The next step in investigating soliton behaviour

In the previous chapters, we introduced a wide variety of solitons. We started with onedimensional ones, depending on x - vt, or else k · x - ωt when embedded in a higher dimensional space. Next, we saw how two or more such entities could combine to produce X shaped solitons, still behaving like their one-dimensional components far from the intersections. Finally, in Chapter 9, we looked at fully two- and three-dimensional compact entities produced both experimentally and theoretically from model equations. Stability was considered in some detail, especially in Chapter 8. When so doing, we performed small perturbation analyses, linearizing around the soliton structure (often treated as a limit of a nonlinear wavetrain). The reader should be warned against being seduced into thinking that linearization tells the whole story. Linear instability, leading to exponential growth, cannot persist. Some possible subsequent scenarios were outlined in Chapter 8. More will come in Chapter 11. Here we will focus on still another fate, that of soliton metamorphisis. If, say, both one- and two-dimensional solitons are known to exist in a given medium and, furthermore, the one-dimensional version is linearly unstable, might it not break up into an array of two-dimensional solitons? Could not two-dimensional solitons likewise produce three-dimensional decay products after a while? Can decay proceed directly from ID to 3D? In a series of papers, it was shown that one-dimensional Zakharov–Kuznietsov solitons, discussed in Chapter 8, can indeed break up into two-dimensional, cylindrical entities, functions of (x - ωt)2 + y2.