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

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) similar to qu(alpha), alpha > 0, q is an element of R. The far field of such solitons may have three options. For q > 0 and alpha > 1 the analysis re-confirmed that all traveling solitons have ``light'' exponentially decaying tails and propagate to the right. If q < 0 and alpha < 1, the traveling solitons (compactons) have a compact support (and thus vanishing tails) and propagate to the left. For more complicated F (u) and alpha > 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. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

The study analyzes the main properties of soliton solutions to the generalized KdV equation under different conditions, showing that solitons exhibit different behaviors depending on the values of q and alpha.