On the existence and evaluation of Stokes phenomena in fluid mechanics

A singularly perturbed, high order KdV-type model, which describes localized travelling waves (“solitons”) is being considered. We focus on the Inner solution, and detect Stokes phenomena that are crucial as to whether we can obtain a suitable solution. We provide a simple proof that the corresponding Stokes constant is non-zero. Also, we evaluate this splitting constant numerically by using two methods that are induced by the underlying theory.

Stokes phenomena in 3d $$ \mathcal{N} $$ = 2 SQED2 and $$ \mathbbm{CP} $$1 models

We propose a novel approach of uncovering Stokes phenomenon exhibited by the holomorphic blocks of $$ \mathbbm{CP} $$ CP 1 model by considering it as a specific decoupling limit of SQED2 model. This approach involves using a ℤ3 symmetry that leaves the supersymmetric parameter space of SQED2 model invariant to transform a pair of SQED2 holomorphic blocks to get two new pairs of blocks. The original pair obtained by solving the line operator identities of the SQED2 model and the two new transformed pairs turn out to be related by Stokes-like matrices. These three pairs of holomorphic blocks can be reduced to the known triplet of $$ \mathbbm{CP} $$ CP 1 blocks in a particular decoupling limit where two of the chiral multiplets in the SQED2 model are made infinitely massive. This reduction then correctly reproduces the Stokes regions and matrices of the $$ \mathbbm{CP} $$ CP 1 blocks. Along the way, we find six pairs of SQED2 holomorphic blocks in total, which lead to six Stokes-like regions covering uniquely the full parameter space of the SQED2 model.

The exact WKB for cosmological particle production

The Bogoliubov transformation in cosmological particle production can be explained by the Stokes phenomena of the corresponding ordinary differential equation. The calculation becomes very simple as far as the solution is described by a special function. Otherwise, the calculation requires more tactics, where the Exact WKB (EWKB) may be a powerful tool. Using the EWKB, we discuss cosmological particle production focusing on the effect of more general interaction and classical scattering. The classical scattering appears when the corresponding scattering problem of the Schrödinger equation develops classical turning points on the trajectory. The higher process of fermionic preheating is also discussed using the Landau-Zener model.

Stokes phenomena in discrete Painlevé II

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.