Existence, Uniqueness, and Stability Criteria for Travelling Waves of
Forced Boussinesq-Type Equations (Nick Costanzino, Brown University
and UNC-Chapel Hill)
Abstract:
We consider the dispersive nonlinear wave equation utt
-(u - f(u) - ε2 uxx)xx =
g(u) for ε small. When f=u2 and the
forcing term g is identically zero there are several results
concerning the existence, stability and instability of travelling
waves. Clearly, for nonzero g the situation is different in that the
exact form of g may introduce an additional dissipation mechanism,
or conversely, amplify nonlinearities. Here we study the simplest
form for g, namely that g has a single simple zero. We give
conditions on the forcing term with guarantee the existence of
travelling waves. We use a geometric singular perturbation theory
approach in the construction of the travelling waves, and an Evans
function analysis for the linear stability and instability of these
solutions. This gives the advantage of being able to relate the
existence and stability of solutions to the equation with ε >
0 to that with ε = 0.
This is joint work with CKRT Jones (UNC -Chapel Hill).
This is an abstract of a poster to
be presented at the 2004 SEAMS Workshop in Charleston, SC. For more
information, visit the workshop's homepage at math.cofc.edu/SEAMS.