Resonant non-linear waves—IV. continuous and discontinuous solutions and an assessment of modal analyses

Abstract

[For the former parts see ibid. 14, 223-233 and 235-246 (1979; Zbl 0448.73037 and Zbl 0459.73017) and the summary above (Zbl 0561.73028).] Modal decomposition for the solution of partial differential equations is a quite commonly used procedure. For the class of problems of resonant systems, this procedure reduces the original partial differential equation to a system of ordinary differential equations describing an ensemble of coupled nonlinear oscillators. Again, a common approximation is the single mode analysis, which truncates the infinite series at its first term. This paper indicates that near resonances many nonlinear systems develop discontinuous solutions and discusses the range of applicability of the single mode analysis. The discussion is based on the construction of an exact solution to a nonlinear integro-differential equation derived by a method due to \textit{W. D. Collins} [Q. J. Mech. Appl. Math. 24, 129-153 (1971; Zbl 0214.354)] expanding on a procedure by \textit{J. B. Keller} and \textit{L. Ting} [Commun. Pure Appl. Math. 19, 371-420 (1966; Zbl 0284.35004)] and applied to various problems in the preceding articles by these authors.