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Generation of large-amplitude solitons in the extended Korteweg-de Vries equation. (English) Zbl 1080.35532

Summary: We study the extended Korteweg-de Vries equation, that is, the usual Korteweg-de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg-de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude “table-top” solitons as well as small-amplitude solitons similar to the solitons of the Korteweg-de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a “sech”-profile.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
76B25 Solitary waves for incompressible inviscid fluids
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