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

Abstract

Exact stationary soliton solutions of the fifth order KdV type equation $$u_t+\alpha u^p{u}_x+\beta u_{3x}+\gamma u_{5x}=0$$ are obtained for any p ($>0$) in case $\alpha \beta >0$, $D\beta >0$, $\beta \gamma <0$ (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case $p\ge 5$. 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.

8 pages, no figures

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• Remarks on existence/nonexistence of analytic solutions to higher order KdV equations
• Inverted solutions of KdV-type and Gardner equations