Finite-amplitude cylindrical and spherical waves in weakly dispersive media

Nonlinear ring waves in a two-layer fluid

“…Weak nonlinearity and dispersion, acting together, yield the generation of a well-developed oscillatory dispersive wave train behind the lead wave of elevation (Q > 0) or depression (Q < 0), which is not captured by the 2D linear wave equation. The amplitude of the lead wave decreases with the increase of the distance from the centre much more rapidly than in the linear solution, which agrees with previous studies (e.g., [4,5]).…”

Solitary waves and their interactions in the cylindrical Korteweg-de Vries equation

“…As well-known, amplitudes of linear waves in cylindrical systems without dispersion vary as A ∼ r −1/2 , and linear waves in cylindrical systems with dispersion vary as A ∼ r −1 . All these amplitude dependencies for pulse-type initial perturbations were observed in experiments with electromagnetic waves in 2D lattices [6,38]. Similar results were obtained in the numerical study of radially spreading axisymmetric intrusions and solitary waves [28].…”

“…These are the laws of parameter variations in the nonlinear outgoing waves which were obtained in the papers cited above [23,38] and in many others (see, for example, Refs. [6,8,32,35]). Both the experimental and numerical data confirm the dependences (2.7) derived in the adiabatical approximation for cylindrical solitons (see, e.g., [8,35] and references therein).…”

Propagation of cylindrical waves in a rotating fluid

“…In application to the waves in nonrotating shallow water, this problem was intensively studied in the 1970s; analogous problems were also considered in plasma physics and in other ÿelds of physics. Summary of the results obtained can be found in the papers (Stepanyants, 1981;Dorfman et al, 1982). A brief resume can be formulated as follows.…”

“…When nonlinearity and dispersion are balanced, there arise cylindrical solitons with gradually changing amplitudes: A ∼ r −2=3 . The law according to which the amplitude of a cylindrical soliton is changed was a matter of active and continuous discussions and was ÿnally conÿrmed experimentally and analytically (see, for instance, Stepanyants, 1981;Dorfman et al, 1982, and the references cited therein). It was revealed (Druma, 1976;Calogero and Degasperis, 1978;Nakamura and Chen, 1981) that the cKdV equation belongs to the class of completely integrable equations for which the inverse scattering method can be applied.…”