By Andrea Macchi

The continual development in the direction of larger and better laser intensities has opened tips on how to new actual regimes and complex functions of laser-plasma interactions, therefore stimulating novel connections with ultrafast optics, astrophysics, particle physics, and biomedical functions. This booklet is essentially orientated in the direction of scholars and younger researchers who have to collect swiftly a simple wisdom of this energetic and quickly altering study box. To this objective, the presentation is targeted on a range of simple types and encouraging examples, and comprises themes which emerged lately reminiscent of ion acceleration, "relativistic engineering" and radiation friction. The contents are offered in a self-contained approach assuming just a simple wisdom of classical electrodynamics, mechanics and relativistic dynamics on the undergraduate (Bachelor) point, with out requiring any past wisdom of plasma physics. for this reason, the publication may perhaps serve in different methods: as a compact textbook for lecture classes, as a quick and obtainable creation for the newcomer, as a short reference for the skilled researcher, and in addition as an creation to a few nonlinear mathematical equipment via examples in their program to laser-plasma modeling.

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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.