Exponential Asymptotics for Partial Differential Equations
The principles are discussed by which asymptotics beyond all orders can be carried to genuine partial differential equations for which the variables cannot be separated. Access to solutions then depends on the canonical equations of the Hamilto-nian operators, and their phase-space trajectories involve complex geometrical optics and branching. To handle the resulting Stokes phenomena reliably in several dimensions requires a connection method resolving them strictly locally at the branch points of individual trajectories. For linear problems, at least, solution approximations can then be formulated which are concrete and exponentially reliable, even if grossly nonuniform, and can be used to predict transcendentally small functionals of the operator that can be observed directly. The general theory is illustrated by examples of spectral degeneracy.
KeywordsBranch Point Geometrical Optic Periodic Potential Invariant Torus Canonical Equation
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