# American Institute of Mathematical Sciences

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. Baldi, M. Berti, E. 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. Baldi, M. Berti, E. 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. Baldi, M. 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. Baldi, M. 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. Berti, L. 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. Eliasson, B. 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. Feola, F. 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. Geng, X. 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. Zhang, M. 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. Baldi, M. Berti, E. 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. Baldi, M. Berti, E. 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. Baldi, M. 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. Baldi, M. 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. Berti, L. 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. Eliasson, B. 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. Feola, F. 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. Geng, X. 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. Zhang, M. 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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