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Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity
1. | Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 |
2. | Department of Mathematics, Institute of Engineering, Academic Assembly, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan |
References:
[1] |
F. Christ and M. 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. |
[2] |
K. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. |
[3] |
Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, in Proc. IUTAM Symp. Geom. Stat. Turbul., 2001, 211-216.
doi: 10.1007/978-94-015-9638-1_25. |
[4] |
M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non linéaie., 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[5] |
M. Hadac, S. Herr, and H. Koch, Errantum to "Well-posedness and scattering for the KP-II equation in a critical space'' [Ann. I. H. Poincaré-AN26 (3) (2009) 917-941], Ann. Inst. H. Poincaré Anal. Non linéaie., 27 (2010), 971-972.
doi: 10.1016/j.anihpc.2010.01.006. |
[6] |
C. Hao, L. Hsiao, and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265.
doi: 10.1016/j.jmaa.2005.06.091. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity,, FUNKCIALAJ EKVACIOJ, ().
|
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
V. Karpman, Stabilization of soliton instabilities by higher order dispersion: fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. |
[15] |
V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[16] |
B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[17] |
J. Segata, Well-posedness for the fourth order nonlinear Schrödinger type equation related to the vortex filament, Diff. and Integral Eqs., 16 (2003), 841-864. |
[18] |
J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568.
doi: 10.1090/S0002-9939-04-07620-8. |
[19] |
J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105.
doi: 10.3934/dcds.2010.27.1093. |
[20] |
Y. Wang, Global well-posedness for the generalized fourth-order Schrödingier equation, Bull. Aust. Math. Soc., 85 (2012), 371-379.
doi: 10.1017/S0004972711003327. |
show all references
References:
[1] |
F. Christ and M. 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. |
[2] |
K. Dysthe, Note on a modification to the nonlinear Schrödinger equation for application to deep water waves, Proc. R. Soc. Lond. Ser. A, 369 (1979), 105-114. |
[3] |
Y. Fukumoto, Motion of a curved vortex filament: higher-order asymptotics, in Proc. IUTAM Symp. Geom. Stat. Turbul., 2001, 211-216.
doi: 10.1007/978-94-015-9638-1_25. |
[4] |
M. Hadac, S. Herr, and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non linéaie., 26 (2009), 917-941.
doi: 10.1016/j.anihpc.2008.04.002. |
[5] |
M. Hadac, S. Herr, and H. Koch, Errantum to "Well-posedness and scattering for the KP-II equation in a critical space'' [Ann. I. H. Poincaré-AN26 (3) (2009) 917-941], Ann. Inst. H. Poincaré Anal. Non linéaie., 27 (2010), 971-972.
doi: 10.1016/j.anihpc.2010.01.006. |
[6] |
C. Hao, L. Hsiao, and B. Wang, Wellposedness for the fourth order nonlinear Schrödinger equations, J. Math. Anal. Appl., 320 (2006), 246-265.
doi: 10.1016/j.jmaa.2005.06.091. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
H. Hirayama, Well-posedness and scattering for nonlinear Schrödinger equations with a derivative nonlinearity at the scaling critical regularity,, FUNKCIALAJ EKVACIOJ, ().
|
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
V. Karpman, Stabilization of soliton instabilities by higher order dispersion: fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. |
[15] |
V. Karpman and A. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[16] |
B. Pausader, Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case, Dynamics of PDE, 4 (2007), 197-225.
doi: 10.4310/DPDE.2007.v4.n3.a1. |
[17] |
J. Segata, Well-posedness for the fourth order nonlinear Schrödinger type equation related to the vortex filament, Diff. and Integral Eqs., 16 (2003), 841-864. |
[18] |
J. Segata, Remark on well-posedness for the fourth order nonlinear Schrödinger type equation, Proc. Amer. Math. Soc., 132 (2004), 3559-3568.
doi: 10.1090/S0002-9939-04-07620-8. |
[19] |
J. Segata, Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation, Discrete Contin. Dyn. Syst., 27 (2010), 1093-1105.
doi: 10.3934/dcds.2010.27.1093. |
[20] |
Y. Wang, Global well-posedness for the generalized fourth-order Schrödingier equation, Bull. Aust. Math. Soc., 85 (2012), 371-379.
doi: 10.1017/S0004972711003327. |
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