Two-soliton solution of ion acoustic solitary waves in nonplanar geometry

Assumptions and ambiguities in nonplanar acoustic soliton theory

“…Almost all authors consider a range of negative times, typically, say, from s ¼ À30 to s ¼ À5 (although Maxon and Viecelli 13 go down as low as ksk ¼ 1:6), regardless of whether the stretching goes as n ¼ e 1=2 ðr À tÞ or as n ¼ Àe 1=2 ðr þ tÞ, or equivalent choices. Most of the earlier papers, [12][13][14][16][17][18][19][20][21][22][23][24][25]30,31,36,37,[40][41][42] but not they alone, seem to follow n ¼ Àe 1=2 ðr þ tÞ, whereas other papers [26][27][28][32][33][34]38,39,[43][44][45][46][47][48][49][50][51][52][54][55][56] follow n ¼ e 1=2 ðr À tÞ, if one sets the linear acoustic speeds equal to unity, where necessary. Exceptions to the general rule, who use a positive s range, are Yasmin et al 35 with n ¼ Àe 1=2 ðr þ tÞ and Ghosh et al, 29,53 based on n ¼ e 1=2 ðr À tÞ.…”

“…We well know that a number of authors have used this assumption in the literature, but unfortunately that does not make it generally correct, at least not without keeping several limitations in mind. Several authors [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] introduce the stretching (4) with a reference to papers by Maxon and Viecelli, 12,13 without mentioning any of the restrictions about its applicability, whereas more often than not other authors do not even advance any motivation whatsoever for its use, as if this were completely selfevident. Because of the fairly large literature on nonplanar waves, we have cited here only a representative selection of those papers addressing electrostatic modes in unmagnetized plasmas, where the description involves only one spatial coordinate, r, without angular dependence.…”

“…However, one finds in the literature other plasma models, which then either lead to an equation like (23), but with more complicated coefficients, [12][13][14][15][16][17][18][19][20][21]24,25,30,32,33,36,[38][39][40][41][42]44,46,49,54 or to modifications of related evolution equations like the Burgers, 23,31,35 KdV-Burgers, 22,37,45,47,48 Gardner, [26][27][28][29]34,43,44,[50][51][52][53]55,56 or Schamel 23 equations. The standard forms of these equations have been well studied in the literature over many decades.…”

“…22,23,28,29,31,35,37,41,42,45,46,53 Another point of criticism is that, beginning again, already with Maxon and Viecelli, 12-14 many authors [16][17][18][19][20][21]30,33,35,36,[38][39][40][41][46][47][48][49]54 present graphs where the initial KdV soliton amplitudes are far too large to be admissible in an iterative scheme like the reductive perturbation analysis, which neglects higher-order nonlinearities and hence requires variables like u 1 to remain significantly smaller than unity in normalized form. Some authors 24,32,37,42,45,55 start with reasonable values for the undisturbed solution of the planar case, but let these evolve to values of ksk that are too small, so that the amplitudes then exceed allowable limits.…”