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Orbital stability of boundary states in the theory of long waves with additional pressure. (English. Russian original) Zbl 0770.35065
Sov. Phys., Dokl. 36, No. 11, 761-762 (1991); translation from Dokl. Akad. Nauk SSSR 321, No. 3, 505-508 (1991).
We formulate the results of a proof of the nonlinear stability of a family of solitons of the equation \[ u_ t+uu_ x+u_{xxx}- u_{xxxxx}=0, \tag{1} \] describing the propagation of long-wave-length, weakly nonlinear waves in media including the effects of an “additional” dispersion. In particular this equation describes wave processes, in which the effects of the “additional” dispersion guarantee the presence of a pressure caused by internal factors: waves on the surface of a liquid, with allowance for the surface tension or waves on the surface of a liquid under an elastic plate in an extended state. Similar equations are encountered when one describes long-wavelength magnetosound waves in a cold quasineutral plasma.
Despite the universality of eq. (1), the problem of the nonlinear stability of solitons so far has not been solved. The main difficulty involving such an investigation is the absence of an explicit analytical expression for the boundary states. To overcome this difficulty, we develope a special approach, using approximating operators. We also apply special numerical calculations with an improved accuracy to obtain the characteristics of these operators.
35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q51 Soliton equations