Coalescence of Ripplons, Breathers, Dromions and Dark Solitons

Abstract

New solutions of several nonlinear evolution equations (NEEs) are obtained by a special limit corresponding to a coalescence or merger of wavenumbers. This technique will yield the multiple pole solutions of NEEs if ordinary solitons are involved. This limiting process will now be applied through the Hirota bilinear transform to other novel solutions of NEEs. For ripplons (self similar explode-decay solutions) such merger yields interacting self similar solitary waves. For breathers (pulsating waves) this coalescence gives rise to a pair of counterpropagating breathers. For dromions (exponentially decaying solutions in all spatial directions) this merger might generate additional localized structures. For dark solitons such coalescence can lead to a pair of anti-dark (localized elevation solitary waves on a continuous wave background) and dark solitons.KEYWORDS: nonlinear evolution equations, Hirota bilinear method, breathers, dromionsof (1.1) is obtained. This agrees with results obtained by the inverse scattering transform. 2) Computer plot of (1.2) (Fig. 1) shows a pair of counterpropagating soliton, anti-soliton in a frame moving to the right with speed m 2 . There is an important difference between (1.2) and the ordinary multi-soliton of mKdV. Distance of separation of ordinary solitons of mKdV after interaction will be 666 proportional to t (time). However, the distance between the soliton and anti-soliton in (1.2) goes like log t. The hnique of coalescence of wavenumbers will now be applied to ripplons, breathers, dromions and dark solitons in the following sections. tec §1. IntroductionNonlinear evolution equations (NEEs) and localized solitons have been studied intensively recently. Novel and sophisticated solutions for NEEs have been obtained by a variety of ingenious techniques. Examples treated in the present work include: (a) ripplons are self similar explode-decay type solutions, 1) (b) breathers are pulsating waves which can be generated from multi-soliton solutions by employing complex conjugate wavenumbers, 2) (c) dromions are localized solutions of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations decaying exponentially in all directions, 3) and (d) dark solitons are localized minima in intensity propagating on a continuous wave background. 4) For most soliton equations double or multiple poles solutions arise as special cases of multi-soliton expressions. In the language of the inverse scattering transform this occurs as a result of the merger of simple poles of the reflection coefficient. 2, 5) Alternatively the same result can be obtained by the 'coalescence' of wavenumbers in the Hirota bilinear solutions of the multi-soliton expressions. 6) The main goal of this paper is to apply this technique of 'coalescence' of wavenumbers in the Hirota formulation of ripplons, breathers, dromions, and dark solitons. New solutions of several famous and widely applicable evolution equations, as well as some less well known cases, are generated. The validity of these new solutions is estab...

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