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Long-time stability of small FPU solitary waves
Almost global existence for cubic nonlinear Schrödinger equations in one space dimension
1. | Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA |
2. | Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA |
We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension.We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^∞$ decay up to time $\exp(C\varepsilon^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.
References:
[1] |
J. Barab,
Nonexistence of asymptotically free solutions of a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.
doi: 10.1090/cln/010. |
[3] |
R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991. |
[4] |
P. Deift and X. Zhou,
Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1029-1077.
doi: 10.1002/cpa.3034. |
[5] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
[6] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 2D quadratic Schrödinger equations, J. Math. Pures Appl.(9), 97 (2012), 505-543.
doi: 10.1016/j.matpur.2011.09.008. |
[7] |
N. Hayashi and P. Naumkin,
Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math., 20 (1998), 369-389.
doi: 10.1353/ajm.1998.0011. |
[8] |
N. Hayashi and P. Naumkin,
Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, Int. J. Pure Appl. Math., 3 (2002), 255-273.
|
[9] |
N. Hayashi and P. Naumkin,
Large time behavior for the cubic nonlinear Schrödinger equation, Canad. J. Math., 54 (2002), 1065-1085.
doi: 10.4153/CJM-2002-039-3. |
[10] |
N. Hayashi and P. Naumkin,
On the asymptotics for cubic nonlinear Schrödinger equations, Complex Var. Theory Appl., 49 (2004), 339-373.
doi: 10.1080/02781070410001710353. |
[11] |
N. Hayashi and P. Naumkin,
Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.
|
[12] |
N. Hayashi and P. Naumkin,
Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces, Differential Integral Equations, 24 (2011), 801-828.
|
[13] |
N. Hayashi and P. Naumkin,
Logarithmic time decay for the cubic nonlinear Schrödinger
equations, Int Math Res Notices, 2015 (2015), 5604-5643.
doi: 10.1093/imrn/rnu102. |
[14] |
M. Ifrim and D. Tataru,
Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.
doi: 10.1088/0951-7715/28/8/2661. |
[15] |
F. John,
Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math., 40 (1987), 79-109.
doi: 10.1002/cpa.3160400104. |
[16] |
F. John and S. Klainerman,
Almost global existence to nonlinear wave equations in three
space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.
doi: 10.1002/cpa.3160370403. |
[17] |
J. Kato and F. Pusateri,
A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations, 24 (2011), 923-940.
|
[18] |
H. Lindblad and A. Soffer,
Scattering and small data completeness for the critical nonlinear Schrödinger equation, Nonlinearity, 19 (2006), 345-353.
doi: 10.1088/0951-7715/19/2/006. |
[19] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele, A Short Proof of the Coifman-Meyer Multilinear Theorem, http://www.math.brown.edu/~jpipher/trilogy1.pdf |
[20] |
P. Naumkin,
Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.
|
[21] |
F. Pusateri and J. Shatah,
Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540.
doi: 10.1002/cpa.21461. |
[22] |
Y. Sagawa and H. Sunagawa,
The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761.
doi: 10.3934/dcds.2016052. |
[23] |
A. Shimomura,
Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.
doi: 10.1080/03605300600910316. |
[24] |
H. Sunagawa,
Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.
|
[25] |
Y. Tsutsumi and K. Yajima,
The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188.
doi: 10.1090/S0273-0979-1984-15263-7. |
show all references
References:
[1] |
J. Barab,
Nonexistence of asymptotically free solutions of a nonlinear Schrödinger equation, J. Math. Phys., 25 (1984), 3270-3273.
doi: 10.1063/1.526074. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.
doi: 10.1090/cln/010. |
[3] |
R. Coifman and Y. Meyer, Ondelettes et Opérateurs. Ⅲ. Opérateurs Multilinéaires, Actualités Mathématiques. Hermann, Paris, 1991. |
[4] |
P. Deift and X. Zhou,
Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser, Comm. Pure Appl. Math., 56 (2003), 1029-1077.
doi: 10.1002/cpa.3034. |
[5] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, (2009), 414-432.
doi: 10.1093/imrn/rnn135. |
[6] |
P. Germain, N. Masmoudi and J. Shatah,
Global solutions for 2D quadratic Schrödinger equations, J. Math. Pures Appl.(9), 97 (2012), 505-543.
doi: 10.1016/j.matpur.2011.09.008. |
[7] |
N. Hayashi and P. Naumkin,
Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math., 20 (1998), 369-389.
doi: 10.1353/ajm.1998.0011. |
[8] |
N. Hayashi and P. Naumkin,
Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, Int. J. Pure Appl. Math., 3 (2002), 255-273.
|
[9] |
N. Hayashi and P. Naumkin,
Large time behavior for the cubic nonlinear Schrödinger equation, Canad. J. Math., 54 (2002), 1065-1085.
doi: 10.4153/CJM-2002-039-3. |
[10] |
N. Hayashi and P. Naumkin,
On the asymptotics for cubic nonlinear Schrödinger equations, Complex Var. Theory Appl., 49 (2004), 339-373.
doi: 10.1080/02781070410001710353. |
[11] |
N. Hayashi and P. Naumkin,
Nongauge invariant cubic nonlinear Schrödinger equations, Pac. J. Appl. Math., 1 (2008), 1-16.
|
[12] |
N. Hayashi and P. Naumkin,
Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces, Differential Integral Equations, 24 (2011), 801-828.
|
[13] |
N. Hayashi and P. Naumkin,
Logarithmic time decay for the cubic nonlinear Schrödinger
equations, Int Math Res Notices, 2015 (2015), 5604-5643.
doi: 10.1093/imrn/rnu102. |
[14] |
M. Ifrim and D. Tataru,
Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.
doi: 10.1088/0951-7715/28/8/2661. |
[15] |
F. John,
Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data, Comm. Pure Appl. Math., 40 (1987), 79-109.
doi: 10.1002/cpa.3160400104. |
[16] |
F. John and S. Klainerman,
Almost global existence to nonlinear wave equations in three
space dimensions, Comm. Pure Appl. Math., 37 (1984), 443-455.
doi: 10.1002/cpa.3160370403. |
[17] |
J. Kato and F. Pusateri,
A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations, 24 (2011), 923-940.
|
[18] |
H. Lindblad and A. Soffer,
Scattering and small data completeness for the critical nonlinear Schrödinger equation, Nonlinearity, 19 (2006), 345-353.
doi: 10.1088/0951-7715/19/2/006. |
[19] |
C. Muscalu, J. Pipher, T. Tao and C. Thiele, A Short Proof of the Coifman-Meyer Multilinear Theorem, http://www.math.brown.edu/~jpipher/trilogy1.pdf |
[20] |
P. Naumkin,
Cubic derivative nonlinear Schrödinger equations, SUT J. Math., 36 (2000), 9-42.
|
[21] |
F. Pusateri and J. Shatah,
Space-time resonances and the null condition for first-order systems of wave equations, Comm. Pure Appl. Math., 66 (2013), 1495-1540.
doi: 10.1002/cpa.21461. |
[22] |
Y. Sagawa and H. Sunagawa,
The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 36 (2016), 5743-5761.
doi: 10.3934/dcds.2016052. |
[23] |
A. Shimomura,
Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), 1407-1423.
doi: 10.1080/03605300600910316. |
[24] |
H. Sunagawa,
Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math., 43 (2006), 771-789.
|
[25] |
Y. Tsutsumi and K. Yajima,
The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11 (1984), 186-188.
doi: 10.1090/S0273-0979-1984-15263-7. |
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