Stationary solitons of the fifth order KdV-type. Equations and their stabilization

Exact stationary soliton solutions of the fifth order KdV type equation, u(t) + alpha(u)u(x), + beta u(3x) + gamma u(5x) = 0, are obtained for any p (> 0) in case alpha beta > 0, oP > 0, Pr < 0 (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case p greater than or equal to 5. Various properties of these solutions are discussed. In particular, it is shown that for any p these solitons are lower and narrower than the corresponding gamma = 0 solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when D, alpha, beta, gamma are all > 0 and discuss its various properties.