In the present work we study the existence and stability of multi-pulses in dynamical systems that arise as traveling-wave equations for a partial differential equation (PDE) with symmetries. The motivation comes from two different models that describe the propagation of pulses in optical fibers.
In the first part of the thesis we consider reversible, ℤ2 symmetric dynamical systems with heteroclinic orbits related via symmetries. The heteroclinic orbits are assumed to undergo an orbit flip bifurcation upon changing appropriate parameters. We construct multi-bump solutions close to the heteroclinic orbits and investigate their PDE stability by using Lin's method and Lyapunov-Schmidt reduction. We apply this abstract theory to a model equation that describes the propagation of pulses in optical fibers with phase sensitive amplifiers. Our results show that stable multi-pulses exist. In the second part we consider parameter-dependent dynamical systems with reflection and SO(2) symmetry, which possess a homoclinic solution to a saddle focus. The reflection symmetry is broken by the second parameter which plays the role of the wave speed. We derive the bifurcation equations for the existence of N-pulse solutions and solve them for N=3. As a result we obtain standing and traveling 3-pulse solutions which we describe through the phase differences and the distances between consecutive bumps. We also investigate stability of these 3-pulses. We derive the stability matrix for multi-bump solutions and compute to the leading order the location of the eigenvalues for the 3-pulses.