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Solitons from sine waves: Analytical and numerical methods for non- integrable solitary and cnoidal waves. (English) Zbl 0611.35080

The ”FKDV” equation, \(u_ t+uu_ x-u_{xxxxx}=0\), is used as a testbed for a variety of analytical and numerical methods that can be applied to solitary waves and cnoidal waves of ”non-integrable” differential equations, that is to say, to equations which cannot be solved by the inverse scattering transform. The basic tools are (i) Padé approximants formed from power series in the amplitude; (ii) a Newton- Kantorovich/pseudospectral Fourier/continuation numerical method; (iii) singular perturbation theory for two interacting solitons of almost identical phase speed; (iv) bifurcation and branch-switching methods; (v) the imbricate-soliton series. A number of new results for the FKDV equation are obtained including extensive numerical calculations of the spatially periodic solutions with one peak (”cnoidal wave”) and two peaks (”bicnoidal wave”) per period, an analytical expression for the double- peaked soliton (”bion”), calculation of both the limit and bifurcation points for the bicnoidal wave, and finally the computation of accurate analytical approximations to the cnoidal wave for all amplitudes. More important, all of these analytical and numerical tools are highly effective for this equation in spite of the fact that it cannot be solved by the inverse scattering transform. Work now in progress will apply these methods to non-integrable equations in two space dimensions.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
76B25 Solitary waves for incompressible inviscid fluids
35B10 Periodic solutions to PDEs
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