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July  2022, 27(7): 3749-3778. doi: 10.3934/dcdsb.2021205

Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation

1. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

2. 

Graduate School of China Academy of Engineering Physics, Beijing 100088, China

* Corresponding author: Jun Wu

Received  February 2021 Revised  June 2021 Published  July 2022 Early access  August 2021

The main purpose of this paper is to study local regularity properties of the fourth-order nonlinear Schrödinger equations on the half line. We prove the local existence, uniqueness, and continuous dependence on initial data in low regularity Sobolev spaces. We also obtain the nonlinear smoothing property: the nonlinear part of the solution on the half line is smoother than the initial data.

Citation: Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205
References:
[1]

J. L. BonaS. M. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[2]

J. L. BonaS. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[3]

J. L. BonaS.-M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.  doi: 10.1016/j.matpur.2017.11.001.

[4]

J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 107–156,209–262. doi: 10.1007/BF01895688.

[5]

R. de A. Capistrano–FilhoM. Cavalcante and F. A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific J. Math., 309 (2020), 35-70.  doi: 10.2140/pjm.2020.309.35.

[6]

M. Chen and S. Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal., 190 (2020), 111608, 23 pp. doi: 10.1016/j.na.2019.111608.

[7]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.  doi: 10.1081/PDE-120016157.

[8]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good'' Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859.  doi: 10.1016/j.jde.2017.02.016.

[9]

M. DanielL. Kavitha and R. Amuda, Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole-dipole interaction, Phys. Rev. B, 59 (1999), 13774-13781.  doi: 10.1103/PhysRevB.59.13774.

[10]

T. A. Davydova and Y. A. Zaliznyak, Schrödinger ordinary solitons and chirped solitons: Fourth-order dispersive effects and cubic-quintic nonlinearity, Phys. D, 156 (2001), 260-282.  doi: 10.1016/S0167-2789(01)00269-X.

[11]

M. B. EdroǧanT. B. Gürel and N. Tzirakis, The derivative nonlinear Schrödinger equation on the half line, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1947-1973.  doi: 10.1016/j.anihpc.2018.03.006.

[12]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the cubic nonlinear Schrödinger equation on the half line, J. Funct. Aanl., 271 (2016), 2539-2568.  doi: 10.1016/j.jfa.2016.08.012.

[13]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[14]

H. Hirayama and M. Okamoto, Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity, Commun. Pure Appl. Anal., 15 (2016), 831-851.  doi: 10.3934/cpaa.2016.15.831.

[15]

J. Holmer, The initial-boundary value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668. 

[16]

J. Holmer, The initial-boundary value problem for Korteweg-de Vries equation, Comm. Partial Differential Equations, 31 (2006), 1151-1190.  doi: 10.1080/03605300600718503.

[17]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.

[18]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.

[19]

Z. Huo and Y. Jia, Well-posedness for the fourth-order nonlinear derivative Schrödinger equation in higher dimension, J. Math. Pures Appl., 96 (2011), 190-206.  doi: 10.1016/j.matpur.2011.01.002.

[20]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.  doi: 10.1103/PhysRevE.53.R1336.

[21]

V. I. Karpman and A. G. Shagalov, Solitons and their stability in high dispersive systems. I. Fourth-order nonlinear Schrödinger-type equations with power-law nonlinearities, Phys. Lett. A, 228 (1997), 59-65.  doi: 10.1016/S0375-9601(97)00063-7.

[22]

C. E. KenigG. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[23]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[24]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[25]

C. Laurey, The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.  doi: 10.1016/S0362-546X(96)00081-8.

[26]

T. Özsari and N. Yolcu, The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line, Commun. Pure Appl. Anal., 18 (2019), 3285-3316.  doi: 10.3934/cpaa.2019148.

[27]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.

[28]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.

[29]

M. RuzhanskyB. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl., 105 (2016), 31-65.  doi: 10.1016/j.matpur.2015.09.005.

[30]

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

show all references

References:
[1]

J. L. BonaS. M. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer. Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[2]

J. L. BonaS. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[3]

J. L. BonaS.-M. Sun and B.-Y. Zhang, Nonhomogeneous boundary-value problems for one-dimensional nonlinear Schrödinger equations, J. Math. Pures Appl., 109 (2018), 1-66.  doi: 10.1016/j.matpur.2017.11.001.

[4]

J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equation, part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 107–156,209–262. doi: 10.1007/BF01895688.

[5]

R. de A. Capistrano–FilhoM. Cavalcante and F. A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific J. Math., 309 (2020), 35-70.  doi: 10.2140/pjm.2020.309.35.

[6]

M. Chen and S. Zhang, Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities, Nonlinear Anal., 190 (2020), 111608, 23 pp. doi: 10.1016/j.na.2019.111608.

[7]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.  doi: 10.1081/PDE-120016157.

[8]

E. Compaan and N. Tzirakis, Well-posedness and nonlinear smoothing for the "good'' Boussinesq equation on the half-line, J. Differential Equations, 262 (2017), 5824-5859.  doi: 10.1016/j.jde.2017.02.016.

[9]

M. DanielL. Kavitha and R. Amuda, Soliton spin excitations in an anisotropic Heisenberg ferromagnet with octupole-dipole interaction, Phys. Rev. B, 59 (1999), 13774-13781.  doi: 10.1103/PhysRevB.59.13774.

[10]

T. A. Davydova and Y. A. Zaliznyak, Schrödinger ordinary solitons and chirped solitons: Fourth-order dispersive effects and cubic-quintic nonlinearity, Phys. D, 156 (2001), 260-282.  doi: 10.1016/S0167-2789(01)00269-X.

[11]

M. B. EdroǧanT. B. Gürel and N. Tzirakis, The derivative nonlinear Schrödinger equation on the half line, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1947-1973.  doi: 10.1016/j.anihpc.2018.03.006.

[12]

M. B. Erdoǧan and N. Tzirakis, Regularity properties of the cubic nonlinear Schrödinger equation on the half line, J. Funct. Aanl., 271 (2016), 2539-2568.  doi: 10.1016/j.jfa.2016.08.012.

[13]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[14]

H. Hirayama and M. Okamoto, Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity, Commun. Pure Appl. Anal., 15 (2016), 831-851.  doi: 10.3934/cpaa.2016.15.831.

[15]

J. Holmer, The initial-boundary value problem for the 1D nonlinear Schrödinger equation on the half-line, Differential Integral Equations, 18 (2005), 647-668. 

[16]

J. Holmer, The initial-boundary value problem for Korteweg-de Vries equation, Comm. Partial Differential Equations, 31 (2006), 1151-1190.  doi: 10.1080/03605300600718503.

[17]

Z. Huo and Y. Jia, The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament, J. Differential Equations, 214 (2005), 1-35.  doi: 10.1016/j.jde.2004.09.005.

[18]

Z. Huo and Y. Jia, A refined well-posedness for the fourth-order nonlinear Schrödinger equation related to the vortex filament, Comm. Partial Differential Equations, 32 (2007), 1493-1510.  doi: 10.1080/03605300701629385.

[19]

Z. Huo and Y. Jia, Well-posedness for the fourth-order nonlinear derivative Schrödinger equation in higher dimension, J. Math. Pures Appl., 96 (2011), 190-206.  doi: 10.1016/j.matpur.2011.01.002.

[20]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.  doi: 10.1103/PhysRevE.53.R1336.

[21]

V. I. Karpman and A. G. Shagalov, Solitons and their stability in high dispersive systems. I. Fourth-order nonlinear Schrödinger-type equations with power-law nonlinearities, Phys. Lett. A, 228 (1997), 59-65.  doi: 10.1016/S0375-9601(97)00063-7.

[22]

C. E. KenigG. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[23]

C. E. KenigG. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[24]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[25]

C. Laurey, The Cauchy problem for a third order nonlinear Schrödinger equation, Nonlinear Anal., 29 (1997), 121-158.  doi: 10.1016/S0362-546X(96)00081-8.

[26]

T. Özsari and N. Yolcu, The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line, Commun. Pure Appl. Anal., 18 (2019), 3285-3316.  doi: 10.3934/cpaa.2019148.

[27]

B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dyn. Partial Differ. Equ., 4 (2007), 197-225.  doi: 10.4310/DPDE.2007.v4.n3.a1.

[28]

B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.

[29]

M. RuzhanskyB. Wang and H. Zhang, Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces, J. Math. Pures Appl., 105 (2016), 31-65.  doi: 10.1016/j.matpur.2015.09.005.

[30]

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

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