A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam

For a nonlinear beam equation with exponential nonlinearity, we prove existence of at least 36 travelling wave solutions for the specific wave speed c=1.3. This complements the result in [Smets, van den Berg, Homoclinic solutions for Swift–Hohenberg and suspension bridge type equations, J. Differential Equations 184 (2002) 78–96.] stating that for almost all there exists at least one solution. Our proof makes heavy use of computer assistance: starting from numerical approximations, we use a fixed point argument to prove existence of solutions “close to” the computed approximations.