Solitary waves generated by the interaction of oscillating nonlinear waves
The paper deals with rigorous weakly nonlinear geometrical optics. It is devoted to the precise description of the solutions of 1D nonlinear hyperbolic systems with high frequency oscillating initial data and brings to the fore a new phenomenon: the creation of solitary waves by nonlinear interaction of oscillating waves. It is a resonance phenomenon called ``weak resonance'' which corresponds to a stationary property of a linear combination of the phases brought in play by the initial data oscillations. Such weak resonances can be the cause of singular behaviors of the solution like cahotic or explosive behaviors. Several detailed examples are given. This work is based on a rigorous analysis of the fluctuations of the solution and their nonlinear evolution, and uses techniques introduced by \textit{J. L. Joly}, \textit{G. Métivier} and \textit{J. Rauch} [J. Funct. Anal. 114, No. 1, 106-231 (1993; Zbl 0851.35023)]. The results concern local in time classical solutions and apply before the shock waves formation in the quasilinear case and the \(L^\infty\) explosion in the semilinear one.
scientific article; zbMATH DE number 891040