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Stationary solitons of the fifth order KdV-type. Equations and their stabilization. (English) Zbl 1037.35502
Summary: 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 $$p\geq5$$. 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.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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##### References:
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