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## Modified scattering for the cubic Schrödinger equation on product spaces and applications (2013)

Citations: | 7 - 3 self |

### Citations

1514 | Mathematical methods of classical mechanics - Arnold - 1978 |

464 |
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
- Bourgain
- 1993
(Show Context)
Citation Context ...|p−1u, p > 1 (1.2) dates back at least to [21]. The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings =-=[4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89]-=-. The conclusion that could be derived from these works is that the geometry of the spatial domain turned out to be of importance in the context of the best possible Strichartz inequalities or the sha... |

407 |
Semilinear Schrödinger Equations
- Cazenave
- 2003
(Show Context)
Citation Context ... defocusing and analytic (p odd integer). In this case, global smooth solutions disperse and in many cases even scatter to a linear state (possibly after modulation by a real phase when d = 1, p = 3) =-=[28, 30, 33, 35, 36, 55, 69, 70, 76, 81, 89, 91]-=-. In contrast, much less is known for compact domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small ope... |

373 |
Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinearmedia
- Zakharov, Shabat
- 1972
(Show Context)
Citation Context ... defocusing and analytic (p odd integer). In this case, global smooth solutions disperse and in many cases even scatter to a linear state (possibly after modulation by a real phase when d = 1, p = 3) =-=[28, 30, 33, 35, 36, 55, 69, 70, 76, 81, 89, 91]-=-. In contrast, much less is known for compact domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small ope... |

242 | MR2461508) Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
- Kenig, Merle
(Show Context)
Citation Context ...|p−1u, p > 1 (1.2) dates back at least to [21]. The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings =-=[4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89]-=-. The conclusion that could be derived from these works is that the geometry of the spatial domain turned out to be of importance in the context of the best possible Strichartz inequalities or the sha... |

173 |
Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity
- Bourgain
- 1998
(Show Context)
Citation Context ...ume that λ ≥ 10µ ≥ 1 and that u(t) = eit∂xxu0, v(t) = eit∂xxv0. Then, we have the bound ‖QλuQµv‖L2x,t(R×R) . λ − 1 2 ‖u0‖L2x(R)‖v0‖L2x(R). (7.14) We refer to [29] for the proof of Lemma 7.2 (see also =-=[13]-=- for the earlier higher dimensional analogue of (7.14) and [48] for recent closely related estimates). Lemma 7.3. Assume that N ≥ 7. Then we have the bound sup x∈R ∑ p∈Zd [ 1 + |p|2] |eit∂xxFp(x)|2 . ... |

138 | Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds
- Burq, Gérard, et al.
(Show Context)
Citation Context ...|p−1u, p > 1 (1.2) dates back at least to [21]. The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings =-=[4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, 57, 58, 59, 60, 61, 62, 69, 71, 72, 77, 84, 89]-=-. The conclusion that could be derived from these works is that the geometry of the spatial domain turned out to be of importance in the context of the best possible Strichartz inequalities or the sha... |

113 | Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3
- Colliander, Keel, et al.
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109 |
Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear pde, IMRN
- BOURGAIN
- 1994
(Show Context)
Citation Context ...act domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori =-=[12, 38, 74, 80]-=- to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is... |

90 | The cubic nonlinear Schrödinger equation in two dimensions with radial data
- Killip, Tao, et al.
(Show Context)
Citation Context ... defocusing and analytic (p odd integer). In this case, global smooth solutions disperse and in many cases even scatter to a linear state (possibly after modulation by a real phase when d = 1, p = 3) =-=[28, 30, 33, 35, 36, 55, 69, 70, 76, 81, 89, 91]-=-. In contrast, much less is known for compact domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small ope... |

83 |
Periodic nonlinear Schrödinger equation and invariant measures
- Bourgain
- 1994
(Show Context)
Citation Context ... even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention =-=[7, 10, 20, 86]-=-, where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d dimensional ball for d = 2, 3 (with radial data). These works establish the existence of a larg... |

82 | Invariant Cantor manifolds of quasiperiodic oscillations for a nonlinear Schr6dinger equation
- KUKSIN, POSCHEL
(Show Context)
Citation Context ...act domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori =-=[12, 38, 74, 80]-=- to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is... |

73 |
Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n > 2
- Ginibre, Ozawa
- 1993
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70 | Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R1+4
- Ryckman, Visan
(Show Context)
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68 |
Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case
- Bourgain
- 1999
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68 | Global Solutions for the Gravity Water Waves Equation in Dimension 3
- Germain, Masmoudi, et al.
(Show Context)
Citation Context ...erent form a näıve 1d vector valued analysis (as is the case in [87]). We also note that although our approach is close in spirit to recent developments in global existence for quasilinear equations =-=[43, 44, 63, 64, 65]-=-, some of the key estimates really pertain to the low-regularity theory (see Lemma 7.1 and Lemma 7.27). Organization of the paper. Section 2 introduces the notations used in this paper. Section 3 prov... |

57 |
Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces
- Burq, Gérard, et al.
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52 |
An instability property of the nonlinear Schrödinger equation on
- Burq, Gérard, et al.
(Show Context)
Citation Context ...works is that the geometry of the spatial domain turned out to be of importance in the context of the best possible Strichartz inequalities or the sharp local in time well-posedness results (see e.g. =-=[6, 22, 23]-=-). However, the analysis in [58, 59, 60, 61, 62, 77] Key words and phrases. Modified Scattering, Nonlinear Schrödinger equation, wave guide manifolds, energy cascade, weak turbulence. Z. H. is suppor... |

51 | Global well-posedness for Schrödinger equations with derivative
- Colliander, Keel, et al.
(Show Context)
Citation Context ...regularity in Fourier space. 3.1. The high frequency estimates. We start with an estimate bounding high frequencies in x. It uses essentially the bilinear Strichartz estimates on R (see Lemma 7.2 and =-=[29]-=-). MODIFIED SCATTERING ON R× Td 13 Lemma 3.2. Assume that T ≥ 1. The following estimates hold uniformly in T : ‖ ∑ A,B,C max(A,B,C)≥T 16 N t[QAF,QBG,QCH]‖Z . T− 76 ‖F‖S‖G‖S‖H‖S , ∀t ≥ T/4, ‖ ∑ A,B,C m... |

49 | Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations
- Hayashi, I
- 1998
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45 |
On turbulence in nonlinear Schrödinger equations
- Kuksin
- 1997
(Show Context)
Citation Context ...act domains. The most studied example is that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori =-=[12, 38, 74, 80]-=- to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is... |

44 |
Invariant measures for the 2D defocusing nonlinear Schrödinger equation
- Bourgain
- 1996
(Show Context)
Citation Context ... even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention =-=[7, 10, 20, 86]-=-, where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d dimensional ball for d = 2, 3 (with radial data). These works establish the existence of a larg... |

39 |
Nonlinear Schrödinger evolution equations
- Brézis, Gallouet
- 1980
(Show Context)
Citation Context ...tion and background. The question of the influence of the geometry on the global behavior of solutions to the nonlinear Schrödinger equation (i∂t + ∆)u = λ|u|p−1u, p > 1 (1.2) dates back at least to =-=[21]-=-. The first natural question is the issue of global existence of solutions, and many works have investigated this problem in different geometric settings [4, 5, 6, 14, 19, 23, 24, 25, 30, 37, 50, 52, ... |

38 | Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
- Colliander, Keel, et al.
(Show Context)
Citation Context ... that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits =-=[31, 47]-=- and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d ... |

34 | Global well-posedness and scattering for the defocusing, L2-critical, nonlinear Schrdinger equation when d
- Dodson
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34 | A one-dimensional model for dispersive wave turbulence
- Majda, McLaughlin, et al.
- 1997
(Show Context)
Citation Context ...les. This energy cascade is a main aspect of the out-of-equilibrium dynamics predicted for (1.1) by the vast literature of physics and numerics falling under the theory of weak (wave) turbulence (cf. =-=[75, 92]-=-). The corresponding result on Td does not directly follow from Corollary 1.4 (nor does it imply it). This is somehow surprising because one would naturally expect that adding a dispersive MODIFIED SC... |

34 |
On the growth of high Sobolev norms of solutions for KdV and Schrödinger
- Staffilani
- 1997
(Show Context)
Citation Context ...ero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we refer to =-=[9, 18, 32, 82, 83]-=-. One can also use Theorem 1.2 to construct other interesting non-scattering dynamics for equation (1.1) as is illustrated in the following result. Corollary 1.6 (Forward compact solutions). Let d ≥ 2... |

33 |
Exponential sums and nonlinear Schrödinger equations
- Bourgain
- 1993
(Show Context)
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33 |
On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE
- Bourgain
- 1996
(Show Context)
Citation Context ...ero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we refer to =-=[9, 18, 32, 82, 83]-=-. One can also use Theorem 1.2 to construct other interesting non-scattering dynamics for equation (1.1) as is illustrated in the following result. Corollary 1.6 (Forward compact solutions). Let d ≥ 2... |

33 |
Global solutions for 3D quadratic Schrödinger equations
- Germain, Masmoudi, et al.
(Show Context)
Citation Context ...erent form a näıve 1d vector valued analysis (as is the case in [87]). We also note that although our approach is close in spirit to recent developments in global existence for quasilinear equations =-=[43, 44, 63, 64, 65]-=-, some of the key estimates really pertain to the low-regularity theory (see Lemma 7.1 and Lemma 7.27). Organization of the paper. Section 2 introduces the notations used in this paper. Section 3 prov... |

31 | Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations
- Burq, Gérard, et al.
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31 | Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K
- Deift, Zhou
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29 |
On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential
- Bourgain
- 1999
(Show Context)
Citation Context ...esonant terms are transformed into quintic terms which scatter, and hence, at least heuristically do not modify the long-term dynamics. Previous results in the spirit of Corollary 1.4 may be found in =-=[15, 16]-=- for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and [40, 41, 4... |

28 |
Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1
- Delort
(Show Context)
Citation Context ....6) depends very much on how simple or complicated Neff (G) is. Previous modified scattering results that we are aware of, only concerned equations (or systems) posed on Rd, quasilinear or semilinear =-=[1, 26, 33, 34, 54, 55, 56, 64, 65, 68, 76, 91]-=- and had an integrable asymptotic system for (1.6). This often allowed for a simple phase conjugation (in physical or Fourier space) to give the modification. In contrast, our limiting system is given... |

24 |
Kolmogorov spectra of turbulence 1. Wave turbulence
- Zakharov, L’vov, et al.
- 1992
(Show Context)
Citation Context ...les. This energy cascade is a main aspect of the out-of-equilibrium dynamics predicted for (1.1) by the vast literature of physics and numerics falling under the theory of weak (wave) turbulence (cf. =-=[75, 92]-=-). The corresponding result on Td does not directly follow from Corollary 1.4 (nor does it imply it). This is somehow surprising because one would naturally expect that adding a dispersive MODIFIED SC... |

23 | Global solutions for the gravity water waves system in 2d. arXiv preprint:1303.5357
- Ionescu, Pusateri
(Show Context)
Citation Context ....6) depends very much on how simple or complicated Neff (G) is. Previous modified scattering results that we are aware of, only concerned equations (or systems) posed on Rd, quasilinear or semilinear =-=[1, 26, 33, 34, 54, 55, 56, 64, 65, 68, 76, 91]-=- and had an integrable asymptotic system for (1.6). This often allowed for a simple phase conjugation (in physical or Fourier space) to give the modification. In contrast, our limiting system is given... |

22 |
Semilinear Schrödinger flows on hyperbolic spaces : Scattering
- Ionescu, Staffilani
- 2009
(Show Context)
Citation Context ... scattering” characterized by a correction to scattering on a larger time-scale. We are interested in this latter regime to which (1.1) belongs. In support of the heuristic H1) we cite the results in =-=[4, 62, 66, 87, 88]-=-. The second heuristic H2) was put to test in [52] where the authors study the quintic NLS equation on R×Td. There, a strong relation is drawn between the large-data scattering theory for the quintic ... |

17 |
Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations, Geom
- Bourgain
- 1995
(Show Context)
Citation Context ...ynamics. Previous results in the spirit of Corollary 1.4 may be found in [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, =-=[8, 11, 51]-=- for NLS with suitably chosen non-local nonlinearities, and [40, 41, 42, 78, 90] for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on t... |

17 |
Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces
- Bourgain
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17 | Global well-posedness and scattering for the defocusing quintic NLS in three dimensions
- Killip, Visan
- 2011
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16 |
Global solutions and asymptotic behavior for two dimensional gravity water waves. arXiv preprint:1305.4090
- Alazard, Delort
(Show Context)
Citation Context ....6) depends very much on how simple or complicated Neff (G) is. Previous modified scattering results that we are aware of, only concerned equations (or systems) posed on Rd, quasilinear or semilinear =-=[1, 26, 33, 34, 54, 55, 56, 64, 65, 68, 76, 91]-=- and had an integrable asymptotic system for (1.6). This often allowed for a simple phase conjugation (in physical or Fourier space) to give the modification. In contrast, our limiting system is given... |

16 |
Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations. Ergodic Theory Dynam
- Bourgain
(Show Context)
Citation Context ...ero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we refer to =-=[9, 18, 32, 82, 83]-=-. One can also use Theorem 1.2 to construct other interesting non-scattering dynamics for equation (1.1) as is illustrated in the following result. Corollary 1.6 (Forward compact solutions). Let d ≥ 2... |

15 | Growth of Sobolev norms in the cubic defocusing nonlinear Schrodinger equation.
- Guardia, Kaloshin
- 2012
(Show Context)
Citation Context ... that of the torus Td. In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits =-=[31, 47]-=- and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d ... |

15 |
Nonlinear fractional Schrödinger equations in one dimension
- AD, Pusateri
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15 |
A new proof of long-range scattering for critical nonlinear Schrödinger equations. Differential Integral Equations
- Kato, Pusateri
- 2011
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15 |
On 2D Nonlinear Schrödinger equations with data on
- Takaoka, Tzvetkov
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15 |
Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions
- Vişan
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14 | Global well-posedness of the energy critical nonlinear Schrödinger equation with small initial data in H1(T3
- Herr, Tataru, et al.
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14 | On the global well-posedness of energy-critical Schrödinger equations in curved spaces
- Ionescu, Pausader, et al.
- 2010
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13 | On scattering for NLS: from Euclidean to hyperbolic space, Discrete Contin
- Banica, Carles, et al.
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13 | Strichartz estimates for partially periodic solutions to Schrödinger equations
- Herr, Tataru, et al.
- 2010
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13 |
Oscillations in space–periodic nonlinear Schrödinger equations
- Kuksin
- 1997
(Show Context)
Citation Context ...ter, and hence, at least heuristically do not modify the long-term dynamics. Previous results in the spirit of Corollary 1.4 may be found in [15, 16] for linear Schrödinger equations with potential, =-=[31, 47, 73]-=- for finite time amplifications of the initial Hs norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and [40, 41, 42, 78, 90] for the zero-dispersion Szegö and half-wave equatio... |

12 | The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
- Faou, Germain, et al.
(Show Context)
Citation Context ... behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 =-=[39]-=-. One may also mention [7, 10, 20, 86], where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d dimensional ball for d = 2, 3 (with radial data). These w... |

12 | Global well-posedness of the energy-critical defocusing NLS
- Ionescu, Pausader
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12 | Explicit formula for the solution of the Szegő equation on the real line and applications, Discrete Contin
- Pocovnicu
(Show Context)
Citation Context ...n [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and =-=[40, 41, 42, 78, 90]-=- for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we... |

11 | The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds
- Herr
- 2011
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10 |
Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case
- Bourgain, Bulut
(Show Context)
Citation Context ... even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics2 [39]. One may also mention =-=[7, 10, 20, 86]-=-, where invariant measures for (1.1) are constructed, when the problem is posed on Td, d = 1, 2 or the d dimensional ball for d = 2, 3 (with radial data). These works establish the existence of a larg... |

10 |
Geometric Optics and Long Range Scattering for One-Dimensional Nonlinear Schrödinger Equations
- Carles
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10 | Global solutions of quasilinear systems of Klein-Gordon equations in 3D
- Ionescu, Pausader
(Show Context)
Citation Context ...erent form a näıve 1d vector valued analysis (as is the case in [87]). We also note that although our approach is close in spirit to recent developments in global existence for quasilinear equations =-=[43, 44, 63, 64, 65]-=-, some of the key estimates really pertain to the low-regularity theory (see Lemma 7.1 and Lemma 7.27). Organization of the paper. Section 2 introduces the notations used in this paper. Section 3 prov... |

9 |
Modified Wave Operators for Nonlinear Schrödinger Equations
- Hayashi, Naumkin, et al.
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9 | Quintic NLS in the exterior of a strictly convex obstacle, preprint arXiv:1208.4904
- Killip, Visan, et al.
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8 | Resonant dynamics for the quintic nonlinear Schrödinger equation - Grébert, Thomann |

8 | Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary
- Hani
- 2010
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8 | On scattering for the quintic defocusing nonlinear Schrödinger equation - Hani, Pausader |

8 |
Procesi C., A KAM algorithm for the resonant nonlinear Schrödinger equation, preprint 2013
- Procesi
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8 | Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on S1. Differential Integral Equations
- Sohinger
- 2011
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8 | data scattering for the nonlinear Schrödinger equation on product spaces, Comm. P.D.E. 37 (2012) 125–135
- Tzvetkov, Visciglia, et al.
(Show Context)
Citation Context ...Td, considerable interest has emerged in the past few years to study questions of long-time behavior on “in between” manifolds, like the ones presented by the non-compact quotients of Euclidean space =-=[52, 59, 60, 84, 85, 87]-=-. In the generality of non-compact Riemannian d−manifolds M , it seems plausible that a key role is played by the parameter α for which solutions to the linear NLS equation ((1.2) with λ = 0) with smo... |

7 |
On growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in Rd. Journal d’analyse mathématique
- Bourgain
- 1997
(Show Context)
Citation Context ...ynamics. Previous results in the spirit of Corollary 1.4 may be found in [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, =-=[8, 11, 51]-=- for NLS with suitably chosen non-local nonlinearities, and [40, 41, 42, 78, 90] for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on t... |

7 |
Modified wave operator for a system of nonlinear Schrödinger equations
- Hayashi, Li, et al.
- 2012
(Show Context)
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6 | Effective integrable dynamics for some nonlinear wave equation. Preprint available at http://arxiv.org/abs/1110.5719
- Grellier, Gerard
- 2011
(Show Context)
Citation Context ...n [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and =-=[40, 41, 42, 78, 90]-=- for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we... |

5 | Problems in Hamiltonian PDE’s, Geom - Bourgain |

5 | A remark on normal forms and the upside-down I-method for periodic NLS: growth of higher Sobolev norms. Preprint available at http://arxiv.org/abs/1010.2501 - Colliander, Kwon, et al. - 2012 |

5 | Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping. preprint. Archived at http://arxiv.org/abs/1408.6213
- HANI, THOMANN
(Show Context)
Citation Context ...al form analysis, and the H1 well-posedness analysis on the sphere of [23, 24] provides the needed substitute of Lemma 7.1. A similar remark applies to the case of a partial harmonic confinement (cf. =-=[53]-=-). On the other hand, the extension of our analysis to an irrational torus is less clear because of the appearance of small denominators in the normal form analysis. As a consequence of Theorem 1.2, a... |

5 | The energy-critical defocusing NLS
- Ionescu, Pausader
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5 | First and second order approximations for a nonlinear wave equation. Preprint available at http://arxiv.org/abs/1111.6060 - Pocovnicu - 2012 |

4 |
Global well-posedness and scattering for the defocusing, energy-critical, nonlinear Schrödinger equation in the exterior of a convex obstacle when d
- Dodson
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4 |
Long-time strong instability and unbounded orbits for some periodic nonlinear Schrödinger equation
- Hani
(Show Context)
Citation Context ...ted for solutions of the resonant system (1.3) have counterparts in the asymptotic behavior of solutions of (1.1). Most notably, given the existence of unbounded Sobolev orbits for (1.3) as proved in =-=[51]-=- for d ≥ 2 (cf. Theorem 4.8 for an explicit construction with quantitative lower bounds on the growth), we have the following. Corollary 1.4 (Existence of infinite cascade solutions). Let d ≥ 2 and s ... |

4 |
Invariant measures for the defocusing
- Tzvetkov
- 2008
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3 | Scattering for nonlinear Schrödinger equation under partial harmonic confinement
- ANTONELLI, CARLES, et al.
(Show Context)
Citation Context ...s 1.1 and 1.2 can be extended to the case of spheres (i.e. R × Sd, d = 2, 3). A good understanding of the corresponding resonant system is presently missing. Finally, we also mention the situation in =-=[2]-=- where the partial periodicity is replaced by adding a (partially) confining potential. 1.3. Overview of proof. 1.3.1. Modified scattering. In order to describe the asymptotic behavior of a nonlinear ... |

3 |
Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems
- Carles, Faou
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Citation Context ...n be seen to be autonomous when written in terms of the slow time scale τ = pi ln t in which it has the form (1.3). Note that this system was previously studied and shown to have interesting dynamics =-=[27, 31, 39, 51]-=-. The upshot of the above formal calculation is that one should expect a solution F (t) to (1.7) to asymptote to some G(pi ln t) where G(τ) solves (1.3). This is the content of Theorem 1.1. 1.3.3. Nor... |

3 |
The Szegö Cubic Equation
- Gérard, Grellier
- 2010
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Citation Context ...n [15, 16] for linear Schrödinger equations with potential, [31, 47, 73] for finite time amplifications of the initial Hs norm, [8, 11, 51] for NLS with suitably chosen non-local nonlinearities, and =-=[40, 41, 42, 78, 90]-=- for the zero-dispersion Szegö and half-wave equations. Concerning the opposite question of obtaining upper bounds on the rate of possible growth of the Sobolev norms of solutions of NLS equations we... |

3 | An explicit formula for the cubic Szegő equation, preprint
- Gérard, Grellier
- 2013
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Beating effects in cubic Schrödinger systems and growth of Sobolev norms
- Grébert, Paturel, et al.
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Citation Context ...d mass) between two disjoint sites in frequency space periodically in time. These correspond to periodic-in-time solutions of (1.3) that exhibit the following “beating effect” (in the nomenclature of =-=[45]-=-): there exists two disjoint subsets R× Λ1 and R× Λ2 in R× Z2, so that for any ε ∈ (0, 1) there exists a solution G(t) of (1.3) that is supported in frequency space on R × Λ1 ∪ R × Λ2 in such a way th... |

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On the 1D cubic NLS in an almost critical space, master thesis
- Guo
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Citation Context ... Then, we have the bound ‖QλuQµv‖L2x,t(R×R) . λ − 1 2 ‖u0‖L2x(R)‖v0‖L2x(R). (7.14) We refer to [29] for the proof of Lemma 7.2 (see also [13] for the earlier higher dimensional analogue of (7.14) and =-=[48]-=- for recent closely related estimates). Lemma 7.3. Assume that N ≥ 7. Then we have the bound sup x∈R ∑ p∈Zd [ 1 + |p|2] |eit∂xxFp(x)|2 . 〈t〉−1(‖F‖2Z + 〈t〉− 14 (‖xF‖2L2 + ‖F‖2HN )). (7.15) Proof. It su... |

2 |
Global regularity for the energy-critical
- Pausader, Tzvetkov, et al.
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2 | Well-posedness and scattering for NLS on Rd×T in the energy space. preprint. Archived at http://arxiv.org/abs/1409.3938 - TZVETKOV, VISCIGLIA - 2014 |

2 | Large time blow up for a perturbation of the cubic Szegő equation, paper in preparation. Université Paris-Sud XI, Laboratoire de Mathématiques d’Orsay, CNRS, UMR 8628, et Institut Universitaire de France E-mail address : Patrick.Gerard@math.u-psud.fr D
- Xu
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1 | The Euler-Maxwell 2 fluid in 3D, preprint arXiv:1303.1060 - Guo, Ionescu, et al. |

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The NLS ground states on product spaces, preprint arXiv:1205.0342
- Terracini, Tzvetkov, et al.
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Citation Context ...Td, considerable interest has emerged in the past few years to study questions of long-time behavior on “in between” manifolds, like the ones presented by the non-compact quotients of Euclidean space =-=[52, 59, 60, 84, 85, 87]-=-. In the generality of non-compact Riemannian d−manifolds M , it seems plausible that a key role is played by the parameter α for which solutions to the linear NLS equation ((1.2) with λ = 0) with smo... |