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Summary: Motion of soliton perturbations in the elastic bodies interacting with
uid
A. K. Abramyan, S. A. Vakulenko
1 Introduction. Statement of problem. We consider a simplest model describing a soliton moving
in an elastic nonlinear system interacting with a
uid. Such problems are important for ...
Our model, given by the following equations (1.1)-(1.3), describes a simple system, consisting of a
elastic nonlinear beam contacting with a
uid. (This beam is a bottom and the
uid occupies a layer
under it). The mathematical model is as follows
Dw xxxx +Kw w xx +mw tt w 3 = P+ (x; t); (1:1)
where w = w(x; t); x 2 (1;1); t 0 is a de
ection of the elastic beam and
g y + tt j y=h = 0; (1:2)
y j y=0 = w t ; (1:3)
c 2 tt = 0 (1:4)
where (x; y; t) is a potential of velocity of the
uid, y 2 [0; h]; x 2 (1;1); t 0. Equations (1.2) and
(1.3) are boundary conditions, respectively on the free surface y = h and on the bottom, where one has
the contact between the beam and the
uid. Such system can give, for example, a simplest model of an
ocean with an elastic nonlinear bottom. The term P+ = t is the pressure of the
uid on the beam and
give a connection between the elastic and
uid components.
The detail mathematical investigation of this model is not quite elementary. To simplify, the statement
will be proceed into some steps. Our main hypothesis is that P+ (x; t) is small. Such idea allows us to use
the theory of perturbations. Our main aim is to describe a connection between the soliton and surface
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