Recent advances associated with an improvement of the popular cold-plasma sharp-boundary model in the theory of nonlinear surface waves in low temperature plasmas are reviewed. The revised boundary model calls for the existence of induced nonlinear surface charges and currents in an infinitely thin (spatially singular) layer at the plasma boundary. It provides a much more accurate description of the physics of the region, especially with respect to the nonlinear effects. Various nonlinear wave phenomena of current interest, such as those of the third-order ponderomotive effect, nonlinear waves, and soliton formation, are discussed. It is demonstrated that for the problems considered, the results from the new model agree well with those obtained from the kinetic theory. The latter inherently includes a more complete description of the boundary conditions since the plasma density can vanish smoothly (and physically consistently) within a boundary layer adjacent to the wall. The layer, or sheath region, has a small but finite dimension usually of the order of the Debye length. Effectively, the new model takes into account the relevant physical processes occuring in this layer, which itself is nevertheless taken to be of zero width. Thus, spatially singular charges and currents appear in the model. Furthermore, a new class of nonlinear cold-plasma surface wave solutions is also reviewed. These solutions are obtained by separating the temporal and spatial dependences of the physical quantities. They are mathematically exact in the sense that starting from the basic cold-plasma equations and boundary conditions, no approximation of any kind needs to be made in obtaining them. Exact nonlinear surface wave solutions in cylindrically and spherically bounded plasmas, as well as the effects of constant and oscillatory external electric fields in the dielectric container, are presented. Modern applications of the results from the above topics in various branches of physics, chemistry, and technology are discussed.

1. Introduction.........................................................................................................................4

2. Nonlinear surface currents.....................................................................................................7

2.1. Electron dynamics...................................................................................................................7
2.2. Dispersion equation for surface waves..................................................................................8
2.3. Surface currents.....................................................................................................................10
2.4. Other third-order effects......................................................................................................12

3. Second-order effects. KdV-like solitons................................................................................13

3.1. Surface solitons in semi-infinite fully ionized plasmas.....................................................13
3.2. Surface solitons in semi-infinite partially ionized plasmas...............................................17
3.3. Coupled surface solitons in plasma slabs...........................................................................19

4. Exact global surface waves solutions......................................................................................21

4.1. Exact solutions in a cylindrical plasma...............................................................................23
4.2. Nonlinear spherical surface waves......................................................................................25
4.3. External electric fields in cylindrical plasmas.....................................................................28
4.4. External electric fields in spherical plasmas.......................................................................33

5. General kinetic theory..........................................................................................................37

5.1. General nonlinear equations................................................................................................37
5.2. Third-order plasma currents...............................................................................................41
5.3. Expressions for third-order responses for semi-infinite plasmas...................................43
5.4. Plasma slabs and plasmas bounded by thin dielectric slabs.............................................47

6. Modulational instability. Envelope solitons................................................................................50

6.1. Instability of monochromatic pump surface wave...............................................................50
6.2. Surface envelope solitons....................................................................................................54
6.3. Envelope solitons in plasma slabs and plasmas adjacent to thin dielectric slabs...........56
6.4. Coupled solitons in plasma slabs........................................................................................58

7. Conclusions........................................................................................................................59

8. Acknowledgements..............................................................................................................61
References............................................................................................................................61

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