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 [47] 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 u_{t} +
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(ε^{k}) 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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