The structure of algebraic solitons and compactons in the generalized Korteweg–de Vries equation

Physica D: Nonlinear Phenomena. 2021. Vol. 419. No. 5. Article 132785.

Pelinovsky E., Talipova T., Soomere T.

We analyze the main properties of soliton solutions to the generalized KdV equation u_t +[F (u)]_x+u_xxx
= 0, where the leading term F (u) ∼ qu^α, α > 0, q ∈ R. The far field of such solitons may have
three options. For q > 0 and α > 1 the analysis re-confirmed that all traveling solitons have
‘‘light’’ exponentially decaying tails and propagate to the right. If q < 0 and α < 1, the traveling
solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left.
For more complicated F (u) and α > 1 (e.g., the Gardner equation) standing algebraic solitons with
‘‘heavy’’ power-law tails may appear. If the leading term of F (u) is negative, the set of solutions may
include wide or table-top solitons (similar to the solutions of the Gardner equation), including algebraic
solitons and compactons with any of the three types of tails. The solutions usually have a single-hump
structure but if F (u) represents a higher-order polynomial, the generalized KdV equation may support
multi-humped pyramidal solitons.

Keywords: солитон, soliton, Korteweg-de Vries equation, обобщенное уравнение Кортевега-де Вриза

Publication based on the results of: Теория динамических систем и ее приложения (2021)