On bidirectional fifth-order nonlinear evolution equations, Lax pairs, and directionally dependent solitary waves

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Abstract
In this paper, Lax pairs are constructed for two fifth-order nonlinear evolution equations of “Boussinesq”-type which govern wave propagation in two opposite directions. One of the equations is related to the well-known Sawada–Kotera (SK) equation and, through its bilinear form, is identified with the Ramani equation. The second equation—about which very little seems to be known—may be considered a bidirectional version of the Kaup–Kupershmidt (KK) equation and is the main focus of this study. The “anomalous” solitary wave of this latter equation is derived and is found to possess the remarkable property that its profile depends on the direction of propagation. This type of directional dependence would appear to be quite unusual and, to our knowledge, has not been reported in the literature before now. By taking an appropriate undirectional (long wave) limit, it is shown that neither the Ramani, nor the bidirectional Kaup–Kupershmidt (bKK) equation can be classified as truly “Boussinesq” in character (a distinction that is made precise in the study). Recursion formulas are given for generating an infinity of conserved densities for both equations. These are used to obtain the first few conservation laws of the bKK and Ramani equations explicitly; not surprisingly, they exhibit the same lacunary behavior as their unidirectional counterparts. In conclusion, a canonical interpretation of the N-soliton solution of the bKK equation is proposed which provides a basis for constructing these anomalous solitons in a future work.

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