The anisotropic collapse of Langmuir waves (Q1113774)

Since Zakharov's work [Sov. Phys. JETP 35, 908 ff. (1972)], it is well known that self-focusing of Langmuir waves in a plasma produces collapsing wave packets. However, although it tends to build a point singularity when dissipation is disregarded, further numerical computations exhibited pancake-shaped solutions rather than spherical ones. In this paper, we first propose an analogy between self-focusing waves and self-gravitating systems which clarifies pancake formation. Then we construct a group transformation that allows to solve for the evolution of pancakes explicitly, for both a homogeneous and an inhomogeneous medium. The transformed problem can be investigated by the inverse scattering method and its subsequent perturbation technique.