Traveling waves in nonlinearly supported beams and plates

Date of Completion

January 2001






The work on traveling wave solutions of the nonlinear beam equation and their stability and interaction properties is continued. The equation is a partial differential equation of fourth order in space and second order in time. While this equation has been previously studied in one space dimension, this is the first time a multidimensional case is considered. A proof of existence of traveling wave solutions in two and three space dimensions for a certain class of nonlinearities is presented using mountain pass theory. A description of the Mountain Pass Algorithm is given. The algorithm is then employed to obtain the traveling wave solutions numerically in two space dimensions. A single and double-pulse waves are found for a range of values of the wave speed. A parallel version of an explicit finite difference scheme is used to study stability properties of the waves. Fission of the double-pulse wave into two single-pulse waves is observed. When two single-pulse waves travel in opposite directions and collide, they emerge from the collision almost unchanged. ^