Solitary Waves With Higher Order Nonlinear Dispersion and Stability of Compacton Solutions

We consider fifth order nonlinear dispersive K(m,n, p) type equations to study the effect of nonlinear dispersion. Using simple scaling arguments we show, how, instead of the conventional solitary waves like solitons, the interaction of the nonlinear dispersion with nonlinear convection generates compactons the compact solitary waves free of exponential tails. This interaction also generates many other solitary wave structures like cuspons, peakons, tipons etc. which are otherwise unattainable with linear dispersion. Various self similar solutions of these higher order nonlinear dispersive equations are also obtained using similarity transformations. Further, it is shown that, like the third order nonlinear K(m,n) equations, the fifth order nonlinear dispersive equations also have the same four conserved quantities and further even any arbitrary odd order nonlinear dispersive K(m,n, p...)