Exponential Asymptotics and Singularities Saleh Tanveer (Ohio State - - PowerPoint PPT Presentation

Exponential Asymptotics and Singularities

Saleh Tanveer (Ohio State University) Joint work with Ovidiu Costin (Rutgers University) Research supported by National Science Foundation

Exponential Asymptotics

Asymptotic Expansion in the Poincare Sense:

• ✁

as ignores possible terms like

. Important sometimes, for instance, if we want , when

real Example in a differential equations context: Solution:

• ☎

for , large, on positive real axis

Background and Applications

Fundamental Work by

Stokes, Dingle, Berry, Kruskal, Ecalle......

Large Body of Work in Applications to

Quantum Tunneling, Viscous Fingering, Water Wave, Chaos, Crystal Growth,..

Exponential Asymptotics for nonlinear ODEs

Example: A Simple Nonlinear Equation

Asymptotics for the solution is still of the form:

• ☎

However, with exponential corrections:

where

Exponential Asymptotics for a Nonlinear Equation

Determination of

Formally, plugging

into ODE and equating powers of

, for :

• ✒

• ✒

We can then use a formal procedure to compute

so that:

• ☞

Borel Plane Transformation

So, ,

On transformation:

analytic for , exponentially bounded as Suppose

known for

. Then near , the unknown part of

• ccurs linearly!

Pole of at , implying

leading order exponential

Exponential Asymptotics for Nonlinear ODEs

General Setting (Costin, 1998)

• ☎

analytic at , satisfying generic conditions: (ii) Eigenvalues

• f

• ☞

linearly independent over and

are all different. W.L.O.G

• diag

diag

constant matrices.

• ✁

as and

Exponential Asymptotics for Nonlinear ODEs

Costin, 1998 results

factorially divergent formal power series:

Resumming Trans-series for Singularity Determination

Most relevant near

when

• ✄

: Periodic array of poles in

PDEs in the complex plane and singularities

Motivation and Scope of Analysis

Singularity formation for nonlinear PDEs of general interest One way is to follow singularities in the complex plane (Moore, 1979) Formal procedure for singularity formation at

known (Tanveer, 1993 a,b, Cowley, Baker & Tanveer, 1999 and others) Many equations allow formal asymptotic similarity solutions. Want to know if actual solutions asymptote to the formal similarity solution. Stability of Similarity Solutions

Formal Singularity analysis for

Consider satisfying:

Formal Taylor Expansion in time for , (using

):

Breaks down when

• ✮

:

• ✮

• ✮

• ✮

To the leading order,

Formal Singularity analysis for

Integrates to:

Additional Change of Variable:

:

for ,

Singularity of

at

implies moving singularity of :

• ✮

Existence and Matching conditions ? Validity of

?

Singularity Formation in Modified Harry-Dym Equation

• ✮

[Costin & Tanveer, 2000]: Unique analytic solution to IVP in sector

with

• ✮

• ✮

• ✶

For

,

• ✮

Taylor Expansion:

converges Near

,

• ✮

Equations for

satisfies:

satisfies linear equation:

known in terms of

, For ,

• ☛

• ☛

Main Theorem Proved:

For a singularity

• f

, with

large enough and with

close to the anti-Stokes line

, there exists a domain that extends to with

for some

, and includes a region around the the singularity

but excludes an open neighborhood such that

is convergent for all sufficiently small . In particular, in the limit

, the singularity of approaches the singularity of

and is of the same type.

Note that convergence implies

If small around

, nonzero contour integral.

Domain

• f convergence

η = −4 π/9 − δ ηs η=−2 π/9 − δ1 η= 2 π/9 − δ1 Arg Arg Arg Arg η = 4 π/9 −δ

D

Scope of Theorem of the type proved

Not just limited to Formal blow-up solution aymptotically:

Can analyze rigorously

Useful in Stability analysis of Blow-ups

Can be useful in analysis as well

Main Ideas of Proof

Lemma 1: There exists and independent of integer so that

Use induction on

Control for large

• needed. WKB solutions to

:

Solution to

essentially

• ✳

• ✮

Necessary property for Domain

Property 1: Given any point , for each =1,2,3, there exists a piecewise differentiable path

from to

so that

• ✮

where is the arc-length parametrization of

and and is independent of , .

This is to be ensured for containing

Definition Let be a region around

,

with

and

small

Flow and Invariant set

Definition We define three distinct flows,

for :

, where

satisfies

Definition:

• ✂

• ☛

Theorem: For

sufficiently large, and appropriate choice of

, domain satisfies property 1

Conclusion

• 1. Resummation of trans-series, used previously for nonlinear

ODEs, can be effectively extended to determine complex singularities of PDEs.

• 2. Methodology for proofs developed here equally applicable to

stability analysis for similarity solutions of PDEs

• 3. A theory for existence of solutions to certain higher-order

partial differential equations developed.

• 4. Closer to proving a long-held conjecture: For generic initial

conditions for a large class of PDEs, any singularity formation

• ccurs at

, though in the complex plane.