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Abstract
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. Onedimensional 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 trappedparticle 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 selfaction but misses the trappedparticle oscillationcenter inertia.
The work was supported through the NNSA SSAA Program through DOE Research Grant No. DE274FG5208NA28553.
I. INTRODUCTION
II. BASIC MODEL
III. WAVEEQUATIONS
IV. WAVE TRANSFORMATIONS IN PLASMA WITH VARYING PARAMETERS
V. PULSE PROPAGATION
A. Group velocity splitting
B. TPMI
VI. DISCUSSION
A. Stability criterion
B. Comparison with the existing theories
VII. CONCLUSIONS
Key Topics
 Plasma waves
 62.0
 Negative resistance
 13.0
 Lagrangian mechanics
 10.0
 Electrons
 9.0
 Electrostatic waves
 7.0
Figures
1D stationary wave with trapped particles in inhomogeneous plasma. Shown is the normalized amplitude of the wave potential, φ/φ_{c} , vs. the plasma normalized density for different ν. Here φ_{c} = 2Θ, with Θ = 0.1 taken as an example; n_{c} is the critical density. Dashed is the linear solution (ν = 0) and the location of the linear cutoff (n = n_{c} ).
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1D stationary wave with trapped particles in inhomogeneous plasma. Shown is the normalized amplitude of the wave potential, φ/φ_{c} , vs. the plasma normalized density for different ν. Here φ_{c} = 2Θ, with Θ = 0.1 taken as an example; n_{c} is the critical density. Dashed is the linear solution (ν = 0) and the location of the linear cutoff (n = n_{c} ).
(Color online) Schematic of the parameter domain assumed for Sec. V in space (κ, ϑ, Ω_{ E }) and (b) in space (κ, ϑ, S). Combined here are the following assumptions: the plasma is cold [Eq. (10)], the wave is sinusoidal [Eq. (14)] and weak enough [Eq. (A7)], the quasistatic field due to trapped particles is negligible [Eq. (A11)]; also, the bulk motion and the nonlinear effects are weak, i.e., [Eq. (30); see also Eqs. (8) and (28)]. The inequalities Eqs. (9) and (A12) reduce to and thus are satisfied automatically.
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(Color online) Schematic of the parameter domain assumed for Sec. V in space (κ, ϑ, Ω_{ E }) and (b) in space (κ, ϑ, S). Combined here are the following assumptions: the plasma is cold [Eq. (10)], the wave is sinusoidal [Eq. (14)] and weak enough [Eq. (A7)], the quasistatic field due to trapped particles is negligible [Eq. (A11)]; also, the bulk motion and the nonlinear effects are weak, i.e., [Eq. (30); see also Eqs. (8) and (28)]. The inequalities Eqs. (9) and (A12) reduce to and thus are satisfied automatically.
(a) Nonlinear group velocities, and [Eq. (31)] vs. κ, for sample Ω_{ E } and ϑ; the dashed line shows the linear group velocity . At S > 1/2, corresponding to , no real solutions exist for , rendering the wave unstable. (b) Closeup at κ ≈ κ_{S} .
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(a) Nonlinear group velocities, and [Eq. (31)] vs. κ, for sample Ω_{ E } and ϑ; the dashed line shows the linear group velocity . At S > 1/2, corresponding to , no real solutions exist for , rendering the wave unstable. (b) Closeup at κ ≈ κ_{S} .
(Color online) Evolution of the perturbation to a homogeneous wave with the initial amplitude a _{0}. The solution is obtained by numerical integration of Eqs. (B1) and (B3), with taken from Eq. (18), for the same parameters as in Fig. 3 and ℓ = 20λ_{D} . Shown is Δ(a ^{2}) (arbitrary color scaling), vs. t and x in the frame moving with the linear group velocity ; the units are and λ_{D} , correspondingly. (a) κ = 0.2, so S = 1/4; the wave is stable, resulting in signal splitting. (b) κ = 0.1, so S = 1; the wave is TPMIunstable.
Click to view
(Color online) Evolution of the perturbation to a homogeneous wave with the initial amplitude a _{0}. The solution is obtained by numerical integration of Eqs. (B1) and (B3), with taken from Eq. (18), for the same parameters as in Fig. 3 and ℓ = 20λ_{D} . Shown is Δ(a ^{2}) (arbitrary color scaling), vs. t and x in the frame moving with the linear group velocity ; the units are and λ_{D} , correspondingly. (a) κ = 0.2, so S = 1/4; the wave is stable, resulting in signal splitting. (b) κ = 0.1, so S = 1; the wave is TPMIunstable.
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Abstract
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. Onedimensional 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 trappedparticle 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 selfaction but misses the trappedparticle oscillationcenter inertia.
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