RESONANT NON-LINEAR WAVES .4. CONTINUOUS AND DISCONTINUOUS SOLUTIONS AND AN ASSESSMENT OF MODAL ANALYSES

The authors investigated the single mode analysis of an acceptable approximation for the solution of partial differential equations for resonant systems and attempted to give a meaning to the multitude of solution branches arising from the solution of the algebraic equations of the modal decomposition. Assessment was made by comparing the exact solution of a non-linear integro-differential equation with its modal decomposition. The equation, obtained by perturbation analysis through extraction of secular terms at resonance, yielded identical algebraic equations as the usual singular perturbation based on modal analysis. Near resonances many non-linear systems develop discontinuous solutions. The range of applicability of single mode analysis is discussed, based on the construction of an exact solution to a non-linear integro-differential equation derived by W. D. Collins' method expanding on J. B. Keller procedure applied to various problems.