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E. n 1. profile such that ∇⊥ Within these assumptions, Eq. 35) which is a nonlinear Schrödinger equation (NLSE), one of the most recurrent and studied equations in nonlinear physics (Sulem and Sulem 1999). 36) where ∂μ = (∂x , ∇ ⊥ ) and L is the functional (also named the Lagrangian density) i 1 1 4 L = − (a˜ ∗ ∂x a˜ − a∂ ˜ . 37) The rule is that a, ˜ ∂t a, ˜ ∇ ⊥ a˜ and their complex conjugates must be considered as independent variables. 35). This formulation generalizes the Lagrangian approach to classical mechanics that should be known to most readers, and is strongly used in the quantum theory of fields.

31) by numerical methods (Sun et al. 1987) shows that in the limit of uniform density n = 1 for which the threshold is evaluated (see next Sect. 3), also η → 1, making the calculation consistent. 3 Nonlinear Schrödinger Equation. Self-focusing Threshold Following the discussion at the end of the preceding Sect. e. n 1. profile such that ∇⊥ Within these assumptions, Eq. 35) which is a nonlinear Schrödinger equation (NLSE), one of the most recurrent and studied equations in nonlinear physics (Sulem and Sulem 1999).

As a case which will be of interest in Chap. 5, let us consider an electrostatic regime with no magnetic field, so that E = −∇Φ, and where we are interested on phenomena occurring on the time scale of ion motion. 71) yields en e ∇Φ − ∇Pe = 0. We further assume that the electrons have an isothermal, perfect gas-like EoS Pe = n e Te with constant and uniform temperature Te . g. Feynman et al. 1963, Sect. 4). 73) and Eqs. 71) for ions yield a consistent model which, when the equations are linearized, provides the dispersion relation for ion-acoustic waves.

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Physics of nonneutral plasmas-New by Davidson
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