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## DECAY ESTIMATES AND SMOOTHNESS FOR SOLUTIONS OF THE DISPERSION MANAGED NON-LINEAR SCHRÖDINGER EQUATION

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702 |
The concentration-compactness principle in the caclulus of variations. The locally compact case
- Lions
- 1984
(Show Context)
Citation Context ...y converge weakly to zero, since thes4 D. HUNDERTMARK AND Y.-R. LEE functional (1.6) is invariant under shifts of f. This non-compactness was overcome using Lions’ concentration compactness principle =-=[18]-=-. In the case of positive dav, every weak solution f ∈ H 1 (R) of (1.5) is automatically C ∞ . This follows from a simple bootstrapping argument. Using that f ↦→ Q(f, f, f) maps the Sobolev spaces H s... |

616 |
Methods of Modern Mathematical Physics. I. Functional Analysis
- Reed, Simon
- 1972
(Show Context)
Citation Context ...ion f : R → C is a Schwartz function if and only if x ↦→ 〈x〉 n f(x) is square integrable for all n ∈ N and all weak derivatives of f are square integrable. Proof. Let D = −i∂x. Lemma 1 on page 141 in =-=[25]-=- tells us that f is a Schwartz function, that is, �f�n,m,∞ = sup |〈x〉 x∈R n D m f(x)| < ∞ for all n, m ∈ N0, if and only if � �f�n,m,2 = � |〈x〉 n D m f(x)| 2 �1/2 dx < ∞ for all n, m ∈ N0. In particul... |

474 |
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
- Bourgain
- 1993
(Show Context)
Citation Context ...rmation on the minimizer besides being square integrable and bounded. Shortly afterwords, Milena Stanislavova showed that Kunze’s minimizer is smooth. Her approach employed the use of Bourgain spaces =-=[2, 3]-=- and Tao’s bilinear estimates [30]. To the best of our knowledge these results are the only known rigorous results concerning solutions of (1.5). For example, nothing is rigorously known so far on the... |

409 | Semilinear Schrödinger Equations - Cazenave - 2003 |

342 |
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
- Strichartz
- 1977
(Show Context)
Citation Context ...is proven. For this we use the one-dimensional Strichartz inequality, � � R R |Ttf(x)| 6 dxdt ≤ S 6 1�f� 6 L2 (R) , (2.6) which holds due to the dispersive properties of the free Schrödiner equation, =-=[10, 28, 29]-=-. The sharp constant in (2.6) is known, S1 = 12−1/12 , one even knows S2 in two space dimensions, see [7, 11], but, so far, not in any other space dimension d ≥ 3. Let �f�2 = 1. Using the Cauchy-Schwa... |

325 | AND P.-L.SULEM, The nonlinear Schrödinger equation: Self-Focusing and Wave Collapse - SULEM - 1999 |

175 |
L.: Oscillatory integrals and regularity of dispersive equations
- Kenig, Ponce, et al.
- 1991
(Show Context)
Citation Context ...with ∆ = �d j=1 ∂2 xj the Laplacian in Rd , the free Schrödinger time evolution, then �Ttf1Ttf2� L 2 (R×R d ,dtdx) ≤ C � dist(supp � f1, supp � f2) for some constant depending on d, see, for example, =-=[12, 6]-=-. �f1� L 2 (R d )�f2� L 2 (R d ). (2.23)s12 D. HUNDERTMARK AND Y.-R. LEE (ii) Similarly, a suitable version of the duality Lemma is valid in all space dimensions. To formulate this, let � f be the d-d... |

122 |
Course of Theoretical Physics
- Landau, Lifshitz
- 1990
(Show Context)
Citation Context ... along the optical waveguide and the varying dispersion, is interpreted, in the spirit of Kapitza’s treatment of the unstable pendulum which is stabilized by fast small oscillations of the pivot, see =-=[17]-=-, as a fast background oscillation, justifying formally the above averaging procedure. The Gabitov–Turitzin model (1.3) for the dispersion managed optical waveguide well-supported by numerical studies... |

108 |
The global Cauchy problem for the non linear Schrödinger equation revisited
- Ginibre, Velo
- 1985
(Show Context)
Citation Context ...is proven. For this we use the one-dimensional Strichartz inequality, � � R R |Ttf(x)| 6 dxdt ≤ S 6 1�f� 6 L2 (R) , (2.6) which holds due to the dispersive properties of the free Schrödiner equation, =-=[10, 28, 29]-=-. The sharp constant in (2.6) is known, S1 = 12−1/12 , one even knows S2 in two space dimensions, see [7, 11], but, so far, not in any other space dimension d ≥ 3. Let �f�2 = 1. Using the Cauchy-Schwa... |

59 | Tao: A refined global well-posedness result for Schrodinger equations with derivative, preprint arXiv:math.AP/0110026,
- Colliander, Keel, et al.
- 2001
(Show Context)
Citation Context ...gging this back into (2.21) gives � � which is 2.19. R |Ttf1Ttf2| R 2 |t| −1 dxdt = 2 R � R � |Tτ R ˇ f1(z)Tτ ˇ f2(z)| 2 dzdτ Proof of Lemma 2.3. This result is known to the experts, see, for example =-=[6, 24]-=-. We give a proof for the convenience of the reader. Using the Fourier representation (2.15) of a solution of the free Schrödinger equation, Ttf1(x)Ttf2(x) = 1 2π In particular, � � |Ttf1Ttf2| 2 dxdt ... |

50 |
Multilinear weighted convolution of L2 functions, and applications to nonlinear dispersive equations
- Tao
(Show Context)
Citation Context ...ng square integrable and bounded. Shortly afterwords, Milena Stanislavova showed that Kunze’s minimizer is smooth. Her approach employed the use of Bourgain spaces [2, 3] and Tao’s bilinear estimates =-=[30]-=-. To the best of our knowledge these results are the only known rigorous results concerning solutions of (1.5). For example, nothing is rigorously known so far on the spatial behavior of dispersion ma... |

40 | Maximizers for the Strichartz inequality
- Foschi
(Show Context)
Citation Context ..., (2.6) which holds due to the dispersive properties of the free Schrödiner equation, [10, 28, 29]. The sharp constant in (2.6) is known, S1 = 12−1/12 , one even knows S2 in two space dimensions, see =-=[7, 11]-=-, but, so far, not in any other space dimension d ≥ 3. Let �f�2 = 1. Using the Cauchy-Schwarz inequality, one gets � 1 � Q(f, f, f, f) = |Ttf| 3+1 dxdt ≤ �� 1 � |Ttf| 6 dxdt �1/2� � 1 � |Ttf| 2 dxdt �... |

35 |
Multiscale pulse dynamics in communication systems with strong dispersion management,”
- Ablowitz, Biondini
- 1998
(Show Context)
Citation Context ...n d(t) along the fiber, which, if the dispersion exactly cancel each other, leads to pulses changing periodically along the fiber. This idea has turned out to be enormously fruitful, see for example, =-=[1, 8, 9, 14, 15, 16, 21, 23]-=- and the references therein, even if one takes small non-linear effects into account and allows for a small residual dispersion along the fiber, the residual dispersion together with the non-linearity... |

32 |
Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation
- Gabitov, Turitsyn
- 1996
(Show Context)
Citation Context ...n d(t) along the fiber, which, if the dispersion exactly cancel each other, leads to pulses changing periodically along the fiber. This idea has turned out to be enormously fruitful, see for example, =-=[1, 8, 9, 14, 15, 16, 21, 23]-=- and the references therein, even if one takes small non-linear effects into account and allows for a small residual dispersion along the fiber, the residual dispersion together with the non-linearity... |

30 | On sharp Strichartz inequalities in low dimensions
- Hundertmark, Zharnitsky
(Show Context)
Citation Context ..., (2.6) which holds due to the dispersive properties of the free Schrödiner equation, [10, 28, 29]. The sharp constant in (2.6) is known, S1 = 12−1/12 , one even knows S2 in two space dimensions, see =-=[7, 11]-=-, but, so far, not in any other space dimension d ≥ 3. Let �f�2 = 1. Using the Cauchy-Schwarz inequality, one gets � 1 � Q(f, f, f, f) = |Ttf| 3+1 dxdt ≤ �� 1 � |Ttf| 6 dxdt �1/2� � 1 � |Ttf| 2 dxdt �... |

29 |
Pulse-overlapped dispersionmanaged data transmission and intrachannel four-wave mixing,”
- Mamyshev, Mamysheva
- 1999
(Show Context)
Citation Context ...n d(t) along the fiber, which, if the dispersion exactly cancel each other, leads to pulses changing periodically along the fiber. This idea has turned out to be enormously fruitful, see for example, =-=[1, 8, 9, 14, 15, 16, 21, 23]-=- and the references therein, even if one takes small non-linear effects into account and allows for a small residual dispersion along the fiber, the residual dispersion together with the non-linearity... |

27 |
Space-time estimates for null gauge forms and nonlinear Schrodinger equations,
- Ozawa, Tsutsumi
- 1998
(Show Context)
Citation Context ...gging this back into (2.21) gives � � which is 2.19. R |Ttf1Ttf2| R 2 |t| −1 dxdt = 2 R � R � |Tτ R ˇ f1(z)Tτ ˇ f2(z)| 2 dzdτ Proof of Lemma 2.3. This result is known to the experts, see, for example =-=[6, 24]-=-. We give a proof for the convenience of the reader. Using the Fourier representation (2.15) of a solution of the free Schrödinger equation, Ttf1(x)Ttf2(x) = 1 2π In particular, � � |Ttf1Ttf2| 2 dxdt ... |

16 |
Demonstration of massive wavelength-division multiplexing over
- Mollenauer, Mamyshev, et al.
- 1999
(Show Context)
Citation Context |

14 |
Breathing solitons in optical fiber links,”
- Gabitov, Turitsyn
- 1996
(Show Context)
Citation Context |

14 |
Suppression of fiber nonlinearities by appropriate dispersion management
- Kurtzke
- 1993
(Show Context)
Citation Context |

14 | Dispersion-managed soliton in a strong dispersion map limit
- Lushnikov
(Show Context)
Citation Context ...cedure. The Gabitov–Turitzin model (1.3) for the dispersion managed optical waveguide well-supported by numerical studies, see, for example, [1] and [32], and theoretical arguments, see, for example, =-=[19, 20]-=-. In addition, this averaging procedure was rigorously justified in [33] where it is shown that in the regime of strong dispersion management, ε ≪ 1, on long scales 0 ≤ t ≤ Cε −1 the solution of the f... |

13 |
On a variational problem with lack of compactness related to the Strichartz inequality,
- Kunze
- 2004
(Show Context)
Citation Context ... of f in real space and in Fourier space, the traditional a-priori bounds from the calculus of variations are not available. The existence of minimizers of (1.6) for dav = 0 was shown by Markus Kunze =-=[13]-=-, using the concentration compactness principle in tandem; first in Fourier and then in x-space. Unfortunately, the bootstrapping argument which shows smoothness of solutions of (1.5) for dav > 0 now ... |

13 |
Quasi-soliton propagation in dispersion-managed optical fibers
- Kumar, Hasegawa
- 1997
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12 |
V.K.: Physics and mathematics of dispersion–managed optical solitons.
- Turitsyn, Shapiro, et al.
- 2003
(Show Context)
Citation Context ...e the dispersion is varied periodically along an optical fiber; it describes the amplitude of a signal transmitted via amplitude modulation of a carrier wave through a fiber-optical cable, see, e.g., =-=[2, 34, 38]-=-. In (1.1) t corresponds to the distance along the fiber, x denotes the (retarded) time, ut = ∂tu = ∂ ∂t u, uxx = ∂2 xu = ∂2 (∂x) 2 u, c a constant determining the strength of the non-linearity which,... |

9 |
Optical pulse equalization and low dispersion transmission in singlemode fibers in the 1.3–1.7 µm spectral region
- Lin, Kogelnik, et al.
- 1980
(Show Context)
Citation Context ... a rapidly varying dispersion d(t) along the fiber, which, if the dispersion exactly cancel each other, leads to pulses changing periodically along the fiber. This idea had been introduced in 1980 in =-=[21]-=-. It has turned out to be enormously fruitful, see for example, [1, 10, 11, 17, 18, 19, 25, 27] and the references therein, even if one takes small non-linear effects into account and allows for a sma... |

8 |
Shape of the stationary pulse in the strong dispersion management regime
- Lakoba, Kaup
- 1998
(Show Context)
Citation Context |

8 |
Regularity of ground state solutions of DMNLS equations
- Stanislavova
(Show Context)
Citation Context ...1.4.ii and Corollary 3.3. This misses the conjectured exponential decay rate, however. (iv) Theorem 1.1 significantly strengthens Stanislavova’s result on smoothness of dispersion managed solitons in =-=[27]-=-. In addition, our proof is technically much simpler than Stanislavova’s. We will deduce the regularity property of dispersion managed solitons given in Theorem 1.1 from a suitable decay estimate on t... |

6 | 8X10 GB/s transmission through 280 Km. of dispersion managed fiber - Chraplyvy |

6 |
Experimental test of dense wavelengthdivision multiplexing using novel, periodic-group-delaycomplemented dispersion compensation and dispersionmanaged solitons
- Mollenauer, Grant, et al.
- 2003
(Show Context)
Citation Context ...h other, allowing the existence of stable soliton-like pulses. Record breaking transmission rates of more than 1 Tbits/s over an 18,000 kilometer optical fiber had been achieved using this technology =-=[22]-=- and the technique of dispersion management is now widely used commercially. Due to the enormous practical implications, there has been a huge literature concerning the numerical and phenomenological ... |

5 |
Oscillating tails of dispersion-managed soliton
- Lushnikov
- 2004
(Show Context)
Citation Context ...cedure. The Gabitov–Turitzin model (1.3) for the dispersion managed optical waveguide well-supported by numerical studies, see, for example, [1] and [32], and theoretical arguments, see, for example, =-=[19, 20]-=-. In addition, this averaging procedure was rigorously justified in [33] where it is shown that in the regime of strong dispersion management, ε ≪ 1, on long scales 0 ≤ t ≤ Cε −1 the solution of the f... |

4 |
Turitsyn: Optimization of dispersion-managed optical fiber lines, to appear in
- Schafer, Laedke, et al.
- 2002
(Show Context)
Citation Context ...onal loss of compactness. Nevertheless, it is very important physically, since certain physical effects which destabilize pulse propagation in optical fibers are minimal for dav near or equal to zero =-=[26, 31]-=-. In this case the constraint minimizing problem is given by P λ = inf � − 1 4 Q(f, f, f, f) : f ∈ L2 (R), �f� 2 � 2 = λ . (1.9) Using the Strichartz inequality, it was shown in [33] that even in this... |

4 |
Tailoring zero chromatic dispersion into the 1.5 - 1.6 µm low-loss spectral region of single-mode fibers
- Cohen, Lin, et al.
- 1979
(Show Context)
Citation Context ...ted to overcome the difficulty that there are no stable pulses in the linear regime. The idea, building on the fact that optical fibers can be engineered to have positive and negative dispersion, see =-=[7]-=-, is to use alternating sections of constant but opposite, or nearly opposite, dispersion. This introduces a rapidly varying dispersion d(t) along the fiber, which, if the dispersion exactly cancel ea... |

3 |
Seikai: Experimental Evaluation of Gordon-Haus timing jitter of dispersion managed solitons,
- Toda, Hamada, et al.
- 1999
(Show Context)
Citation Context ...onal loss of compactness. Nevertheless, it is very important physically, since certain physical effects which destabilize pulse propagation in optical fibers are minimal for dav near or equal to zero =-=[26, 31]-=-. In this case the constraint minimizing problem is given by P λ = inf � − 1 4 Q(f, f, f, f) : f ∈ L2 (R), �f� 2 � 2 = λ . (1.9) Using the Strichartz inequality, it was shown in [33] that even in this... |

3 | Nonlinear Fiber Optics. Second Edition (Optics and Photonics - Agrawal - 1995 |

3 | Veng: Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion managed solitons, - Mollenauer, Mamyshev, et al. - 2000 |

2 |
Stabilzing effects of dispesion management, Physica D 152-153 (2001), 794–817. Department of Mathematics, Altgeld Hall, and Institute for Condensed Matter Theory at the University of Illinois at Urbana–Champaign, 1409
- Zharnitsky, Grenier, et al.
(Show Context)
Citation Context ...waveguide well-supported by numerical studies, see, for example, [1] and [32], and theoretical arguments, see, for example, [19, 20]. In addition, this averaging procedure was rigorously justified in =-=[33]-=- where it is shown that in the regime of strong dispersion management, ε ≪ 1, on long scales 0 ≤ t ≤ Cε −1 the solution of the full equation (1.2) stays ε-close to a solution of (1.3) with the same in... |

2 | 8 10 Gb/s transmission through 280 km of dispersion-managed fiber - Chraplyvy, Gnauck, et al. - 1993 |

1 |
On the existence of dispersion managed solitons for vanishing residual dispersion
- Hundertmark, Lee
(Show Context)
Citation Context ... immediately transforms Fourier-space bilinear estimates into corresponding x-space bilinear bounds seems to be new.DECAY AND SMOOTHNESS OF DISPERSION MANAGED SOLITONS 13 (v) In a forthcoming paper, =-=[13]-=-, we use the Fourier and x-space bilinear Strichartz estimates to give a simple proof of existence of minimizers of the minimization problem (1.10) which avoids the use of Lion’s concentration compact... |

1 |
Elementary introduction to the theory of speudodifferential operators
- Raymond
- 1991
(Show Context)
Citation Context ...ce ε〈x〉 ≤ 〈εx〉 for all x and all 0 ≤ ε ≤ 1. This proves (3.15) since 1 ≤ 〈εx〉 ≤ 〈x〉 for all x, and 0 ≤ ε ≤ 1. Remark 3.9. Using the multidimensional Binomial and Leibnitz formulas, see Theorem 1.2 in =-=[29]-=-, the corresponding statement of Lemma 3.7 and Lemma 3.8 hold also on R d with virtually identical proofs. Acknowledgment: It is a pleasure to thank Vadim Zharnitsky for instructive discussions on the... |

1 | G.: Tailoring zero chromatic dispersion into the 1.5 - Cohen, Lin, et al. |

1 | Turitsyn, S.K.: Optimization of dispersion-managed optical fiber lines - Schäfer, Laedke, et al. |

1 | Turitsyn, S.K: Stabilzing effects of dispesion management - Zharnitsky, Grenier, et al. - 2001 |