scholarly journals ZN graded discrete Lax pairs and Yang–Baxter maps

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160946 ◽  
Author(s):  
Allan P. Fordy ◽  
Pavlos Xenitidis
Keyword(s):  
Integrable Systems ◽  
Boussinesq Equation ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
New Family ◽  
Modified Boussinesq Equation ◽  
Lower Dimensional

We recently introduced a class of Z N graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang–Baxter maps. Many well-known examples belong to this scheme for N =2, so, for N ≥3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang–Baxter maps, which include H B III (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N =2, and that associated with the discrete modified Boussinesq equation, for N =3. For N ≥5 we introduce a new family of Yang–Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang–Baxter maps F IV and F V (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967–1007. (doi:10.4310/CAG.2004.v12.n5.a1)).

scholarly journals Direct linearization approach to discrete integrable systems associated with ℤ 𝒩 graded Lax pairs

2020 ◽  
Vol 476 (2237) ◽  
pp. 20200036
Author(s):  
Wei Fu
Keyword(s):  
Difference Equations ◽  
Bilinear Form ◽  
Integrable Systems ◽  
Solution Structure ◽  
Discrete Systems ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
Direct Linearization ◽  
Linear Integral ◽  
Linear Integral Equations

Fordy and Xenitidis [ J. Phys. A: Math. Theor. 50 (2017) 165205. ( doi:10.1088/1751-8121/aa639a )] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.

A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms

2015 ◽  
Vol 27 (2) ◽  
pp. 194-212 ◽  
Author(s):  
YI HE ◽  
XING-BIAO HU ◽  
HON-WAH TAM ◽  
YING-NAN ZHANG
Keyword(s):  
Integrable Systems ◽  
Numerical Experiments ◽  
Boussinesq Equation ◽  
Volterra Equation ◽  
Algebraic Method ◽  
Convergence Acceleration ◽  
New Method ◽  
Discrete Integrable Systems ◽  
Convergence Acceleration Algorithm ◽  
Autonomous Version

In this paper, we propose a new algebraic method to construct non-autonomous discrete integrable systems. The method starts from constructing generalizations of convergence acceleration algorithms related to discrete integrable systems. Then the non-autonomous version of the corresponding integrable systems are derived. The molecule solutions of the systems are also obtained. As an example of the application of the method, we propose a generalization of the multistep ϵ-algorithm, and then derive a non-autonomous discrete extended Lotka–Volterra equation. Since the convergence acceleration algorithm from the lattice Boussinesq equation is just a particular case of the multistep ϵ-algorithm, we have therefore arrived at a generalization of this algorithm. Finally, numerical experiments on the new algorithm are presented.

Three new species, two new genera, and new family Biphragmosagittidae (Сhaetognatha) from Southwest Pacific Ocean

2011 ◽  
Vol 20 (1) ◽  
pp. 161-173
Author(s):  
A.P. Kassatkina
Keyword(s):  
New Species ◽  
Type Species ◽  
Morphological Characters ◽  
The Body ◽  
New Order ◽  
New Genera ◽  
New Family ◽  
The Family ◽  
Three New Species

Resuming published and own data, a revision of classification of Chaetognatha is presented. The family Sagittidae Claus & Grobben, 1905 is given a rank of subclass, Sagittiones, characterised, in particular, by the presence of two pairs of sac-like gelatinous structures or two pairs of fins. Besides the order Aphragmophora Tokioka, 1965, it contains the new order Biphragmosagittiformes ord. nov., which is a unique group of Chaetognatha with an unusual combination of morphological characters: the transverse muscles present in both the trunk and the tail sections of the body; the seminal vesicles simple, without internal complex compartments; the presence of two pairs of lateral fins. The only family assigned to the new order, Biphragmosagittidae fam. nov., contains two genera. Diagnoses of the two new genera, Biphragmosagitta gen. nov. (type species B. tarasovi sp. nov. and B. angusticephala sp. nov.) and Biphragmofastigata gen. nov. (type species B. fastigata sp. nov.), detailed descriptions and pictures of the three new species are presented.

scholarly journals Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Matteo Petrera ◽  
Yuri B. Suris ◽  
Kangning Wei ◽  
René Zander
Keyword(s):  
Integrable Systems ◽  
Geometric Description ◽  
Discrete Integrable Systems ◽  
Birational Maps ◽  
Base Points ◽  
Special Geometry ◽  
Low Degree ◽  
Geometric Study

AbstractWe contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals Discrete integrable systems generated by Hermite-Padé approximants

Nonlinearity ◽  
2016 ◽  
Vol 29 (5) ◽  
pp. 1487-1506 ◽  
Author(s):  
Alexander I Aptekarev ◽  
Maxim Derevyagin ◽  
Walter Van Assche
Keyword(s):  
Integrable Systems ◽  
Padé Approximants ◽  
Discrete Integrable Systems ◽  
Pade Approximants

scholarly journals PseudoGTPase domains in p190RhoGAP proteins: a mini-review

2018 ◽  
Vol 46 (6) ◽  
pp. 1713-1720 ◽  
Author(s):  
Amy L. Stiegler ◽  
Titus J. Boggon
Keyword(s):  
Family Members ◽  
Small Gtpases ◽  
Small Gtpase ◽  
Signaling Proteins ◽  
New Family ◽  
Gtpase Activating Proteins ◽  
Rho Family ◽  
Biochemical Analyses ◽  
Catalytic Cleft

Pseudoenzymes generally lack detectable catalytic activity despite adopting the overall protein fold of their catalytically competent counterparts, indeed ‘pseudo’ family members seem to be incorporated in all enzyme classes. The small GTPase enzymes are important signaling proteins, and recent studies have identified many new family members with noncanonical residues within the catalytic cleft, termed pseudoGTPases. To illustrate recent discoveries in the field, we use the p190RhoGAP proteins as an example. p190RhoGAP proteins (ARHGAP5 and ARHGAP35) are the most abundant GTPase activating proteins for the Rho family of small GTPases. These are key regulators of Rho signaling in processes such as cell migration, adhesion and cytokinesis. Structural biology has complemented and guided biochemical analyses for these proteins and has allowed discovery of two cryptic pseudoGTPase domains, and the re-classification of a third, previously identified, GTPase-fold domain as a pseudoGTPase. The three domains within p190RhoGAP proteins illustrate the diversity of this rapidly expanding pseudoGTPase group.

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