Nonlinear equations and wavelets

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Abstract

We use a multi-scale similarity analysis which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations, whose exact solutions are generally difficult to obtain.

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  • A repository of equations with cosine/sine compactons

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  • Nonlinear dispersion relations

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    Citation Excerpt :

    Often in these settings, it is of particular interest to examine the dynamics of localized in space, and possibly traveling in time solutions, which can represent bits of information, Bose-Einstein condensates, flame fronts, elementary particles or water waves [16]. When we linearize these equations around the solutions of interest (to examine their stability and the elementary excitations around them), one tool that is very customary is the use of the (linear) dispersion relations that relate the wave-number and frequency of propagating excitations [10]. It is well-known that for nonlinear models, exact solutions satisfy nonlinear dispersion relations(NLDR) that connect their speed to their amplitude and width [3].

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