The nonlinear beam equation$$u\sb{tt} + u\sb{xxxx} + f(u) = 0$$on the real line was studied in this dissertation. We proved the existence of the traveling wave solutions for both $f(u)$ = $u\sp+$ $-$ 1 and $f(u)$ = $u\sp+$ $-$ 1 + $g(u)$. We showed that these solutions can be obtained as saddle points in a variational formulation.
The numerical solutions can be found by the Mountain Pass algorithm, and be analyzed by the central finite difference scheme. During the numerical experiments, these traveling wave solutions appear to be extremely stable and behave like "multitons", that is, these traveling waves will emerge almost intact after interacting nonlinearly with each other.
My Research