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Emergence of champion solitons from two-solitary-wave interactions in the fourth-order generalized Korteweg–de Vries equation
Chaos, Solitons and Fractals. 2026. Vol. 208. No. 3. Article 118271.
Flamarion M., Pelinovsky E.
Two-solitary-wave interactions are investigated within the fourth-order generalized Korteweg–
de Vries equation. This equation is closely related to the classical Korteweg–de Vries equation
but includes a quartic nonlinear term. We show that, although collisions between two solitary
waves are not perfectly elastic, only a small amount of radiation is generated during the
interaction. This allows a clear characterization of the collision type based on the number of
local maxima observed during the interaction, following the Lax geometric categorization. Our
results indicate that, in contrast to several non-integrable systems such as the Schamel equation
and Whitham-type equations, the collision type depends solely on the ratio of the initial
solitary-wave amplitudes. Moreover, after the interaction, the larger solitary wave increases
its amplitude while the smaller one decreases. This behavior suggests that extreme, or freak
waves or champion solitons may arise from the interaction of multiple solitary waves
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Pyramidal solitons (solitons with more than two inflection points) of the generalized Korteweg–de Vries (KdV) equation are investigated. Necessary and sufficient conditions for the existence of such structures are presented. Within the framework of the generalized Gardner equation — the simplest model that admits pyramidal solitons as solutions — it is shown that these solutions ...
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