We study the existence and stability of periodic travelling-wave solutions for generalized Benjamin-Bona-Mahony and Camassa-Holm equations. To prove orbital stability, we use the abstract results of Grillakis-Shatah-Strauss and the Floquet theory for periodic eigenvalue problems.
In this work a class of nondissipative curves in Hilbert spaces whose correlation functions have a limit as t -+ :t=oo is presented. These curves correspond to a class ~ of nondissipative basic operators that are a coupling of a dissipative operator and an antidissipative one. The wave operators and the scattering operator for the couple (A*, A) (A E ~R) are obtained. The present work is a continuation and a generalization of the investigations of K. Kirchev and V.Zolotarev [1,2,3] on the model representations of curves in Hilbert spaces where the respective semigroup generator is a dissipative operator. This article includes four parts. A new form of the triangular model of M.S.Liv~ic ([4, 5]) for the considered operators is introduced in the first part by the help of a suitable representation of the selfadjoint operator L. This allows us to describe the studied class of nondissipative curves. The second part studies some results concerning the application of the analogue for multiplicative integrals of the well-known Privalov's theorem ([6]) about the limit values in the scalar case. This analogue is a reconstruction of measure by limit values in Stieltjes-Perron's style and it is obtained by L. A.Sakhnovich ([7]). Another problem considered in the second part is the analogue in C m of the classical gamma-function and several properties for further consideration. In the third part the asymptotics of the studied curves corresponding to the nondissipative operators -couplings of a dissipative and an antidissipative operator with absolutely continuous real spectra and the limits of their correlation functions are obtained. In the fourth part a scattering theory of a couple (A*, A) with a nondissipative operator A from ~R is constructed as in the selfadjoint case ([8, 9, 10]) and in the dissipative case ([7]). These results show an interesting new effect: the studied nondissipative case is near to the dissipative one.
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