A test of asymptotic integrability of \((1+1)\) wave equations (Q2734879)

The author propose a far-going generalization of the Poincaré perturbation scheme for the wave equations \(Lu=G(u)\), where \(L\) is a linear, first-order in time differential operator with constant coefficients, and \(G(u)\) an analytic function of dependent variable \(u=u(x,t)\) and its \(x\)-derivatives. They indroduce the expansions \(u=\sum^\infty_{-\infty} u^{(\alpha)} E^{\alpha)}\) with plane-waves \(E^{(\alpha)}\) and coefficients \(u^{(\alpha)}\) depending on the slow variables \(\xi=\varepsilon x\) and time variables \(t_n(n=1, 2,\dots)\). Unfortunately, the exposition is rather brief and the main concept of the asymptotic integrability cannot be easily understood for the lack of particular examples. The authors refer to a forthcoming paper.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00047].