connected with the velocity shift of the Langmuir waves, the spatial profile of waves will become smooth and the solution will be close to that obtained in the case of uniform plasma. This result agrees well with the qualitative results of ([@Coste75]).

For application to the theory of type III bursts, special interest is presented by a combination of the two main properties of the solutions.

On one hand, the electron beam can propagate in a plasma over large distances, and is a source of a high level of Langmuir waves. A portion of these Langmuir waves can easily be transformed into observable radio emission via nonlinear plasma processes ([@Ginzburg58]). At a scale much greater than the size of the beam, electrons and Langmuir waves propagate as a BPS that may be the source of type III bursts. The BPS propagates in inhomogeneous plasma with velocity $\approx v_0/2$ that can explain the almost-constant speed of the type III source. The finite size of the structure, the spatial expansion of the structure, and conservation of the particle number, are promising results for the theory of type III bursts.

On the other hand, plasma inhomogeneity brings additional results. The spatial distribution of 