<<

. 21
( 24 .)



>>

142. , Double wells, Commun. Math. Phys., 75 (1980), pp. 239“261. Expo-
nentially small splitting of eigenvalues.
143. F. H. Hildebrand, Introduction to Numerical Analysis, Dover, New York,
1974. Numerical; asymptotic-but-divergent series in h for errors.
144. D. B. Hinton and J. K. Shaw, Absolutely continuous spectra of 2d order
di¬erential operators with short and long range potentials, Quart. J. Math.,
36 (1985), pp. 183“213. Exponential smallness in eigenvalues.
145. P. Holmes, J. Marsden, and J. Scheurle, Exponentially small splitting
of separatrices with applications to KAM theory and degenerate bifurcations,
Comtemporary Mathematics, 81 (1988), pp. 214“244.
146. D. C. Hong and J. S. Langer, Analytic theory of the selection mechanism
in the Sa¬man-Taylor problem, Phys. Rev. Letters, 56 (1986), pp. 2032“2035.
147. C. J. Howls, Hyperasymptotics for multidimensional integrals, exact remain-
der terms and the global connection problem, Proc. Roy. Soc. London A, 453
(1997), pp. 2271“2294.
148. C. J. Howls and S. A. Trasler, Weyl™s wedges, J. Physics A-Math. Gen.,
31 (1998), pp. 1911“1928. Hyperasymptotics for quantum billiards with non-
smooth boundary.
149. J. Hu, Asymptotics beyond all orders for a certain type of nonlinear oscillator,
Stud. Appl. Math., 96 (1996), pp. 85“109.
150. J. Hu and M. Kruskal, Re¬‚ection coe¬cient beyond all orders for singular
problems, 1, separated critical-points on the nearest critical-level line, J. Math.
Phys., 32 (1991), pp. 2400“2405.
151. , Re¬‚ection coe¬cient beyond all orders for singular problems, 2, close-
spaced critical-points on the nearest critical-level line, J. Math. Phys., 32
(1991), pp. 2676“2678.
152. J. Hu and M. D. Kruskal, Re¬‚ection coe¬cient beyond all orders for sin-
gular problems, in Asymptotics Beyond All Orders, H. Segur, S. Tanveer, and
H. Levine, eds., Plenum, Amsterdam, 1991, pp. 247“253.
153. J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solu-
tions to a model equation for water waves, Physica D, 32 (1988), pp. 253“268.
FKdV nonlocal solitons.




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.96
97
Exponential Asymptotics

154. M. Jardine, H. R. Allen, R. E. Grundy, and E. R. Priest, A fami-
ly of two-dimensional nonlinear solutions for magnetic ¬eld annihilation, J.
Geophys. Res. ” Space Physics, 97 (1992), pp. 4199“4207.
155. D. S. Jones, Uniform asymptotic remainders, in Asymptotic Comput. Anal.,
R. Wong, ed., Marcel Dekker, New York, 1990, pp. 241“264.
156. , Asymptotic series and remainders, in Ordinary and Partial Di¬eren-
tial Equations, Volume IV, B. D. Sleeman and R. J. Jarvis, eds., Longman,
London, 1993, pp. 12“.
157. , Asymptotic remainders, SIAM J. Math. Anal., 25 (1994), pp. 474“490.
Hyperasymptotics; shows that the remainders in a variety of asymptotic series
can be uniformly approximated by the same integral.
158. D. S. Jones, Introduction to Asymptotics: a Treatment Using Nonstandard
Analysis, World Scienti¬c, Singapore, 1997. 160 pp.; includes a chapter on
hyperasymptotics.
159. S. Kaplun, Low Reynolds number ¬‚ow past a circular cylinder, J. Math.
Mech., 6 (1957), pp. 595“603. Log-plus-power expansions.
160. , Fluid Mechanics and Singular Perturbations, Academic Press, New
York, 1967. ed. by P. A. Lagerstrom, L. N. Howard and C. S. Liu; Analyzed
di¬culties of log-plus-power expansions.
161. S. Kaplun and P. A. Lagerstrom, Asymptotic expansions of Navier-Stokes
solutions for small Reynolds number, J. Math. Mech., 6 (1957), pp. 585“593.
162. W. L. Kath and G. A. Kriegsmann, Optical tunnelling: radiation losses
in bent ¬bre-optic waveguides, IMA J. Appl. Math., 41 (1988), pp. 85“103.
Radiation loss is exponentially small in the small parameter, so “beyond all
orders” perturbation theory is developed here.
163. D. A. Kessler, J. Koplik, and H. Levine, Pattern selection in ¬ngered
growth phenomena, Adv. Phys., 37 (1988), pp. 255“339.
164. J. Killingbeck, Quantum-mechanical perturbation theory, Reports on
Progress in Theoretical Physics, 40 (1977), pp. 977“1031. Divergence of
asymptotic series.
165. , A polynomial perturbation problem, Phys. Lett. A, 67 (1978), pp. 13“15.
166. , Another attack on the sign-change argument, Chem. Phys. Lett., 80
(1981), pp. 601“603.
167. Y. S. Kivshar and B. A. Malomed, Comment on “nonexistence of small-
amplitude breather solutions in φ4 theory, Phys. Rev. Lett., 60 (1988), pp. 164“
164. Exponentially small radiation from perturbed sine-Gordon solitons and
other species.
168. , Dynamics of solitons in nearly integrable systems, Revs. Mod. Phys.,
61 (1989), pp. 763“915. Exponentially small radiation from perturbed sine-
Gordon solitons and other species.
169. V. Kowalenko, M. L. Glasser, T. Taucher, and N. E. Frankel, Gen-
eralised Euler-Jacobi inversion formula and asymptotics beyond all orders,
vol. 214 of London Mathematical Society Lecture Note Series, Cambridge
University Press, Cambridge, 1995.
170. M. C. A. Kropinski, M. J. Ward, and J. B. Keller, A hybrid asymptotic-
numerical method for low Reynolds number ¬‚ows past a cylindrical body, SIAM
J. Appl. Math., 55 (1995), pp. 1484“1510. Log-and-power series in Re.
171. M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a mod-
el of crystal growth, Tech. Rep. 85-25, Aeronautical Research Associates of
Princeton, 1985.
172. , Asymptotics beyond all orders in a model of crystal growth, Stud. Appl.
Math., 85 (1991), pp. 129“181.
173. J. G. L. Laforgue and R. E. O™Malley, Jr., Supersensitive boundary
value problems, in Asymptotic and Numerical methods for Partial Di¬eren-




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.97
98 John P. Boyd

tial Equations with Critical Parameters, H. G. Kaper and M. Garbey, eds.,
Kluwer, Dordrecht, 1993, pp. 215“223.
174. , On the motion of viscous shocks and the supersensitivity of their steady-
state limits, Methods and Applications of Analysis, 1 (1994), pp. 465“487.
Exponential smallness in shock movement.
175. , Shock layer movement for Burgers™ equation, SIAM J. Appl. Math., 55
(1995), pp. 332“347.
176. , Viscous shock motion for advection-di¬usion equations, Stud. Appl.
Math., 95 (1995), pp. 147“170.
177. C. Lanczos, Trigonometric interpolation of empirical and analytical func-
tions, Journal of Mathematics and Physics, 17 (1938), pp. 123“199. The
origin of both the pseudospectral method and the tau method. Lanczos is to
spectral methods what Newton was to calculus.
178. , Discourse on Fourier Series, Oliver and Boyd, Edinburgh, 1966.
179. C. G. Lange and H. J. Weinitschke, Singular perturbations of elliptic prob-
lems on domains with small holes, Stud. Appl. Math., 92 (1994), pp. 55“93.
Log-and-power series for eigenvalues with comparisons with numerical solu-
tions; demonstrates the surprisingly large sensitivity of eigenvalues to small
holes in the membrane.
180. V. F. Lazutkin, I. G. Schachmannski, and M. B. Tabanov, Splitting of
separatrices for standard and semistandard mappings, Physica D, 40 (1989),
pp. 235“248.
181. J. C. Le Guillou and J. Zinn-Justin, eds., Large-Order Behaviour of Per-
turbation Theory, North-Holland, Amsterdam, 1990. Exponential corrections
to power series, mostly in quantum mechanics.
182. R. Lim and M. V. Berry, Superadiabatic tracking for quantum evolution, J.
Phys. A, 24 (1991), pp. 3255“3264.
183. J. Liu and A. Wood, Matched asymptotics for a generalisation of a model
equation for optical tunnelling, European J. Appl. Math., 2 (1991), pp. 223“
231. Compute the exponentially small imaginary part of the eigenvalue »,
(») ∼ exp(’1/ 1/n ), for the problem uxx + (» + xn )u = 0 with outward
radiating waves on the semi-in¬nite interval.
184. E. N. Lorenz and V. Krishnamurthy, On the nonexistence of a slow man-
ifold, J. Atmos. Sci., 44 (1987), pp. 2940“2950. Weakly non-local in time.
185. C. Lozano and R. E. Meyer, Leakage and response of waves trapped by
round islands, Phys. Fluids, 19 (1976), pp. 1075“1088. Leakage is exponen-
tially small in the perturbation parameter.
186. C. Lu, A. D. MacGillivray, and S. P. Hastings, Asymptotic behavior of
solutions of a similarity equation for laminar ¬‚ows in channels with porous
walls, IMA J. Appl. Math., 49 (1992), pp. 139“162. Beyond-all-orders matched
asymptotics.
187. Y. L. Luke, The Special Functions and Their Approximations, vol. I & II,
Academic Press, New York, 1969.
188. J. N. Lyness, Adjusted forms of the Fourier Coe¬cient Asymptotic Expan-
sion, Mathematics of Computation, 25 (1971), pp. 87“104.
189. , The calculation of trigonometric Fourier coe¬cients, Journal of Com-
putational Physics, 54 (1984), pp. 57“73. A good review article which discusses
the integration-by-parts series for the asymptotic Fourier coe¬cients.
190. J. N. Lyness and B. W. Ninham, Numerical quadrature and asymptotic
expansions, Math. Comp., 21 (1967), pp. 162“. Shows that the error is a
power series in the grid spacing h plus an integral which is transcendentally
small in 1/h.
191. A. D. MacGillivray, A method for incorporating transcendentally small
terms into the method of matched asymptotic expansions, Stud. Appl. Math.,
99 (1997), pp. 285“310. Linear example has a general solution which is the sum




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.98
99
Exponential Asymptotics

of an antisymmetric function A(x) which is well-approximated by a second-
derivative-dropping outer expansion plus a symmetric part which is expo-
nentially small except at the boundaries, and can be approximated only by
a WKB method (with second derivative retained). His nonlinear example is
the Carrier-Pearson problem whose exact solution is a KdV cnoidal wave,
but required to satisfy Dirichlet boundary conditions. Matching fails because
each soliton peak can be translated with only an exponentially small error;
MacGillivray shows that the peaks, however many are ¬t between the bound-
aries, must be evenly spaced.
192. A. D. MacGillivray, B. Liu, and N. D. Kazarinoff, Asymptotic anal-
ysis of the peeling-o¬ point of a French duck, Methods and Applications of
Analysis, 1 (1994), pp. 488“509. Beyond-all-orders theory.
193. A. D. MacGillivray and C. Lu, Asymptotic solution of a laminar ¬‚ow
in a porous channel with large suction: A nonlinear turning point problem,
Methods and Applications of Analysis, 1 (1994), pp. 229“248. Incorporation
of exponentially small terms into matched asymptotics.
194. B. A. Malomed, Emission from, quasi-classical quantization, and stochastic
decay of sine-gordon solitons in external ¬elds, Physica D, 27 (1987), pp. 113“
157. Explicit calculations of exponentially small radiation.
195. , Perturbation-induced radiative decay of solitons, Phys. Lett. A, 123
(1987), pp. 459“468. Explicit calculations of exponentially small radiation
for perturbed sine-Gordon solitons, ¬‚uxons, kinks and coupled double-sine-
Gordon equations.
196. M. Marion and R. Temam, Nonlinear Galerkin methods, SIAM Journal of
´
Numerical Analysis, 26 (19), pp. 1139“1157.
197. O. Martin and S. V. Branis, Solitary waves in self-induced transparency,
in Asymptotics Beyond All Orders, H. Segur, S. Tanveer, and H. Levine, eds.,
Plenum, Amsterdam, 1991, pp. 327“336.
198. Ele-
J. Martinet and J.-P. Ramis,
mentary acceleration and multisummability-I, Ann. Inst. H. Poincar´-Phys.
e
Theor., 54 (1991), pp. 331“401.
199. V. P. Maslov, The Complex WKB Method for Nonlinear Equations I: Linear
Theory, Birkhauser, Boston, 1994. Calculation of exponentially small terms.
200. J. B. McLeod, Smoothing of Stokes discontinuities, Proc. Roy. Soc. London,
Series A, 437 (1992), pp. 343“354.
201. J. D. Meiss and W. Horton, Solitary drift waves in the presence of magnetic
shear, Phys. Fluids, 26 (1983), pp. 990“997. Show that plasma modons leak
radiation for large |x|, and therefore are nanopterons.
202. R. E. Meyer, Adiabatic variation. Part I: Exponential property for the simple
oscillator, J. Appl. Math. Phys. ZAMP, 24 (1973), pp. 517“524.
203. , Adiabatic variation. Part II: Action change for simple oscillator, J.
Appl. Math. Phys. ZAMP, (1973).
204. , Exponential action of a pendulum, Bull. Amer. Math. Soc., 80 (1974),
pp. 164“168.
205. , Adiabatic variation. Part IV: Action change of a pendulum for general
frequency, J. Appl. Math. Phys. ZAMP, 25 (1974), pp. 651“654.
206. , Gradual re¬‚ection of short waves, SIAM J. Appl. Math., 29 (1975),
pp. 481“492.
207. , Adiabatic variation. Part V: Nonlinear near-periodic oscillator, J.
Appl. Math. and Physics ZAMP, 27 (1976), pp. 181“195.
208. , Quasiclassical scattering above barriers in one dimension, J. Math.
Phys., 17 (1976), pp. 1039“1041.
209. , Surface wave re¬‚ection by underwater ridges, J. Phys. Oceangr., 9
(1979), pp. 150“157.
210. , Exponential asymptotics, SIAM Rev., 22 (1980), pp. 213“224.




ActaApplFINAL_OP92.tex; 21/08/2000; 16:16; no v.; p.99
100 John P. Boyd

211. , Wave re¬‚ection and quasiresonance, in Theory and Application of Sin-
gular Perturbation, no. 942 in Lecture Notes in Mathematics, Springer-Verlag,
New York, 1982, pp. 84“112.
212. , Quasiresonance of long life, J. Math. Phys., 27 (1986), pp. 238“248.
213. , A simple explanation of Stokes phenomenon, SIAM Rev., 31 (1989),
pp. 435“444.
214. , Observable tunneling in several dimensions, in Asymptotic and Compu-
tational Analysis, R. Wong, ed., Marcel Dekker, New York, 1990, pp. 299“328.
215. , On exponential asymptotics for nonseparable wave equations I: Com-
plex geometrical optics and connection, SIAM J. Appl. Math., 51 (1991),
pp. 1585“1601.
216. , On exponential asymptotics for nonseparable wave equations I: EBK
quantization, SIAM J. Appl. Math., 51 (1991), pp. 1602“1615.
217. , Exponential asymptotics for partial di¬erential equations, in Asymp-
totics Beyond All Orders, H. Segur, S. Tanveer, and H. Levine, eds., Plenum,
Amsterdam, 1991, pp. 29“36.
218. , Approximation and asymptotics, in Wave Asymptotics, D. A. Mar-
tin and G. R. Wickham, eds., Cambridge University Press, New York, 1992,
pp. 43“53. Blunt and perceptive review.
219. R. E. Meyer and E. J. Guay, Adiabatic variation. Part III: A deep mirror
model, J. Appl. Math. and Physics ZAMP, 25 (1974), pp. 643“650.
220. R. E. Meyer and J. F. Painter, Wave trapping with shore absorption, J.
Engineering Math., 13 (1979), pp. 33“45.
221. , New connection method across more general turning points, Bull. Amer.
Math. Soc., 4 (1981), pp. 335“338.
222. , Irregular points of modulation, Adv. Appl. Math., 4 (1982), pp. 145“
174.
223. , Connection for wave modulation, SIAM J. Math. Anal., 14 (1983),
pp. 450“462.
224. , On the Schroedinger connection, Bull. Amer. Math. Soc., 8 (1983),
pp. 73“76.
225. R. E. Meyer and M. C. Shen, On Floquet™s theorem for nonseparable partial
di¬erential equations, in Eleventh Dundee Conference in Ordinary and Partial
Di¬erential Equations, B. D. Sleeman, ed., Pitman Advanced mathematical
Research Notes, Longman-Wiley, New York, 1991, pp. 146“167.
226. , On exponential asymptotics for nonseparable wave equations III:
Approximate spectral bands of periodic potentials on strips, SIAM J. Appl.
Math., 52 (1992), pp. 730“745.
227. G. F. Miller, On the convergence of Chebyshev series for functions possess-
ing a singularity in the range of representation, SIAM J. Numer. Anal., 3
(1966), pp. 390“409.

<<

. 21
( 24 .)



>>