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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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