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Adiabatic nonlinear waves with trapped particles. III. Wave dynamics
2. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974), Chaps. 14 and 15.
3. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012102 (2012).
4. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012103 (2012).
6. V. I. Karpman, Non-Linear Waves in Dispersive Media (Pergamon, New York, 1974), Chap. V.
7. The indexes a, k, ω, A, and (except in ) denote differentiation with respect to the corresponding variables.
8. From the standpoint of time-averaged dynamics, particle detrapping is an irreversible process, which thus cannot be described by a Lagrangian in principle. Of course, the plasma true Lagrangian still exists; however, its wave part cannot be represented in the Whitham’s form, .
9. Of course, an alternative is the kinetic approach, which yet leads to solutions that are not easily tractable and are generally specific to particular settings. See Ref. 23 and V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 95, 1951 (1989) [Sov. Phys. JETP 68, 1129 (1989)]. Also, see references therein and more recent Ref. 27 and A. I. Matveev, Fiz. Plazmy 34, 1001 (2008) [Plasma Phys. Rep. 34, 924 (2008)]; A. I. Matveev, Russ. Phys. J. 52, 885 (2009); A. I. Matveev, Fiz. Plazmy 35, 351 (2009) [Plasma Phys. Rep. 35, 315 (2009)]; A. I. Matveev, Russ. Phys. J. 53, 369 (2010); A. I. Matveev, Russ. Phys. J. 53, 657 (2010).
11. V. L. Krasovsky, Fiz. Plazmy 18, 739 (1992)
11. V. L. Krasovsky, [Sov. J. Plasma Phys. 18, 382 (1992)].
18. V. I. Karpman and E. M. Krushkal, Zh. Eksp. Teor. Fiz. 55, 530 (1968)
18. V. I. Karpman and E. M. Krushkal, [Sov. Phys. JETP 28, 277 (1969)].
23. V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 107, 741 (1995)
23. V. L. Krasovskii, [JETP 80, 420 (1995)].
27. A. I. Matveev, Fiz. Plazmy 34, 114 (2008)
27. A. I. Matveev, [Plasma Phys. Rep. 34, 95 (2008)].
28. Notice, in particular, that Eq. (18) can be written as . Yet, this form holds only for a δ-shaped trapped-particle distribution; for other distributions, there is a minimum a below which is undefined.
31. T. H. Stix, Waves in Plasmas (AIP, New York, 1992), Sec. 4.2.
32. From S ∼ 1, one gets another estimate, φ ∼ (ν/κ3)1/2. Thus, and .
37. Considered here is nondissipative dynamics only. For dissipative effects, see D. Bénisti, O. Morice, L. Gremillet, E. Siminos, and D. J. Strozzi, Phys. Plasmas 17, 082301 (2010);
38. L. D. Landau and E. M. Lifshitz, Fluid Dynamics (Pergamon, New York, 1987), Secs. 103 and 104.
40. To our knowledge, the results of Ref. 13 have never been associated with the TPMI before.
41. P. E. Masson-Laborde, W. Rozmus, Z. Peng, D. Pesme, S. Hüller, M. Casanova,V. Yu. Bychenkov, T. Chapman, and P. Loiseau, Phys. Plasmas 17, 092704 (2010).
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The evolution of adiabatic waves with autoresonant trapped particles is described within the Lagrangianmodel developed in Paper I, under the assumption that the action distribution of these particles is conserved, and, in particular, that their number within each wavelength is a fixed independent parameter of the problem. One-dimensional nonlinear Langmuir waves with deeply trapped electrons are addressed as a paradigmatic example. For a stationary wave, tunneling into overcritical plasma is explained from the standpoint of the action conservation theorem. For a nonstationary wave, qualitatively different regimes are realized depending on the initial parameter S, which is the ratio of the energy flux carried by trapped particles to that carried by passing particles. At S < 1/2, a wave is stable and exhibits group velocity splitting. At S > 1/2, the trapped-particle modulational instability (TPMI) develops, in contrast with the existing theories of the TPMI yet in agreement with the general sideband instability theory. Remarkably, these effects are not captured by the nonlinear Schrödinger equation, which is traditionally considered as a universal model of wave self-action but misses the trapped-particle oscillation-center inertia.
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