We study models of Fei et al. and of Forinash et al of kinks in the sine-Gordon equation, and solitons in the nonlinear Schrodinger equation interacting with point defects. The models are two and three degree-of-freedom Hamiltonian systems. Using dynamical systems methods, we show that they exhibit interesting behaviors including transverse heteroclinic orbits to degenerate equilibria at infinity, chaotic dynamics and complex and delicate structures describing the interaction of traveling waves with the defect. We interpret the behavior in terms of invariant manifolds and phase space transport theory.