Related papers: Microscopic conservation laws for integrable latti…
We propose new algorithms for numerical integration of the equations of motion for classical spin systems with fixed spatial site positions. The algorithms are derived on the basis of a mid-point scheme in conjunction with the multiple time…
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable…
The $2$-dimensional Toda lattice ($2$D Toda) is a completely integrable semi-discrete wave equation with the KP-II equation in its continuous limit. Using Darboux transformations, we prove the linear stability of $1$-line solitons for $2$D…
Consider a strictly hyperbolic $n\times n$ system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup…
A complete classification of integrable conservative hydrodynamic chains is presented. These hydrodynamic chains are written via special coordinates -- moments, such that right hand sides of these infinite component systems depend linearly…
In this paper we prove that the unique entropy solution to a scalar nonlinear conservation law with strictly monotone velocity and nonnegative initial condition can be rigorously obtained as the large particle limit of a microscopic…
We consider a Markov chain on non-negative integer arrays of a given shape (and satisfying certain constraints) which is closely related to fundamental $SL(r+1,\mathbb{R})$ Whittaker functions and the Toda lattice. In the index zero case…
The main purpose of the paper is to demonstrate that condition of invariance with respect to the Legendre transformations allows effectively isolate the class of integrable difference equations on the triangular lattice, which can be…
A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of…
We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences…
The paper compares computational aspects of four approaches to compute conservation laws of single differential equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer…
We show for a variety of classes of conservative PDEs that discrete gradient methods designed to have a conserved quantity (here called energy) also have a time-discrete conservation law. The discrete conservation law has the same conserved…
Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, self-avoiding polymers, ... This has led to numerous exact (but non-rigorous) predictions of their scaling…
Originally a model for wave propagation on the line, the Toda lattice is a wonderful case study in mechanics and symplectic geometry. In Flaschka's variables, it becomes an evolution given by a Lax pair on the vector space of real,…
In this short note, we extend the conservation law of Lamm-Rivi\`ere [Comm. PDEs 2008] for a fourth order elliptic system to supercritical dimensions, under certain Lorentz-Sobolev integrability assumptions on the associated coefficient…
A family of modified Kadomtsev-Petviashvili equations (mKP) in 2+1 dimensions is studied. This family includes the integrable mKP equation when the coefficients of the nonlinear terms and the transverse dispersion term satisfy an algebraic…
Reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model. We introduce projection-based hyper-reduced models of nonlinear conservation laws which are…
The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of…
For a class of generalized integrable hierarchies associated with affine (twisted or untwisted) Kac-Moody algebras, an explicit representation of their local conserved densities by means of a single scalar tau-function is deduced. This…
The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place…