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Eleonskij V. M., Korolev V. G., Kulagin N. E. On the dynamic system generated by Whitham's equation with oscillating kernel. Izvestiya VUZ. Applied Nonlinear Dynamics, 1993, vol. 1, iss. 3, pp. 72-86.

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Russian
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Article
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53:51

On the dynamic system generated by Whitham's equation with oscillating kernel

Autors: 
Eleonskij Vladimir Markovich, P.L. Kapitza Institute for Physical Problems of Russian Academy of Sciences
Korolev Vadim Germanovich, P.L. Kapitza Institute for Physical Problems of Russian Academy of Sciences
Kulagin Nikolaj Evgenevich, P.L. Kapitza Institute for Physical Problems of Russian Academy of Sciences
Abstract: 

The Whitham's equation possessing the fast—vanishing oscillating kernel is investigated. A possibility of the reducing of this equation is shown either to the differential equation of the fourth rate or to the nonintegrable dynamic system with two degrees of freedom. which permits an existence of singular point of «saddle—focus» type. The series of homoclinic loops of that point are found numerically. These loops are images of self-localized oscillating solutions of the Whitham's equation. The curves of points on the parametre plane are found, where the system permits an existence of «sharpened» solutions. These solutions are investigated analytically.

Key words: 
Acknowledgments: 
We thank L.P. Shilnikov, who drew our attention to this problem and stimulated the study of loops with peaks in problems with singular points of the saddle-focus type.
Reference: 
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Received: 
03.08.1992
Accepted: 
18.12.1992
Published: 
12.01.1994