8 research outputs found
Symmetries of a class of nonlinear fourth order partial differential equations
In this paper we study symmetry reductions of a class of nonlinear fourth
order partial differential equations \be u_{tt} = \left(\kappa u + \gamma
u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2,
\ee where , , , and are constants. This
equation may be thought of as a fourth order analogue of a generalization of
the Camassa-Holm equation, about which there has been considerable recent
interest. Further equation (1) is a ``Boussinesq-type'' equation which arises
as a model of vibrations of an anharmonic mass-spring chain and admits both
``compacton'' and conventional solitons. A catalogue of symmetry reductions for
equation (1) is obtained using the classical Lie method and the nonclassical
method due to Bluman and Cole. In particular we obtain several reductions using
the nonclassical method which are no} obtainable through the classical method
Extreme values of elastic strain and energy in sine-Gordon multi-kink collisions
In our recent study the maximal values of kinetic and potential energy
densities that can be achieved in the collisions of slow kinks in the
sine-Gordon model were calculated analytically (for , and 3) and
numerically (for ). However, for many physical applications it is
important to know not only the total potential energy density but also its two
components (the on-site potential energy density and the elastic strain energy
density) as well as the extreme values of the elastic strain, tensile
(positive) and compressive (negative). In the present study we give (i) the two
components of the potential energy density and (ii) the extreme values of
elastic strain. Our results suggest that in multi-soliton collisions the main
contribution to the potential energy density comes from the elastic strain, but
not from the on-site potential. It is also found that tensile strain is usually
larger than compressive strain in the core of multi-soliton collision.Comment: 9 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1605.0976