Stability of compacton solutions of fifth-order nonlinear dispersive equations

We consider fifth-order nonlinear dispersive K(m, n, p) type equations to s tudy the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves such as solitons, the interaction of the nonlinear dispersion with nonlinear convection generate s compactons-the compact solitary waves free of exponential tails. This int eraction also generates many other solitary wave structures such as cuspons , peakons, tipons etc which are otherwise unattainable with linear dispersi on. Various self-similar solutions of these higher-order nonlinear dispersi ve equations are also obtained using similarity transformations. Further, i t is shown that, like the third-order nonlinear K(m, n) equations, the fift h-order nonlinear dispersive equations also have the same four conserved qu antities and, furthermore even any arbitrary odd-order nonlinear dispersive K(m, n, p, ...) type equations also have the same three (and most likely t he four) conserved quantities. Finally, the stability of the compacton solu tions for the fifth-order nonlinear dispersive equations are studied using linear stability analysis. From the results of the linear stability analysi s it follows that, unlike solitons, all the allowed compacton solutions are stable, since the stability conditions are satisfied for arbitrary values of the nonlinear parameters.