Stationary Solitons of the Fifth Order KdV-type Equations and their Stabilization

Abstract

Exact stationary soliton solutions of the fifth order KdV type equation $$ u_t +αu^p u_x +βu_{3x}+γu_{5x} = 0$$ 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 $p\geq 5$. 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.