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Adiabatic nonlinear waves with trapped particles. II. Wave dispersion
2. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974), Chap. 14 and 15.
3. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012102 (2012).
5. V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 95, 1951 (1989) [Sov. Phys. JETP 68, 1129 (1989)].
6. V. L. Krasovskii, Zh. Eksp. Teor. Fiz. 107, 741 (1995) [JETP 80, 420 (1995)].
12. I. Y. Dodin and N. J. Fisch, Phys. Plasmas 19, 012104 (2012).
13. For example, for anharmonic oscillations, can be constructed iteratively, along the lines of existing iterative approaches to NDR (Refs. 16 and 18). Yet, in practice, the spatial profile is most often assumed to be prescribed, e.g., sinusoidal. Within this commonly accepted approach, our method becomes particularly useful, because then is known immediately.
19. Some of the fluid nonlinearities, yielding , are retained in Eq. (10) due to the nonlinear dependence J(p). However, describing these effects accurately would require accounting for the contribution of the wave second harmonic, which is of the same order.
20. T. H. Stix, Waves in Plasmas (AIP, New York, 1992), Secs. 8-6.
23. To reduce the results of Ref. 10 to Eq. (11), the former must be amended in two aspects. First, rather than ni, it is f that must be adjusted to ensure that (i.e., must be fixed); this changes Fi. Second, when summing over the contributions of multiple waterbags in Eqs. (24)–(26) of Ref. 10, the weight must be F(J)ΔJ rather than f′Δu.
24. Nonzero ∂ta and ∂xa may also yield collisionless dissipation. [See Ref. 7 and also
24. D. D. Ryutov and V. N. Khudik, Zh. Teor. Eksp. Fiz. 64, 1252 (1973) [Sov. Phys. JETP 37, 637 (1973)];
31. In a homogeneous wave with varying u, passing particles conserve their canonical momentum [Eq. (5)] instead, as shown, e.g., in Refs. 10 and 18 .
33. A. V. Timofeev, Zh. Eksp. Teor. Fiz. 75, 1303 (1978) [Sov. Phys. JETP 48, 656 (1978)].
35. Remember that we consider quasistationary waves, on time scales large compared to the bounce period (Paper I). No Landau damping exists in this case.
35.[R. K. Mazitov, Prikl. Mekh. Tekh. Fiz. 1, 27 (1965);
36. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, New York, 1981), Sec. 29.
51. The results reported in the Appendix were facilitated by Mathematica © 1988-2009 Wolfram Research, Inc., version number 184.108.40.206.
52. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., National Bureau of Standards Applied Mathematics Series Vol. 55 (U.S. Dept. of Commerce, U.S. Gov. Printing Office, Washington, DC, 1972), Sec. 17.3.
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A general nonlinear dispersion relation is derived in a nondifferential form for an adiabatic sinusoidal Langmuir wave in collisionless plasma, allowing for an arbitrary distribution of trapped electrons. The linear dielectric function is generalized, and the nonlinear kinetic frequency shift ωNL is found analytically as a function of the wave amplitude a. Smooth distributions yield , as usual. However, beam-like distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation. Such beams are formed whenever the phase velocity changes, because the trapped distribution is in autoresonance and thus evolves differently from the passing distribution. Hence, even adiabatic ωNL(a) is generally nonlocal.
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