Assumptions and ambiguities in nonplanar acoustic soliton theory

There have been many recent theoretical investigations of the nonlinear evolution of electrostatic modes with cylindrical or spherical symmetry. Through a reductive perturbation analysis based on a quasiplanar stretching, a modified form of the Korteweg-de Vries or related equation is derived, containing an additional term which is linear in the electrostatic potential and singular at time t = 0. Unfortunately, these analyses contain several restrictive assumptions and ambiguities which are normally neither properly explained nor discussed, and severely limit the applicability of the technique. Most glaring are the use of plane-wave stretchings, the assumption that shape-preserving cylindrical modes can exist and that, although time is homogeneous, the origin of time (which can be chosen arbitrarily) needs to be avoided. Hence, only in the domain where the nonlinear modes are quasiplanar, far from the axis of cylindrical or from the origin of spherical symmetry can acceptable but unexciting results be obtained. Nonplanar nonlinear modes are clearly an interesting topic of research, as some of these phenomena have been observed in experiments. However, it is argued that a proper study of such modes needs numerical simulations rather than ill-suited analytical approximations. (C) 2014 AIP Publishing LLC.