In this paper, we consider a Cauchy problem for the modified Korteweg--de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and the nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of $t^{-1/(2n+1)}$ with smooth coefficients of the variable $(-1)^{n+1}x((2n+1) t)^{-1/(2n+1)}$ in the self-similarity region for the solution of the $n$th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlevé II hierarchy, which generalize the classical Ablowitz--Segur solution for the Painlevé II equation and appear in a variety of random matrix and statistical physics models. We establish the connection formulas for this family of solutions. In the special case that the reflection coefficient vanishes at the origin, the solutions of the Painlevé II hierarchy in the leading coefficient vanishes as well, and the leading and subleading terms in the asymptotic expansion are instead given explicitly in terms of derivatives of the generalized Airy function.


  1. long-time asymptotics
  2. modified Korteweg--de Vries hierarchy
  3. Painlevé II transcendents
  4. Riemann--Hilbert problems
  5. nonlinear steepest descent method

MSC codes

  1. 37K15
  2. 41A60
  3. 35Q15
  4. 35Q53

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Information & Authors


Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 5291 - 5334
ISSN (online): 1095-7154


Submitted: 22 September 2021
Accepted: 21 June 2022
Published online: 13 September 2022


  1. long-time asymptotics
  2. modified Korteweg--de Vries hierarchy
  3. Painlevé II transcendents
  4. Riemann--Hilbert problems
  5. nonlinear steepest descent method

MSC codes

  1. 37K15
  2. 41A60
  3. 35Q15
  4. 35Q53



Funding Information

Fudan University https://doi.org/10.13039/501100003347 : EZH1411513
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11901141
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11822104
Shanghai Education Development Foundation https://doi.org/10.13039/501100003024
Program for Professor of Special Appointment at Shanghai Institutions of Higher Learning at Shanghai Institutions of Higher Learning) https://doi.org/10.13039/501100013285

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