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 occurs linearly! Pole of at , implying leading order exponential
☞✙ ✒ ✙ ✒ ✁ ☞ � ✡ ✒ � ✖ ✝ ☎ ✔ ✓ ✒ ✒ � ✁ ✁ ☎ � ✙ Exponential Asymptotics for Nonlinear ODEs General Setting (Costin, 1998) analytic at , satisfying generic conditions: (ii) Eigenvalues of 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 of , 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 of convergence Arg η = 4 π/9 −δ Arg η= 2 π/9 − δ1 D Arg η=−2 π/9 − δ1 η s Arg η = −4 π/9 − δ
✸ ✼ ✸ ☎ ✸ 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 occurs at , though in the complex plane.
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