2,386 research outputs found
Solitary Wave and Shock Wave Solutions of the Variants of Boussinesq Equations
This paper obtains the solitary wave as well as the shock wave solutions of the variants of the Boussinesq equations in both (1+1) and (1+2) dimensions. The domain restrictions are also identified in the process
Finite volume schemes for dispersive wave propagation and runup
Finite volume schemes are commonly used to construct approximate solutions to
conservation laws. In this study we extend the framework of the finite volume
methods to dispersive water wave models, in particular to Boussinesq type
systems. We focus mainly on the application of the method to bidirectional
nonlinear, dispersive wave propagation in one space dimension. Special emphasis
is given to important nonlinear phenomena such as solitary waves interactions,
dispersive shock wave formation and the runup of breaking and non-breaking long
waves.Comment: 41 pafes, 20 figures. Other author's papers can be downloaded at
http://www.lama.univ-savoie.fr/~dutykh
Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation
The stability properties of line solitary wave solutions of the
(2+1)-dimensional Boussinesq equation with respect to transverse perturbations
and their consequences are considered. A geometric condition arising from a
multi-symplectic formulation of this equation gives an explicit relation
between the parameters for transverse instability when the transverse
wavenumber is small. The Evans function is then computed explicitly, giving the
eigenvalues for transverse instability for all transverse wavenumbers. To
determine the nonlinear and long time implications of transverse instability,
numerical simulations are performed using pseudospectral discretization. The
numerics confirm the analytic results, and in all cases studied, transverse
instability leads to collapse.Comment: 16 pages, 8 figures; submitted to Phys. Rev.
On Boussinesq-type models for long longitudinal waves in elastic rods
In this paper we revisit the derivations of model equations describing long
nonlinear longitudinal bulk strain waves in elastic rods within the scope of
the Murnaghan model in order to derive a Boussinesq-type model, and extend
these derivations to include axially symmetric loading on the lateral boundary
surface, and longitudinal pre-stretch. We systematically derive two forced
Boussinesq-type models from the full equations of motion and non-zero surface
boundary conditions, utilising the presence of two small parameters
characterising the smallness of the wave amplitude and the long wavelength
compared to the radius of the waveguide. We compare the basic dynamical
properties of both models (linear dispersion curves and solitary wave
solutions). We also briefly describe the laboratory experiments on generation
of bulk strain solitary waves in the Ioffe Institute, and suggest that this
generation process can be modelled using the derived equations.Comment: 19 pages, 5 figures, submitted to the Special Issue of Wave Motion,
"Nonlinear Waves in Solids", in Memory of Professor Alexander M. Samsono
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