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Concept# Wave equation

Summary

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics – such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media.
Introduction
The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions u = u (x1, x2, ..., xn;

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We investigate low-frequency electromagnetic wave propagation and absorption properties in 2D and 3D plasma configurations. For these purposes, we have developed a new full-wave 3D code LEMan that determines a global solution of the wave equation in bounded stellarator plasmas excited with an external antenna. No assumption on the wavelength compared to the plasma size is made, all the effects of the 3D geometry and finite plasma extent are included. The equation is formulated in terms of electromagnetic potentials in order to avoid numerical pollution effects. The code utilises linear and Hermite cubic finite element discretisation in the radial direction and Fourier series in the poloidal and toroidal variables. The full cold plasma model including finite electron inertia and, thus, mode conversion effects is implemented. The code uses Boozer magnetic coordinates and has an interface to the TERPSICHORE code. Special care is taken to treat the magnetic axis and to ensure the unicity of the numerical solution. The discretisation, interpolation and numerical derivation methods specifically adapted for our problem avoid the energy sink in the origin and provide a very good local and global energy conservation. A special algorithm has been developed to analytically expand the wave equation coefficients in the full 3D stellarator geometry. The code has been specifically optimised for vector computing platform, reaching close to maximum average performances on the NEC SX5 machine. The code has been applied in 1D, 2D, and 3D geometries. No unphysical solutions have been observed. LEMan successfully recovers all the fundamental properties of the Alfvén spectrum (gaps, eigenmodes). Benchmarks have been made against the 2D LION code and JET experimental measurements, showing a good agreement between the results.

I will try to explain, without going into too much detail, how one can
consider a non-linear wave equation as a dynamical system and what it brings to the study of its
solutions. We begin by considering our model case, the non-linear Klein-Gordon equation and state
its basic properties. We will then see what happens for solutions with energies below that of the
ground state. After that, we place ourselves energetically around the ground state and we show the
apparition of the so-called invariant manifolds. Finally, we consider the critical "pure" (without
the mass term) wave equation and describe some of its interesting solutions. The last part will be
concerned with an attempt to rely what we have learn so far with the critical case.

We are interested in the well posedness of quasilinear partial differential equations of order two. Motivated by the study of the Einstein equation in relativity theory, there are a number of works dedicated to the local well-posedness issue for the quasilinear wave equation. We will focus on local well-posedness for the wave equation ; more precisely we are looking at the smallest Sobolev index such that the local well-posedness holds true for initial data in this space. In 2005, D. Tataru and Hart. F. Smith provided the current best upper bound for the smallest index in low dimension. In 1998, Hans Lindblad constructed a counter example for s=3 in dimension three, thus revealing the sharpness of Tataru and Smith's criteria in this dimension. Here, our goal is to obtain sharp counterexamples to local well-posedness for quasilinear wave equations of geometric character. First, we check how the construction by Lindblad translates to dimension two. Next, we shall try to see if a similar breakdown result applies to the vanishing mean curvature problem in Minkowski space. Finally, as a more long term goal, we may try to find explicit singular solutions of this problem, starting with smooth data, by following the constructions of Krieger-Schlag-Tataru.

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