Joly, Jean-Luc; Rauch, Jeffrey Justification of multidimensional single phase semilinear geometric optics. (English) Zbl 0771.35010 Trans. Am. Math. Soc. 330, No. 2, 599-623 (1992). Summary: For semilinear strictly hyperbolic systems \(Lu=f(x,u)\), we construct and justify high frequency nonlinear asymptotic expansions of the form \[ u^ \varepsilon(x)\sim\sum_{j\geq 0} \varepsilon^ j U_ j(x,\varphi(x)/\varepsilon), \qquad Lu^ \varepsilon-f(x,u^ \varepsilon)\sim 0. \] The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature.We show (i) formal expansions with periodic profiles \(U_ j\) can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion. Cited in 19 Documents MSC: 35C20 Asymptotic expansions of solutions to PDEs 35L60 First-order nonlinear hyperbolic equations Keywords:semilinear oscillating waves; semilinear strictly hyperbolic systems; high frequency nonlinear asymptotic expansions; nonlinear geometric optics × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Yvonne Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires, J. Math. Pures Appl. (9) 48 (1969), 117 – 158 (French). · Zbl 0177.36404 [2] Ronald J. DiPerna and Andrew Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws, Comm. Math. Phys. 98 (1985), no. 3, 313 – 347. · Zbl 0582.35081 [3] W. E, Propagation of oscillations in the solutions of \( 1\)-\( d\) compressible fluid equations (preprint). 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