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

Abstract

We analyze the main properties of soliton solutions to the generalized KdV equation ut+[F(u)]x+uxxx=0, where the leading term F(u)∼quα, 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 Fu and α>1 (e.g., the Gardner equation) standing algebraic solitons with “heavy” power-law tails may appear. If the leading term of Fu 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 Fu represents a higher-order polynomial, the generalized KdV equation may support multi-humped pyramidal solitons.