On Soliton Instabilities In 1+1 Dimensional Integrable Systems

Author(s):  
F. Lambert ◽  
J. Springael ◽  
R. Willox
Keyword(s):  
2017 ◽  
Vol 72 (4) ◽  
pp. 331-337 ◽  
Author(s):  
Zhao-Wen Yan

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.


2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Andrea Cavaglià ◽  
Nikolay Gromov ◽  
Fedor Levkovich-Maslyuk

Abstract The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in $$ \mathcal{N} $$ N = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of $$ \mathcal{N} $$ N = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the $$ \mathcal{N} $$ N = 4 SYM case, as we speculate in the last part of the article.


2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Matteo Petrera ◽  
Yuri B. Suris ◽  
Kangning Wei ◽  
René Zander

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.


2017 ◽  
Vol 4 (1) ◽  
pp. 263-272 ◽  
Author(s):  
Niccolò Lora Lamia Donin

Abstract In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].


Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.


1997 ◽  
Vol 11 (26n27) ◽  
pp. 3093-3124
Author(s):  
A. Marshakov

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.


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