Integrals and differential equations

On adapted coordinate systems

“…V. I. Arnol'd conjectured in [1] that the asymptotic behavior of the oscillatory integral I(λ) is determined by the Newton polyhedron associated to the phase function f in a so-called "adapted" coordinate system. For some special cases this conjecture was then indeed verified by means of Arnol'd's classification of singularities (see [2]). Later, however, A. N. Varchenko [8] disproved Arnol'd's conjecture in dimensions three and higher.…”

“…Now, either the new coordinates y are adapted, in which case we are finished, or we can apply the same procedure tof. Composing the change of coordinates from the first step with the one from the second step, we see that we then can find a change of coordinates x = ϕ (2) (y) of the form ∈ R, such that the following holds: If the function f (2) (2) expresses the function f in the new coordinates, then the principal face of the Newton polyhedron of f (2) is either associated to a cluster of roots which corresponds to a cluster of roots β β 2 · λ λ 2 λ 3 in the original coordinates or is a horizontal, unbounded edge (so that the new coordinates are adapted).…”

“…But the point (Ã λ ,B λ ) will be contained in the Newton diagrams of f associated to all subsequent systems of coordinates that we constructed by our algorithm (compare (4.14), applied to the coordinates y), which means that the principal face of f in our final, adapted coordinate system must lie in the half-space t 2 Proof. Indeed, the algorithm that we devised in the proof of Theorem 4.2 in order to construct an adapted coordinate system shows that we can arrive at an adapted coordinate system with a polynomial function ψ, unless we have to choose for ψ one of the roots r with infinitely many non-trivial terms in its Puiseux series expansion.…”

“…Moreover, we have κ 2 Indeed, recall that the effect of our change of coordinates ϕ on the Newton polyhedron is such that it preserves all lines κ 1 t 1 + κ 2 t 2 = c. Let us therefore choose m ∈ N so big that we have j + k ≤ m for every point (j, k) lying on any such line passing through any point in the Newton diagram N d (f ) off, and denote by F the Taylor polynomial of order m of f. Then it is clear that f and F have the same principal faces and parts, and the same applies tof andF := F • ϕ. I.e., we have f p = F p andf p =F p .…”

An Eulerian Method for Capturing Caustics

“…Section 5 presents 2-D and 3-D numerical caustic capturing for a problem modeling laser beam propagation in a plasma. The Lagrangian method then consists in solving the Hamilton system formed by the following set of ordinary differential equations (ODEs) [2,27]:…”