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Motion of soliton perturbations in the elastic bodies interacting with uid A. K. Abramyan, S. A. Vakulenko
 

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

  

Source: Abramian, Andrei K. - Institute for Problems in Mechanical Engineering, Russian Academy of Sciences

 

Collections: Engineering