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Asymptotics for the higher-order derivative nonlinear Schrödinger equation
1. | Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán, MEXICO |
2. | Universidad Politécnica de Uruapan, CP 60210 Uruapan Michoacán, MEXICO |
$ \begin{cases} i\partial_{t}v+\dfrac{a}{2}\partial_{x}^{2}v-\dfrac{b}{4}\partial_{x} ^{4}v = \left( \overline{\partial_{x}v}\right) ^{2},\text{ }t>1,\text{ } x\in\mathbb{R},\\ v\left( 1,x\right) = v_{0}\left( x\right) ,\text{ }x\in\mathbb{R}\text{,} \end{cases} $ |
$ a,b>0. $ |
References:
[1] |
M. Ben-Artzi, H. Koch and J. C. Saut,
Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., 330 (2000), 87-92.
doi: 10.1016/S0764-4442(00)00120-8. |
[2] |
A. P. Calderon and R. Vaillancourt,
A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci., 69 (1972), 1185-1187.
doi: 10.1073/pnas.69.5.1185. |
[3] |
Th. Cazenave, Semilinear Schrödinger equations, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[4] |
S. Cohn,
Resonance and long time existence for the quadratic semilinear Schrödinger equation, Commun. Pure Appl. Math., 45 (1992), 973-1001.
doi: 10.1002/cpa.3160450804. |
[5] |
R. R. Coifman and Y. Meyer, Au dela des operateurs pseudo-differentiels, Societe Mathematique de France, Paris, 1978. |
[6] |
H. O. Cordes,
On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., 18 (1975), 115-131.
doi: 10.1016/0022-1236(75)90020-8. |
[7] |
K. B. 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.
|
[8] |
Y. Fukumoto and H. K. Mofatt,
Motion and expansion of a viscous vortex ring: I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45.
doi: 10.1017/S0022112000008995. |
[9] |
N. Hayashi and P. I. Naumkin,
A quadratic nonlinear Schrödinger equation in one space dimension, J. Differ. Equ, 186 (2002), 165-185.
doi: 10.1016/S0022-0396(02)00010-4. |
[10] |
N. Hayashi and P. I. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
doi: 10.1007/s00033-007-7008-8. |
[11] |
N. Hayashi and P. I. Naumkin, Asymptotic behavior for a quadratic nonlinear Schrödinger equation, Electron. J. Differential Equations, 15 2008, 38 pp. |
[12] |
N. Hayashi and P.I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 25 pp.
doi: 10.1063/1.4929657. |
[13] |
N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(R^{n})$ spaces for some Sc rödinger equations, Ann. I. H. P. (Phys. Théor.), 48 (1988), 17-37. |
[14] |
I. L. Hwang,
The $L^{2}$ -boundedness of pseudodifferential operators, Trans. Amer. Math. Soc., 302 (1987), 55-76.
doi: 10.2307/2000896. |
[15] |
V. L. Karpman,
Stabilization of soliton instabilities by high-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1016/0375-9601(95)00752-0. |
[16] |
V. L. Karpman and A. G. Shagalov,
Stabilitiy of soliton described by nonlinear Schrödinger-type equations with high-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[17] |
T. Ozawa,
Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232.
|
[18] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
show all references
References:
[1] |
M. Ben-Artzi, H. Koch and J. C. Saut,
Dispersion estimates for fourth order Schrödinger equations, C. R. Math. Acad. Sci., 330 (2000), 87-92.
doi: 10.1016/S0764-4442(00)00120-8. |
[2] |
A. P. Calderon and R. Vaillancourt,
A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci., 69 (1972), 1185-1187.
doi: 10.1073/pnas.69.5.1185. |
[3] |
Th. Cazenave, Semilinear Schrödinger equations, Courant Institute of Mathematical Sciences, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[4] |
S. Cohn,
Resonance and long time existence for the quadratic semilinear Schrödinger equation, Commun. Pure Appl. Math., 45 (1992), 973-1001.
doi: 10.1002/cpa.3160450804. |
[5] |
R. R. Coifman and Y. Meyer, Au dela des operateurs pseudo-differentiels, Societe Mathematique de France, Paris, 1978. |
[6] |
H. O. Cordes,
On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators, J. Funct. Anal., 18 (1975), 115-131.
doi: 10.1016/0022-1236(75)90020-8. |
[7] |
K. B. 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.
|
[8] |
Y. Fukumoto and H. K. Mofatt,
Motion and expansion of a viscous vortex ring: I. A higher-order asymptotic formula for the velocity, J. Fluid. Mech., 417 (2000), 1-45.
doi: 10.1017/S0022112000008995. |
[9] |
N. Hayashi and P. I. Naumkin,
A quadratic nonlinear Schrödinger equation in one space dimension, J. Differ. Equ, 186 (2002), 165-185.
doi: 10.1016/S0022-0396(02)00010-4. |
[10] |
N. Hayashi and P. I. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
doi: 10.1007/s00033-007-7008-8. |
[11] |
N. Hayashi and P. I. Naumkin, Asymptotic behavior for a quadratic nonlinear Schrödinger equation, Electron. J. Differential Equations, 15 2008, 38 pp. |
[12] |
N. Hayashi and P.I. Naumkin, On the inhomogeneous fourth-order nonlinear Schrödinger equation, J. Math. Phys., 56 (2015), 25 pp.
doi: 10.1063/1.4929657. |
[13] |
N. Hayashi and T. Ozawa, Scattering theory in the weighted $L^{2}(R^{n})$ spaces for some Sc rödinger equations, Ann. I. H. P. (Phys. Théor.), 48 (1988), 17-37. |
[14] |
I. L. Hwang,
The $L^{2}$ -boundedness of pseudodifferential operators, Trans. Amer. Math. Soc., 302 (1987), 55-76.
doi: 10.2307/2000896. |
[15] |
V. L. Karpman,
Stabilization of soliton instabilities by high-order dispersion: fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1016/0375-9601(95)00752-0. |
[16] |
V. L. Karpman and A. G. Shagalov,
Stabilitiy of soliton described by nonlinear Schrödinger-type equations with high-order dispersion, Physica D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[17] |
T. Ozawa,
Remarks on quadratic nonlinear Schrödinger equations, Funkcial. Ekvac., 38 (1995), 217-232.
|
[18] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
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