Reactive Atom-Beam Etching of GaAs.

Okamoto, Kotaro; Iwase, Masayuki

doi:10.3131/jvsj.30.22

Cited by 7 publications

(6 citation statements)

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“…This system can be obtained from (3.7) by the variable transformations t = ρ 2 , y = ρy * and x = ρ −1 x * . Proposition 4.2 [5,10]. Define the functions τ ν n (n ∈ Z 0 , ν ∈ C) by…”

Section: The J -Matrix For the Classical Transcendental Solutions To ...mentioning

confidence: 99%

The anti-self-dual Yang–Mills equation and the Painlevé III equation

Masuda¹

2007

J. Phys. A: Math. Theor.

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Section: The J -Matrix For the Classical Transcendental Solutions To ...mentioning

confidence: 99%

The anti-self-dual Yang–Mills equation and the Painlevé III equation

Masuda¹

2007

J. Phys. A: Math. Theor.

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“…For C = 0 or J = −1 this equation takes a form of a conventional P III equation [24]. One can show that this result extends to any C = −1 by a simple rescaling of variables while for C = −1 (or J = 1) the model becomes one of 44 solvable Painlevé equations [2].…”

Section: Review Of P Iii−v Equations and Their Symmetriesmentioning

confidence: 99%

Solutions of mixed Painlevé P_III—Vmodel

Alves

Aratyn

Gomes

et al. 2019

J. Phys. A: Math. Theor.

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We review the construction of the mixed Painlevé P III−V system in terms of a 4-boson integrable model and discuss its symmetries. Such a mixed system consist of an hybrid differential equation that for special limits of its parameters reduces to either Painlevé P III or P V . The aim of this paper is to describe solutions of P III−V model. In particular, we determine and classify rational, power series and transcendental solutions of P III−V . A class of power series solutions is shown to be convergent in accordance with the Briot-Bouquet theorem. Moreover, the P III−V equations are reduced to Riccati equations and solved for special values of parameters. The corresponding Riccati solutions can be expressed as Whittaker functions or alternatively confluent hypergeometric and Laguerre functions and are given by ratios of polynomials of order n when the parameter of P III−V equation is quantized by integer n ∈ Z.Painlevé P II equation. Painlevé P V equations appeared in the context of impenetrable Bose gas model [13]. More recently the scattering on two Aharonov-Bohm vortices was exactly solved in terms of solutions of the Painlevé III equation [7]. Ablowitz Ramani and Segur (ARS) showed a connection between the mKdV integrable model and the Painlevé P II equation when self-similarity limit was implemented. They suggested that such connection could be generalized to other integrable models [1] meaning that the self-similarity limit would lead for integrable models to equations with the Painlevè property. More recently, it was found [3] that the 2n boson integrable model obtained as particular reductions (Drinfeld-Sokolov) of KP integrable models connected to Toda lattice hierarchy gives rise to higher Painlevé equations invariant under extended affine Weyl groups. In particular, investigation of various Dirac reduction schemes applied to the 4-boson (n = 2) integrable model with Weyl symmetry structure A (1) 4 led to emergence of a new mixed P III−V model [4]. The second order P III−V equation for a canonical variable q is given by:

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“…One sees that actions of π 0 , s 2 , π 2 on parameters (v 1 , v 2 ) realize a representation of the extended affine Weyl group for the root system B 2 1 ( ) [7,11,18,23]. To see the connection to the B 2 1 ( ) root lattice consider a two-dimensional vector space V consisting of vectors v=v 1 e 1 +v 2 e 2 , with v 1 , v 2 being parameters of the Painlevé III equation and e 1 , e 2 being a canonical basis of V. Define next a symmetric bilinear form • • in V such that e e .…”

Section: Symmetric Piii-pv Equationsmentioning

confidence: 99%

“…(1) 2 [18,23,11,7]. To see the connection to the B (1) 2 root lattice consider a 2-dimensional vector space V consisting of vectors v = v 1 e 1 + v 2 e 2 , with v 1 , v 2 being parameters of the Painlevé III equation and e 1 , e 2 being a canonical basis of V. Define next a symmetric bilinear form ·|· in V such that e i |e j = δ ij .…”

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confidence: 99%