Coherent nonlinear waves and the Wiener algebra
We study oscillatory solutions of semilinear first order symmetric hyperbolic system \(Lu= f(t,x,u, \overline{u})\), with real analytic \(f\). The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in \(T\), \(X\) with only the natural hypothesis of coherence. In the special case where \(L\) has constant coefficients and the phases are linear, the solutions have asymptotic description \[ u^ \varepsilon= U(t,x,t/\varepsilon, x/\varepsilon)+ o(1) \] where the profile \(U(t,x,T,X)\) is almost periodic in \(T\), \(X\). The main novelty in the analysis is the space of profiles which have the form \[ U= \sum_{\tau,\omega\in \mathbb{R}^{1+d}} U_{\tau,\omega}(t,x) e^{i(\tau T+\omega X)}, \qquad \sum \| U_{\tau,\omega} (t,x)\|_{C([0, \underline{t}]: H^ s(\mathbb{R}^ d))} <\infty. \] Thus, \(U\) is an element of the Wiener algebra as a function of the fast variables. The profile \(U\) is uniquely determined from the initial data of \(u^ \varepsilon\) by profile equations of standard form. An application to conical refraction where the characteristics have variable multiplicity is presented.
semilinear first order symmetric hyperbolic system
scientific article; zbMATH DE number 566968
scientific article