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"Review of exact solutions to the nonlinear Vlasov equation", dr hab. Piotr Goldstein, NCBJ

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The lecture is a review of exact solutions to the Vlasov equation, including author’s contribution to the subject.

In spite of enormous power of numerical simulations, analytical exact solutions are still important. For simulations and numerical algorithms, they provide excellent tests for consistency, stability, precision and asymptotic behaviour. They also reveal the range of applicability of symbolic manipulation packages. Besides, they play important role in understanding qualitative properties of the equations, and make a good source for designing experiments. Finally, they make good illustrations of basic phenomena for teaching purposes.

Vlasov’s much more precise description of plasmas, compared to fluid models, is achieved at the cost of complexity. Few exact solutions are known.

Their review will begin with static solutions, starting from the oldest (and simplest) Bernstein-Green-Kruskal equilibria (1957) and magnetic equilibria, which are important in geophysical applications. Next some time-dependent models will be discussed, beginning with the water-bag model with its later refinements. A few systematic methods follow, like the Lie-group approach, which provides self-similar solutions (describing real physical situations, e.g., plasma expansion to vacuum). The lecture will be concluded with a short summary of author’s method, which combines singularity analysis with the Lagrangian formalism.

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Research projects carried out at the IPPLM are funded by the Polish Ministry of Education and Science, the National Science Centre and by the European Commission within the framework of EUROfusion Consortium under grant agreement No 633053. Financial support comes also from the International Atomic Energy Agency, European Space Agency and LaserLab Consortium as well as from the Fusion for Energy Agency.

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