Related papers: Symmetry and Integrability of Classical Field Equa…
We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of…
We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund…
The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand…
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and…
We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…
We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically…
An integrable field theory, due to path-independence on the space-time plane, should yield together with an infinite set of independent conserved charges also similar dual charges determining the boundary and defect contributions. On the…
An algebraic method is devised to look for non-local symmetries of the pseudopotential type of nonlinear field equations. The method is based on the use of an infinite-dimensional subalgebra of the prolongation algebra $L$ associated with…
Solutions to scalar theories with derivative self-couplings often have regions where non-linearities are important. Given a classical source, there is usually a region, demarcated by the Vainshtein radius, inside of which the classical…
A non-isospectral linear problem for an integrable 2+1 generalization of the non linear Schr\"odinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the…
We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to…
Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two…
A general method for the construction of solutions of the d'Alambertian and double d'Alambertian (harmonic and bi-harmonic) equations with local dependence of arbitrary functions upon two independent arguments is proposed. The connection…
This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We…
A recent investigation of the SU(3) Yang-Mills field equations found several classical solutions which exhibited a type of confinement due to gauge fields which increased without bound as $r \to \infty$. This increase of the gauge fields…
Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a…
This is an introductory survey written in Japanese on classical integrability from the viewpoint of symmetry reduction of 4-dimensional anti-self-dual Yang-Mills equations. Twistor construction methods are discussed in detail in…
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.