### Citations

145 |
Instability and nonexistence of global solutions of nonlinear wave equations of the form
- Levine
- 1974
(Show Context)
Citation Context ...+ |∇χω|2 + nω|χω|2 + c 2 2 (∇ψω)2 + 1 2 (nω)2 , (4.6) where the function ψω expresses as before by simply replacing the densities |Eω|2 → |χω|2. Following the concavity method introduced by Levine =-=[15]-=-, we now multiply (4.4) by χ̄ω and integrate in space the real part of the resulting equation to get 1 2ω2 ∂2 ∂t2 ∫ Rd |χω|2 = 2 ω2 ∫ Rd ∣∣∣∣∣∂χω∂t ∣∣∣∣∣ 2 −E ′ω+ 1 2 ∫ Rd (nω)2+ c2 2 ∫ Rd (∇ψω)2. (4.... |

128 |
On nonlinear Schrödinger equations
- Kato
- 1995
(Show Context)
Citation Context ...days known about this family of systems : i) The first case is ω = c = +∞ and leads to the cubic Schrödinger equation which has been extensively studied in the past (see Ginibre and Velo [6] or Kato =-=[7]-=-). ii) The case ω = +∞ and c < +∞ corresponds to the Zakharov equations whose well-posedness of the Cauchy problem has been addressed by Sulem and Sulem [8] and Ozawa and Tsutsumi [9]. Furthermore, th... |

102 |
Introduction to Plasma Theory,
- Nicholson
- 1983
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Citation Context ... two decades, the strong Langmuir turbulence (SLT) has often been emphasized as being a natural way for saturating the sudden growth of electron plasma waves interacting with ion density fluctuations =-=[1]-=- : the typical scenario for damping ”energetic” plasma waves (also called Langmuir waves) is that these high-frequency (HF) waves, carried by electron oscillations, couple with low-frequency (LF) ion ... |

93 |
On a class of nonlinear schrödinger equation. I. The Cauchy problems; II. Scattering theory, general case
- Ginibre, Velo
(Show Context)
Citation Context ...lts are nowadays known about this family of systems : i) The first case is ω = c = +∞ and leads to the cubic Schrödinger equation which has been extensively studied in the past (see Ginibre and Velo =-=[6]-=- or Kato [7]). ii) The case ω = +∞ and c < +∞ corresponds to the Zakharov equations whose well-posedness of the Cauchy problem has been addressed by Sulem and Sulem [8] and Ozawa and Tsutsumi [9]. Fur... |

85 |
Collapse of Langmuir waves
- Zakharov
- 1972
(Show Context)
Citation Context ...sponds to the elementary radius of the screening cloud formed by electrons surrounding an isolated ion). The key-mechanism of this physical phenomenon is governed by the well-known Zakharov equations =-=[2]-=- that describe the slowly-varying motions of the complex-valued envelope of the Langmuir electric field which nonlinearly interacts with large-scale density fluctuations through the ponderomotive forc... |

42 |
Note on loss of regularity for solutions of the 3D incompressible Euler and related equations,
- Constantin
- 1986
(Show Context)
Citation Context ...nts as in [4] and using the fact that for ω large enough, Kω, Lω, P ω and P ωt are controlled by the solution of the limit system (3.6)-(3.7). Results similar to ii) have been proved by P. Constantin =-=[11]-=- in the context of the Euler and Navier-Stokes equations and adapted to the Zakharov equations in [10]. Furthermore, the theorem remains valid if (K0, L0, P0, P1) depend on ω and converge in Hs ×Hs ×H... |

28 |
Existence and smoothing effect of solutions for the Zakharov equations
- Ozawa, Tsutsumi
- 1992
(Show Context)
Citation Context ...Velo [6] or Kato [7]). ii) The case ω = +∞ and c < +∞ corresponds to the Zakharov equations whose well-posedness of the Cauchy problem has been addressed by Sulem and Sulem [8] and Ozawa and Tsutsumi =-=[9]-=-. Furthermore, the limit c → ∞ has been treated by Schochet and Weinstein [10]. Some partial results on time blow-up can also be found in Glangetas and Merle [12] and Merle [13]. iii) As recalled in s... |

24 |
The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence
- Schochet, Weinstein
- 1986
(Show Context)
Citation Context ...olution of the limit system (3.6)-(3.7). Results similar to ii) have been proved by P. Constantin [11] in the context of the Euler and Navier-Stokes equations and adapted to the Zakharov equations in =-=[10]-=-. Furthermore, the theorem remains valid if (K0, L0, P0, P1) depend on ω and converge in Hs ×Hs ×Hs+1 ×Hs as ω tends to ∞. Proof of i) Let K and P be respectively solution to K = e− i 2 ∆tK0 + ∫ t 0 e... |

22 | Strong turbulence of plasma waves - Goldman - 1984 |

19 |
Existence of self-similar blow-up solutions for Zakharov equation in dimension two
- Glangetas, Merle
- 1994
(Show Context)
Citation Context ...m and Sulem [8] and Ozawa and Tsutsumi [9]. Furthermore, the limit c → ∞ has been treated by Schochet and Weinstein [10]. Some partial results on time blow-up can also be found in Glangetas and Merle =-=[12]-=- and Merle [13]. iii) As recalled in section 1.3, the case ω < +∞ and c = +∞ has been recently studied by Bergé and Colin in [4] where the asymptotics ω → +∞ has been accurately described. In what fo... |

13 |
Quelques résultats de régularité pour les équations de la turbulence de Langmuir
- Sulem, Sulem
- 1979
(Show Context)
Citation Context ... the past (see Ginibre and Velo [6] or Kato [7]). ii) The case ω = +∞ and c < +∞ corresponds to the Zakharov equations whose well-posedness of the Cauchy problem has been addressed by Sulem and Sulem =-=[8]-=- and Ozawa and Tsutsumi [9]. Furthermore, the limit c → ∞ has been treated by Schochet and Weinstein [10]. Some partial results on time blow-up can also be found in Glangetas and Merle [12] and Merle ... |

10 |
Blow-up results of virial type for Zakharov equations
- Merle
- 1996
(Show Context)
Citation Context ... and Ozawa and Tsutsumi [9]. Furthermore, the limit c → ∞ has been treated by Schochet and Weinstein [10]. Some partial results on time blow-up can also be found in Glangetas and Merle [12] and Merle =-=[13]-=-. iii) As recalled in section 1.3, the case ω < +∞ and c = +∞ has been recently studied by Bergé and Colin in [4] where the asymptotics ω → +∞ has been accurately described. In what follows, we fix c... |

8 |
A singular perturbation problem for an envelope equation in plasma physics
- Bergé, Colin
- 1995
(Show Context)
Citation Context ...t, equation (1.14) simplifies into n = −|E|2 and (1.13) reduces to a nonlinear Schrödinger equation, the solutions of which have been recently examined in the time-envelope limit by Bergé and Colin =-=[4]-=-. From a physical point of view, the applicability domain of the subsonic regime concerns Langmuir envelopes coupled with ion cavities whose typical extensions in space and in time - denoted by L and ... |

6 | On a nonlocal Zakharov equation
- Bidégaray
- 1995
(Show Context)
Citation Context ...of the form [grad(∆−1)div] be applied to the coupling term −nE, all the results of the present paper can be extended to the vectorial version of (1.19) which has been studied for ω = +∞ by Bidégaray =-=[14]-=-. Acknowledgements: We wish to thank J.M. Ghidaglia for having suggested us to use a transformation like (2.6). We also acknowledge the referee for his detailed and valuable comments on the present in... |

1 |
see for a review
- Bergé, Pesme
- 1993
(Show Context)
Citation Context ... + δn/N0) satisfying δn/N0 << 1. An alternative way of recovering (1.1) is to take formally the Fourier transform of the characteristic dispersion relation of Langmuir waves with frequency ω̃, namely =-=[3]-=- ω̃2 = ω2pe[N0 + δn](1 + 3k 2λ2D) (1.2) 2 where the wavenumber k usually obeys the condition k2λ2D << 1 when regarding large-scale fluctuations on which the Landau damping remains unefficient. In (1.2... |