Related papers: Abstract formulation of the Miura transform
Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations…
We consider systems, which conserve the particle number and are described by Schr\"odinger equations containing complex nonlinearities. In the case of canonical systems, we study their main symmetries and conservation laws. We introduce a…
The general approach to chain equations derivation for the function generated by a Miura transformation analog is developing to account evolution (second Lax equation) and illustrated for Sturm-Liouville differential and difference…
The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of…
Quasi double Casoratian solutions are derived for a bilinear system reformulated from the coupled semi-discrete modified Korteweg-de Vries equations with nonzero backgrounds. These solutions, when applied with the classical and nonlocal…
We define and study an $\ell$-adic Fourier transform for a relative version of Banach-Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic…
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
Bracket preserving gauge equivalence is established between several two-boson generated KP type of hierarchies. These KP hierarchies reduce under symplectic reduction (via Dirac constraints) to KdV, mKdV and Schwarzian KdV hierarchies.…
We study the relation between the linear stability of almost-symmetries and the geometry of the Banach spaces on which these transformations are defined. We show that any transformation between finite dimensional Banach spaces that…
We investigate self-similar solutions of the extended discrete KP hierarchy. It is shown that corresponding ansatzes lead to purely discrete equations with dependence on some number of parameters together with equations governing…
For certain class of perturbations of the equation $u_t=f(u) u_x$, we prove the existence of change of coordinates, called quasi-Miura transformations, that reduce these perturbed equations to the unperturbed ones. As an application, we…
We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation…
The algebra of diffeomorphisms derived from general coordinate transformations on commuting coordinates is represented by differential operators on noncommutative spaces. The algebra remains unchanged, the comultiplication however is…
The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a H\"older continuous function. The Fourier transform is written as the difference of $\int_{-1}^1…
This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary…
Meta-conformal transformations are constructed as dynamical symmetries of the linear transport equation in $d$ spatial dimensions. In one and two dimensions, the associated Lie algebras are infinite-dimensional and isomorphic to the direct…
We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator $B$ is…
We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the…
A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat $\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant…
Discrete and q-difference deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by a central system of discrete or q-difference equations…