Initial conditions for the cylindrical Korteweg‐de Vries equation

Abstract

In this paper, we find suitable initial conditions for the cylindrical Korteweg-de Vries equation by first solving exactly the initial-value problem for localized solutions of the underlying axisymmetric linear long-wave equation. The far-field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg-de Vries equation. This initial condition is associated only with the leading wave front of the farfield limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the "mass" conservation law for the cylindrical Korteweg-de Vries equation. The outcome is that in the linear initial-value problem all the mass is carried behind the wave front, and then the "mass" in the initial condition for the cylindrical Korteweg-de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg-de Vries equation has zero "mass." This situation arises because, unlike the well-known unidirectional Korteweg-de Vries equation, the solution of the initial-value problem for the axisymmetric linear long-wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated. K E Y W O R D S cylindrical korteweg-de vries equation, axisymmetric linear wave equation, korteweg-de vries solitons 176

Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

“…Alternative analytical and numerical approaches to such problems, and important experimental work, have been developed, in particular, for surface waves, by Rabaud & Moisy (2013), Darmon, Benzaquen & Raphaël (2014), Ellingsen (2014a,b), Svirkunov & Kalashnik (2014), Arkhipov, Khabakhpashev & Zakharov (2015), Akselsen & Ellingsen (2019), Li & Ellingsen (2019) and Smeltzer, Esoy & Ellingsen (2019), and for internal waves, by Vlasenko et al (2009), Stashchuk & Vlasenko (2009), Arkhipov, Safarova & Khabakhpashev (2014), Grue (2015), Bulatov & Vladimirov (2015) and Bulatov & Vladimirov (2020) (see also the references therein). General approaches to the solution of initial-value problems with the help of cylindrical KdV-type models have been discussed by Weidman & Zakhem (1988), Ramirez, Renouard & Stepanyants (2002), McMilan & Sutherland (2010), Khusnutdinova & Zhang (2016b) and Grimshaw (2019).…”

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

“…In 1959 Iordansky derived the cylindrical version of the Korteweg-de Vries (cKdV) equation [15] for surface waves in a fluid. A similar equation was later derived for water and plasma waves by various authors [12,25,27,29,33,40,41]. Currently, the cylindrical KdV equation is one of the basic equations of contemporary mathematical physics.…”

Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

“…Alternative approaches to such problems, and important experimental work, have been developed, in particular, for surface waves, by Ellingsen (2014a,b); Svirkunov & Kalashnik (2014); Arkhipov, Khbakhpashev & Zakharov (2015); Akselsen & Ellingsen (2019); Li & Ellingsen (2019); Smelser, Esoy & Ellingsen (2019), and for internal waves, by Arkhipov, Safarova & Khabakhpashev (2014); Bulatov & Vladimirov (2015, 2020 (see also the references therein). General approaches to the solution of initial-value problems with the help of cylindrical Korteweg-de Vries -type models have been discussed by Weidman & Zakhem (1988); Ramirez, Renouard & Stepanyants (2002); McMillan & Sutherland (2010); Khusnutdinova & Zhang (2016b); Grimshaw (2019).…”