Related papers: Two-Field Integrable Evolutionary Systems of the T…
We consider here the class of fully-nonlinear symmetry-integrable third-order evolution equations in 1+1 dimensions that were proposed recently in the journal Open Communications in Nonlinear Mathematical Physics, vol. 2, 216--228 (2022).…
We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds…
We find all scalar second order evolution equations possessing an sl$_2$-valued zero curvature representation that is not reducible to a proper subalgebra of sl$_2$. None of these zero-curvature representations admits a parameter.
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a…
We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three…
We classify three dimensional evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.
We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant…
The derivation of a new family of magnetic fields inducing exactly solvable spin evolutions is presented. The conditions for which these fields generate the evolution loops (dynamical processes for which any spin state evolves cyclically)…
Exactly integrable systems connected to semisimple algebras of second rank with an arbitrary choice of grading are presented in explicit form. General solutions of these systems are expressed in terms of matrix elements of two fundamental…
Integrability of the differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are…
We present a classification of (2,2) free field compactifations with one twist in which only 95 distinct models (generations and antigenerations) are found. Models with three generations and no antigenerations are given.
We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to…
The structural constants of an evolution algebra is given by a quadratic matrix $A$. In this work we establish equivalence between nil, right nilpotent evolution algebras and evolution algebras, which are defined by upper triangular matrix…
The new class of integrable mappings and chains is introduced. Corresponding (1+2) integrable systems invariant with respect to such discrete transformations are represented in explicit form. Soliton like solutions of them are represented…
All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…
After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on…
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…
A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional…