Asymptotics beyond all orders

228

3rd workshop on nonlinear evolution equations

THE DARBOUX T R A N S F O R M A T I O N

METHOD FOR FINDING SOLUTIONS

Y. S. Li, Department of Mathematics, Anhui, China

University

OF SOLITON EQUATIONS

of S c i e n c e and Technology,

Hefei,

We have proposed a technique for finding the D a r b o u x t r a n s f o r m a t i o n for a series of eigenvalue problems such that this method can be used to find solutions of some soliton equations.

ASYMPTOTICS

BEYOND ALL ORDERS

M. D. Kruskal, H. Segur*, M a t h e m a t i c s Department, P r i n c e t o n U n i v e r s i t y Princeton, NJ 08544, USA * A e r o n a u t i c Res. A s s o c i a t e s of P r i n c e t o n Inc., Princeton, NJ, 08544, USA The real nonlinear

ordinary differential

E

2~

+

~ =

equation

g(e)

arises as a model in the study of dendrite growth in s u p e r s a t u r a t e d media (J. Langer, et al.); here 8(s) is the angle of a steadily growing dendrite surface as a function of arclength s, g(0) is a given s y m m e t r i c analytic function of 9, positive in an interval containing e = 0 bounded by simple zeros e = + a, and ~ is a small positive parameter. Of interest is the solution determined by the conditions (I) e(s) ~ - a as s ÷ - ~, (2) 0(s) > O, (3) 9(0) = O, which are easily seen to determine it uniquely. The problem is to determine the behaviour of e(s) as s ~ + ~,

in

particular

e(O),

to see whether

it oscillates,

or e q u i v a l e n t l y

in particular to see whether 0(0) = 0. The difficulty is that the obvious p e r t u r b a t i o n

e(s) - 00(s) which describes in general

the solution

diverges,

has

+ E201(s)

asymptotically

0n(S)

~ 0

as

+ E492(s) (as

s ~ +~,

series r e p r e s e n t a t i o n

÷...

E ~ O) and

to determine

to all orders

en(O)

: O,

but

for all

n; thus the desired behaviour is "beyond all orders". It is shown how to detour through the complex s-plane to find the desired information, similarly to the method of Pokrovskii and K u l a t n i k o v for an analogous byt simpler linear equation (the wave reflection problem).