Related papers: Non-Point Invertible Transformations and Integrabi…

We prove higher integrability of the gradient of weak solutions to nonlinear parabolic systems whose prototype is \[ \partial_t u-\mathrm{div}\Big(\frac{\varphi'(z, |\nabla u|)}{|\nabla u|}\nabla u\Big) =0, \qquad u=(u^1,\dots,u^N), \]…

The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on some non-abelian Lie group, is studied and the solutions are shown to be expressed in terms of quasideterminants. As an…

The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff…

A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence…

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent paper [De Lellis, De…

We extend the generalised hodograph method to regular non- diagonalisable integrable systems of hydrodynamic type, in light of the relation between such systems and F-manifolds with compatible connection. The method allows the construction…

We introduce (binary) Darboux transformation for general differential equation of the second order in two independent variables. We present a discrete version of the transformation for a 6-point difference scheme. The scheme is appropriate…

We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As…

Using the generalized symmetry method we finish a classification, started in the article [R.N. Garifullin, R.I. Yamilov and D. Levi, Classification of five-point differential-difference equations, J. Phys. A: Math. Theor. 50 (2017) 125201…

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using…

In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II…

INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…

We propose the notion of integrable boundary in the context of discrete integrable systems on quad-graphs. The equation characterizing the boundary must satisfy a compatibility equation with the one characterizing the bulk that we called…

These lecture notes are devoted to the integrability of discrete systems and their relation to the theory of Yang-Baxter (YB) maps. Lax pairs play a significant role in the integrability of discrete systems. We introduce the notion of Lax…

We establish some higher differentiability results of integer and fractional order for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_{\Omega}f(x, Dv(x))\,:\, v\in…

This paper establishes the symmetries of Darboux's equations (1882) on tori. We extend Ince's work (1940) by developing new infinite series expansions in terms of Jacobi elliptic functions around each of the four regular singular points of…

We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of…

We show essentially that the differential equation $\frac{\partial (P,Q)}{\partial (x,y)} =c \in {\mathbb C}$, for $P,\,Q \in {\mathbb C}[x,y]$, may be "integrated", in the sense that it is equivalent to an algebraic system of equations…

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a…

We construct birational maps that satisfy the parametric set-theoretical Yang-Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable Nonlinear Schr\"odinger type equations…