Related papers: Integrable deformations of a polygon
In analogy with the well-known 2-linkage tractor-trailer problem, we define a 2-linkage problem in the plane with novel non-holonomic ``no-slip'' conditions. Using constructs from sub-Riemannian geometry, we look for geodesics corresponding…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial…
A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…
We show that the nonlinear 2+1--dimensional Three--Wave Resonant Interaction equations, describing several important physical phenomena, can be generated starting from incomplete Lie algebras in the framework of multidimensional…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of…
For systems of evolutionary partial differential equations the tau-structure is an important notion which originated from the deep relation between integrable systems and quantum field theories. We show that, under a certain non-degeneracy…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
Nonlinear dispersionless equations arise as the dispersionless limit of well know integrable hierarchies of equations or by construction, such as the system of hydrodynamic type. Some of these equations are integrable in the Hamiltonian…
We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.
Moving boundary problems of Stefan-type for a novel third order nonlinear evolution equation with temporal modulation are here shown to be amenable to exact Airy-type solution via a classical Ermakov equation with its admitted nonlinear…
The B\"acklund transformations and the superposition formulas for two sub--equations of the general elliptic equation are constructed from the Riccati equation by using an indirect mapping method.The thirty-six previously known solutions of…
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…
In this article we study an integrable deformation of the Kapustin-Witten equations. Using the Weyl-Wigner-Moyal-Groenewold description an integrable $\star$-deformation of a Kapustin-Witten system is obtained. Starting from known solutions…
The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
We introduce deformations of Kazhdan-Lusztig elements and specialised nonsymmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We…
In this paper, the concept of algebroid mappings of complex manifolds is introduced based on that a large number of complex systems of PDEs admit multi-valued solutions that can be defined by a system of independent algebraic equations over…
We investigate basic features of Bianchi's B\"acklund transformation of quadrics to see if it can be obtained under weaker assumptions and if it can be generalized to deformations of other surfaces.
Treating an integrable quad-equation along with its two generalised symmetries as a compatible system allows one to construct an auto-B\"acklund transformation for solutions of the related NLS-type system. A fixed periodic reduction of the…