American Institute of Mathematical Sciences

Advanced
January  2019, 39(1): 219-239. doi: 10.3934/dcds.2019009

Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion

Jean-Claude Saut 1, and  Jun-Ichi Segata 2,

1. 

Laboratoire de Mathématiques, CNRS and Université Paris-Sud, 91405 Orsay, France
2. 

Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan

Received  October 2017 Revised  March 2018 Published  October 2018

We consider the asymptotic behavior in time of solutions to the nonlinear Schrödinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS) as $t\to∞$.

Keywords: Schrödinger equation with higher order dispersion, scattering problem.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.
Citation: Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009
References:
[1]

A. B. Aceves, C. De Angelis, A. M. Rubenchik and S. K. Turitsyn, Multidimensional solitons in fiber arrays, Optical Letters, 19 (1995), 329-331.   Google Scholar

[2]

K. Aoki, N. Hayashi and P. I. Naumkin, Global existence of small solutions for the fourth-order nonlinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 65, 18 pp. doi: 10.1007/s00030-016-0420-z.  Google Scholar

[3]

M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[4]

L. Bergé, Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.  doi: 10.1016/S0370-1573(97)00092-6.  Google Scholar

[5]

D. Bonheure, J.-B. Castera, T. Gou and L. Jeanjean, Strong instability of ground states to a fourth order Schrödinger equation, preprint, available at arXiv: 1703.07977v2 (2017). Google Scholar

[6]

O. Bouchel, Remarks on NLS with higher order anisotropic dispersion, Adv. Differential Equations, 13 (2008), 169-198.   Google Scholar

[7]

T. Boulenger and E. Lenzmann, Blowup for biharmonic NLS, Annales Scientifiques de l' ENS, 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[10]

G. Fibich and B. Ilan, Optical light bullets in a pure Kerr medium, Optics Letters, 29 (2004), 887-889.   Google Scholar

[11]

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

[12]

G. Fibich, B. Ilan and S. Schochet, Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, 16 (2003), 1809-1821.  doi: 10.1088/0951-7715/16/5/314.  Google Scholar

[13]

C. Hao, L. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83.  doi: 10.1016/j.jmaa.2006.05.031.  Google Scholar

[14]

N. Hayashi, A. Mendez-Navarro Jesus and P. I. Naumkin, Scattering of solutions to the fourth-order nonlinear Schrödinger equation, Commun. Contemp. Math., 18 (2016), 1550035, 24 pp. doi: 10.1142/S0219199715500352.  Google Scholar

[15]

N. Hayashi, A. Mendez-Navarro Jesus and P. I. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, J. Differential Equations, 261 (2016), 5144-5179.  doi: 10.1016/j.jde.2016.07.026.  Google Scholar

[16]

N. Hayashi and P. I. Naumkin, Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity, Comm. Math. Phys., 267 (2006), 477-492.  doi: 10.1007/s00220-006-0057-6.  Google Scholar

[17]

N. Hayashi and P. I. Naumkin, Asymptotic properties of solutions to dispersive equation of Schrödinger type, J. Math. Soc. Japan, 60 (2008), 631-652.  doi: 10.2969/jmsj/06030631.  Google Scholar

[18]

N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905.  doi: 10.1016/j.jde.2014.10.007.  Google Scholar

[19]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131.  doi: 10.1016/j.na.2014.12.024.  Google Scholar

[20]

N. Hayashi and P. I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 093502, 25 pp. doi: 10.1063/1.4929657.  Google Scholar

[21]

N. Hayashi and P. I. Naumkin, Factorization technique for the fourth-order nonlinear Schr${\rm{\ddot d}}$inger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.  doi: 10.1007/s00033-015-0524-z.  Google Scholar

[22]

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.  Google Scholar

[23]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336-R1339.   Google Scholar

[24]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[25]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[26]

C. E. Kenig, G. 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.  Google Scholar

[27]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energycritical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[28]

C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions d≥9, J. Differetial Equations, 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2011.08.009.  Google Scholar

[29]

C. Miao and J. Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity, 29 (2016), 692-736.  doi: 10.1088/0951-7715/29/2/692.  Google Scholar

[30]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991), 479-493.  doi: 10.1007/BF02101876.  Google Scholar

[31]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292.  doi: 10.3934/dcds.2009.24.1275.  Google Scholar

[32]

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.  Google Scholar

[33]

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

[34]

B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191.  doi: 10.1088/0951-7715/26/8/2175.  Google Scholar

[35]

J. Segata, A remark on asymptotics of solutions to Schrödinger equation with fourth order dispersion, Asymptotic Analysis, 75 (2011), 25-36.   Google Scholar

[36]

E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.  Google Scholar

[37]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[38]

M. Visan, The Defocusing Energy-critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D. Thesis. UCLA, 2007. Google Scholar

[39]

N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.  doi: 10.4310/MRL.2009.v16.n5.a14.  Google Scholar

[40]

S. Wen and D. Fan, Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher order dispersions, J. Opt. Soc. Am. B, 19 (2002), 1653-1659.   Google Scholar

show all references

References:
[1]

A. B. Aceves, C. De Angelis, A. M. Rubenchik and S. K. Turitsyn, Multidimensional solitons in fiber arrays, Optical Letters, 19 (1995), 329-331.   Google Scholar

[2]

K. Aoki, N. Hayashi and P. I. Naumkin, Global existence of small solutions for the fourth-order nonlinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 65, 18 pp. doi: 10.1007/s00030-016-0420-z.  Google Scholar

[3]

M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.  Google Scholar

[4]

L. Bergé, Wave collapse in physics: Principles and applications to light and plasma waves, Phys. Rep., 303 (1998), 259-370.  doi: 10.1016/S0370-1573(97)00092-6.  Google Scholar

[5]

D. Bonheure, J.-B. Castera, T. Gou and L. Jeanjean, Strong instability of ground states to a fourth order Schrödinger equation, preprint, available at arXiv: 1703.07977v2 (2017). Google Scholar

[6]

O. Bouchel, Remarks on NLS with higher order anisotropic dispersion, Adv. Differential Equations, 13 (2008), 169-198.   Google Scholar

[7]

T. Boulenger and E. Lenzmann, Blowup for biharmonic NLS, Annales Scientifiques de l' ENS, 50 (2017), 503-544.  doi: 10.24033/asens.2326.  Google Scholar

[8]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. doi: 10.1090/cln/010.  Google Scholar

[9]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[10]

G. Fibich and B. Ilan, Optical light bullets in a pure Kerr medium, Optics Letters, 29 (2004), 887-889.   Google Scholar

[11]

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

[12]

G. Fibich, B. Ilan and S. Schochet, Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion, Nonlinearity, 16 (2003), 1809-1821.  doi: 10.1088/0951-7715/16/5/314.  Google Scholar

[13]

C. Hao, L. Hsiao and B. Wang, Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces, J. Math. Anal. Appl., 328 (2007), 58-83.  doi: 10.1016/j.jmaa.2006.05.031.  Google Scholar

[14]

N. Hayashi, A. Mendez-Navarro Jesus and P. I. Naumkin, Scattering of solutions to the fourth-order nonlinear Schrödinger equation, Commun. Contemp. Math., 18 (2016), 1550035, 24 pp. doi: 10.1142/S0219199715500352.  Google Scholar

[15]

N. Hayashi, A. Mendez-Navarro Jesus and P. I. Naumkin, Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case, J. Differential Equations, 261 (2016), 5144-5179.  doi: 10.1016/j.jde.2016.07.026.  Google Scholar

[16]

N. Hayashi and P. I. Naumkin, Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity, Comm. Math. Phys., 267 (2006), 477-492.  doi: 10.1007/s00220-006-0057-6.  Google Scholar

[17]

N. Hayashi and P. I. Naumkin, Asymptotic properties of solutions to dispersive equation of Schrödinger type, J. Math. Soc. Japan, 60 (2008), 631-652.  doi: 10.2969/jmsj/06030631.  Google Scholar

[18]

N. Hayashi and P. I. Naumkin, Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905.  doi: 10.1016/j.jde.2014.10.007.  Google Scholar

[19]

N. Hayashi and P. I. Naumkin, Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131.  doi: 10.1016/j.na.2014.12.024.  Google Scholar

[20]

N. Hayashi and P. I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 093502, 25 pp. doi: 10.1063/1.4929657.  Google Scholar

[21]

N. Hayashi and P. I. Naumkin, Factorization technique for the fourth-order nonlinear Schr${\rm{\ddot d}}$inger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.  doi: 10.1007/s00033-015-0524-z.  Google Scholar

[22]

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.  Google Scholar

[23]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), R1336-R1339.   Google Scholar

[24]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[25]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[26]

C. E. Kenig, G. 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.  Google Scholar

[27]

C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the focusing energycritical nonlinear Schrödinger equations of fourth order in the radial case, J. Differential Equations, 246 (2009), 3715-3749.  doi: 10.1016/j.jde.2008.11.011.  Google Scholar

[28]

C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrödinger equations of fourth order in dimensions d≥9, J. Differetial Equations, 251 (2011), 3381-3402.  doi: 10.1016/j.jde.2011.08.009.  Google Scholar

[29]

C. Miao and J. Zheng, Scattering theory for the defocusing fourth-order Schrödinger equation, Nonlinearity, 29 (2016), 692-736.  doi: 10.1088/0951-7715/29/2/692.  Google Scholar

[30]

T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991), 479-493.  doi: 10.1007/BF02101876.  Google Scholar

[31]

B. Pausader, The focusing energy-critical fourth-order Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 24 (2009), 1275-1292.  doi: 10.3934/dcds.2009.24.1275.  Google Scholar

[32]

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.  Google Scholar

[33]

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

[34]

B. Pausader and S. Xia, Scattering theory for the fourth-order Schrödinger equation in low dimensions, Nonlinearity, 26 (2013), 2175-2191.  doi: 10.1088/0951-7715/26/8/2175.  Google Scholar

[35]

J. Segata, A remark on asymptotics of solutions to Schrödinger equation with fourth order dispersion, Asymptotic Analysis, 75 (2011), 25-36.   Google Scholar

[36]

E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.  Google Scholar

[37]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

[38]

M. Visan, The Defocusing Energy-critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D. Thesis. UCLA, 2007. Google Scholar

[39]

N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett., 16 (2009), 919-926.  doi: 10.4310/MRL.2009.v16.n5.a14.  Google Scholar

[40]

S. Wen and D. Fan, Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher order dispersions, J. Opt. Soc. Am. B, 19 (2002), 1653-1659.   Google Scholar

[1]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1447-1478. doi: 10.3934/cpaa.2021028
[2]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[3]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113
[4]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012
[5]

Aliang Xia, Jianfu Yang. Normalized solutions of higher-order Schrödinger equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 447-462. doi: 10.3934/dcds.2019018
[6]

Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part II. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021037
[7]

Ahmet Batal, Türker Özsarı, Kemal Cem Yılmaz. Stabilization of higher order Schrödinger equations on a finite interval: Part I. Evolution Equations & Control Theory, 2021, 10 (4) : 861-919. doi: 10.3934/eect.2020095
[8]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[9]

Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149
[10]

Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027
[11]

Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[12]

Editorial Office. Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3785-3785. doi: 10.3934/cpaa.2020167
[13]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205
[14]

Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
[15]

Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131
[16]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[17]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
[18]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037
[19]

Chuang Zheng. Inverse problems for the fourth order Schrödinger equation on a finite domain. Mathematical Control & Related Fields, 2015, 5 (1) : 177-189. doi: 10.3934/mcrf.2015.5.177
[20]

Naoufel Ben Abdallah, Yongyong Cai, Francois Castella, Florian Méhats. Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential. Kinetic & Related Models, 2011, 4 (4) : 831-856. doi: 10.3934/krm.2011.4.831

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (226)
  • HTML views (145)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy