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

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 $t\to∞$.

Citation: Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009
References:
[1]

A. B. AcevesC. De AngelisA. M. Rubenchik and S. K. Turitsyn, Multidimensional solitons in fiber arrays, Optical Letters, 19 (1995), 329-331. 

[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.

[3]

M. Ben-ArtziH. 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.

[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.

[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).

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

G. FibichB. 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.

[13]

C. HaoL. 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.

[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.

[15]

N. HayashiA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[24]

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

[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.

[26]

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.

[27]

C. MiaoG. 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.

[28]

C. MiaoG. 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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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. 

show all references

References:
[1]

A. B. AcevesC. De AngelisA. M. Rubenchik and S. K. Turitsyn, Multidimensional solitons in fiber arrays, Optical Letters, 19 (1995), 329-331. 

[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.

[3]

M. Ben-ArtziH. 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.

[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.

[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).

[6]

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

[7]

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

[8]

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

[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.

[10]

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

[11]

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

[12]

G. FibichB. 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.

[13]

C. HaoL. 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.

[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.

[15]

N. HayashiA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[24]

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

[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.

[26]

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.

[27]

C. MiaoG. 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.

[28]

C. MiaoG. 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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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. 

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