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

In this paper, Lax pairs are constructed for two fifth-order nonlinear evol ution equations of "Boussinesq"-type which govern wave propagation in two o pposite directions. One of the equations is related to the well-known Sawad a-Kotera (SK) equation and, through its bilinear form, is identified with t he Ramani equation. The second equation-about which very little seems to be known-may be considered a bidirectional version of the Kaup-Kupershmidt (K K) equation and is the main focus of this study. The "anomalous" solitary w ave of this latter equation is derived and is found to possess the remarkab le 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 a n appropriate undirectional (long wave) limit, it is shown that neither the Ramani, nor the bidirectional Kaup-Kupershmidt (bKK) equation can be class ified as truly "Boussinesq" in character (a distinction that is made precis e 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 surp risingly, they exhibit the same lacunary behavior as their unidirectional c ounterparts. In conclusion, a canonical interpretation of the N-soliton sol ution of the bKK equation is proposed which provides a basis for constructi ng these anomalous solitons in a future work. (C) 2001 American Institute o f Physics.