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

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  • 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 $t\to∞$.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.


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  •   A. B. Aceves , C. De Angelis , A. M. Rubenchik  and  S. K. Turitsyn , Multidimensional solitons in fiber arrays, Optical Letters, 19 (1995) , 329-331. 
      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.
      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.
      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.
      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).
      O. Bouchel , Remarks on NLS with higher order anisotropic dispersion, Adv. Differential Equations, 13 (2008) , 169-198. 
      T. Boulenger  and  E. Lenzmann , Blowup for biharmonic NLS, Annales Scientifiques de l' ENS, 50 (2017) , 503-544.  doi: 10.24033/asens.2326.
      T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. American Mathematical Society, 2003. doi: 10.1090/cln/010.
      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.
      G. Fibich  and  B. Ilan , Optical light bullets in a pure Kerr medium, Optics Letters, 29 (2004) , 887-889. 
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      V. I. Karpman , Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996) , R1336-R1339. 
      M. Keel  and  T. Tao , Endpoint Strichartz estimates, Amer. J. Math., 120 (1998) , 955-980.  doi: 10.1353/ajm.1998.0039.
      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.
      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.
      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.
      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.
      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.
      T. Ozawa , Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991) , 479-493.  doi: 10.1007/BF02101876.
      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.
      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.
      B. Pausader , he cubic fourth-order Schrödinger equation, J. Funct. Anal., 256 (2009) , 2473-2517.  doi: 10.1016/j.jfa.2008.11.009.
      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.
      J. Segata , A remark on asymptotics of solutions to Schrödinger equation with fourth order dispersion, Asymptotic Analysis, 75 (2011) , 25-36. 
      E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.
      Y. Tsutsumi , $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987) , 115-125. 
      M. Visan, The Defocusing Energy-critical Nonlinear Schrödinger Equation in Dimensions Five and Higher, Ph.D. Thesis. UCLA, 2007.
      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.
      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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