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Volume 2,     Number 2,     Spring 1994

 

SOLVABILITY AND NONLINEAR GEOMETRICAL
OPTICS FOR SYSTEMS OF CONSERVATION LAWS
HAVING SPATIALLY DEPENDENT FLUX FUNCTIONS
YUANPING HE AND T. BRYANT MOODIE

Abstract. In a recent series of articles [4-7] the present authors developed a relatively complete theory of nonlinear geometrical optics for signaling problems associated with hyperbolic systems of conservation laws of the form ut + f(u, x)x = 0. As well as allowing for explicit spatial dependence of the flux function, f = f(u, x), the associated characteristic field was permitted to have a local linear degeneracy of arbitrary order. We constructed our asymptotic representation for the nonlinear hyperbolic wave by directly employing a nonlinear phase in a perturbation scheme involving the small amplitude ε. In the process of carrying out this iterative procedure we either implicitly or explicitly employed the fact that for each O(εk) problem, an associated algebraic system was solvable. This is equivalent to assuming a set of solvability conditions. We are now in a position to prove that these solvability conditions are indeed satisfied, thereby establishing the validity of our perturbation procedure.

 

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