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Justification of multidimensional single phase semilinear geometric optics


Authors: Jean-Luc Joly and Jeffrey Rauch
Journal: Trans. Amer. Math. Soc. 330 (1992), 599-623
MSC: Primary 35C20; Secondary 35L40, 35L60, 78A05
DOI: https://doi.org/10.1090/S0002-9947-1992-1073774-7
MathSciNet review: 1073774
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Abstract: For semilinear strictly hyperbolic systems $ Lu= f(x,u)$, we construct and justify high frequency nonlinear asymptotic expansions of the form

$\displaystyle {u^\varepsilon }(x)\sim\sum\limits_{j\, \geq \,0} {{\varepsilon ^... ...phi \,(x)/\varepsilon}, \quad L{u^\varepsilon } - f(x,{u^\varepsilon })\sim 0 .$

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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1073774-7
Keywords: Semilinear geometric optics, semilinear oscillating waves, high frequency asymptotic expansions
Article copyright: © Copyright 1992 American Mathematical Society

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