation, J. Differential Equations, 267 (2019), 1322-1375.  doi: 10.1016/j.jde.2019.02.010.

[14]

J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702, 27 pp. doi: 10.1063/1.4754822.

[15]

J. GengX. Xu and J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.  doi: 10.1016/j.aim.2011.01.013.

[16]

J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542.  doi: 10.1016/j.jde.2006.07.027.

[17]

J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372.  doi: 10.1007/s00220-005-1497-0.

[18]

T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[19]

S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math., 33 (1980), 241-263.  doi: 10.1002/cpa.3160330304.

[20]

S. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22-37. 

[21]

S. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271.  doi: 10.1007/PL00001476.

[22]

S. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys., 10 (1998), ii+64 pp.

[23]

S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179.  doi: 10.2307/2118656.

[24]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154.

[25]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Comm. Pure Appl. Math., 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314.

[26]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3.

[27]

J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652.  doi: 10.1016/j.jde.2013.11.007.

[28]

Z. Lou and J. Si, Quasi-periodic solutions for the reversible derivative nonlinear Schrödinger equations with periodic boundary conditions, J. Dynam. Differential Equations, 29 (2017), 1031-1069.  doi: 10.1007/s10884-015-9481-7.

[29]

Z. Lou and J. Si, Periodic and quasi-periodic solutions for reversible unbounded perturbations of linear Schrödinger equations, J. Dynam. Differential Equations, 32 (2020), 117-161.  doi: 10.1007/s10884-018-9722-7.

[30]

L. Mi and W. Cui, Invariant tori for the Schrödinger equation in the Heisenberg ferromagnetic chain, Appl. Anal., 98 (2019), 2440-2453.  doi: 10.1080/00036811.2018.1460817.

[31]

C. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.  doi: 10.1007/BF02104499.

[32]

Y. Wu and X. Yuan, A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications (in memory of Professor Walter Craig), J. Dynam. Differential Equations, 33 (2021), 1427-1474.  doi: 10.1007/s10884-021-09972-6.

[33]

X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274.  doi: 10.1016/j.jde.2005.12.012.

[34]

J. ZhangM. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.  doi: 10.1088/0951-7715/24/4/010.

show all references

References:
[1]

P. BaldiM. BertiE. Haus and R. Montalto, KAM for gravity water waves in finite depth, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 29 (2018), 215-236.  doi: 10.4171/RLM/802.

[2]

P. BaldiM. BertiE. Haus and R. Montalto, Time quasi-periodic gravity water waves in finite depth, Invent. Math., 214 (2018), 739-911.  doi: 10.1007/s00222-018-0812-2.

[3]

P. BaldiM. Berti and R. Montalto, KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation, Math. Ann., 359 (2014), 471-536.  doi: 10.1007/s00208-013-1001-7.

[4]

P. BaldiM. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1589-1638.  doi: 10.1016/j.anihpc.2015.07.003.

[5]

M. Berti, L. Biasco and M. Procesi, KAM theory for the Hamiltonian derivative wave equation, Ann. Sci. Éc. Norm. Supér., 46 (2013), 301–373. doi: 10.24033/asens.2190.

[6]

M. BertiL. Biasco and M. Procesi, KAM for reversible derivative wave equations, Arch. Ration. Mech. Anal., 212 (2014), 905-955.  doi: 10.1007/s00205-014-0726-0.

[7]

J. Bourgain, Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations, Ann. of Math., 148 (1998), 363-439.  doi: 10.2307/121001.

[8]

L. Chierchia and J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys., 211 (2000), 497-525.  doi: 10.1007/s002200050824.

[9]

M. Daniel and E. Gutkin, The dynamics of generalized Heisenberg ferromagnetic spin chain, Chaos, 5 (1995), 439-442.  doi: 10.1063/1.166114.

[10]

L. EliassonB. Grébert and S. Kuksin, KAM for the nonlinear beam equation, Geom. Funct. Anal., 26 (2016), 1588-1715.  doi: 10.1007/s00039-016-0390-7.

[11]

L. Eliasson and S. Kuksin, KAM for the nonlinear Schrödinger equation, Ann. of Math., 172 (2010), 371-435.  doi: 10.4007/annals.2010.172.371.

[12]

R. FeolaF. Giuliani and M. Procesi, Reducible KAM tori for the Degasperis-Procesi equation, Comm. Math. Phys., 377 (2020), 1681-1759.  doi: 10.1007/s00220-020-03788-z.

[13]

M. Gao and J. Liu, Invariant Cantor manifolds of quasi-periodic solutions for the derivative nonlinear Schrödinger equation, J. Differential Equations, 267 (2019), 1322-1375.  doi: 10.1016/j.jde.2019.02.010.

[14]

J. Geng and J. Wu, Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations, J. Math. Phys., 53 (2012), 102702, 27 pp. doi: 10.1063/1.4754822.

[15]

J. GengX. Xu and J. You, An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation, Adv. Math., 226 (2011), 5361-5402.  doi: 10.1016/j.aim.2011.01.013.

[16]

J. Geng and Y. Yi, Quasi-periodic solutions in a nonlinear Schrödinger equation, J. Differential Equations, 233 (2007), 512-542.  doi: 10.1016/j.jde.2006.07.027.

[17]

J. Geng and J. You, A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces, Comm. Math. Phys., 262 (2006), 343-372.  doi: 10.1007/s00220-005-1497-0.

[18]

T. Kappeler and J. Pöschel, KdV & KAM, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-08054-2.

[19]

S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string, Comm. Pure Appl. Math., 33 (1980), 241-263.  doi: 10.1002/cpa.3160330304.

[20]

S. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funktsional. Anal. i Prilozhen., 21 (1987), 22-37. 

[21]

S. Kuksin, On small-denominators equations with large variable coefficients, Z. Angew. Math. Phys., 48 (1997), 262-271.  doi: 10.1007/PL00001476.

[22]

S. Kuksin, A KAM-theorem for equations of the Korteweg-de Vries type, Rev. Math. Math. Phys., 10 (1998), ii+64 pp.

[23]

S. Kuksin and J. Pöschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math., 143 (1996), 149-179.  doi: 10.2307/2118656.

[24]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613.  doi: 10.1063/1.1704154.

[25]

J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Comm. Pure Appl. Math., 63 (2010), 1145-1172.  doi: 10.1002/cpa.20314.

[26]

J. Liu and X. Yuan, A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations, Comm. Math. Phys., 307 (2011), 629-673.  doi: 10.1007/s00220-011-1353-3.

[27]

J. Liu and X. Yuan, KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions, J. Differential Equations, 256 (2014), 1627-1652.  doi: 10.1016/j.jde.2013.11.007.

[28]

Z. Lou and J. Si, Quasi-periodic solutions for the reversible derivative nonlinear Schrödinger equations with periodic boundary conditions, J. Dynam. Differential Equations, 29 (2017), 1031-1069.  doi: 10.1007/s10884-015-9481-7.

[29]

Z. Lou and J. Si, Periodic and quasi-periodic solutions for reversible unbounded perturbations of linear Schrödinger equations, J. Dynam. Differential Equations, 32 (2020), 117-161.  doi: 10.1007/s10884-018-9722-7.

[30]

L. Mi and W. Cui, Invariant tori for the Schrödinger equation in the Heisenberg ferromagnetic chain, Appl. Anal., 98 (2019), 2440-2453.  doi: 10.1080/00036811.2018.1460817.

[31]

C. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys., 127 (1990), 479-528.  doi: 10.1007/BF02104499.

[32]

Y. Wu and X. Yuan, A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications (in memory of Professor Walter Craig), J. Dynam. Differential Equations, 33 (2021), 1427-1474.  doi: 10.1007/s10884-021-09972-6.

[33]

X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274.  doi: 10.1016/j.jde.2005.12.012.

[34]

J. ZhangM. Gao and X. Yuan, KAM tori for reversible partial differential equations, Nonlinearity, 24 (2011), 1189-1228.  doi: 10.1088/0951-7715/24/4/010.

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