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Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations
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 |
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 $t\to∞$.
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.
|
[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-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. |
[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. Fibich, B. Ilan and G. Papanicolaou,
Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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. Aceves, C. De Angelis, A. 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-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. |
[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. Fibich, B. Ilan and G. Papanicolaou,
Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. |
[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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