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 + αu^pu_x + βu_{3 x} + γu_{5 x} = 0, are obtained for any p (> 0) in case αβ > 0, Dβ > 0, βγ < 0 (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case p ≥ 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 γ = 0 solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when D, α, β, γ are all > 0 and discuss its various properties.