Related papers: Commuting Flows and Conservation Laws for Noncommu…
We discuss extension of soliton theories and integrable systems into noncommutative spaces. In the framework of noncommutative integrable hierarchy, we give infinite conserved quantities and exact soliton solutions for many noncommutative…
Diffusion with multipole-moment conservation gives rise to transport laws that generalize Fick's law and has attracted growing attention following experimental advances in strongly tilted optical lattices. It was recently shown that…
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
In a recent paper Dargis and Mathieu introduced integrodifferential odd flows for the supersymmetric KdV equation. These flows are obtained from the nonlocal conservation laws associated with the fourth root of its Lax operator. In this…
We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes: \partial_t\rho+\partial_xF(x,\rho)=0. The main feature of such a conservation law is the discontinuity of the flux function in the space variable…
We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this…
We construct an infinite hierarchy of nonlocal conservation laws for the $ABC$ equation $A u_t\,u_{xy}+B u_x\,u_{ty}+C u_y\,u_{tx} = 0$, where $A,B,C$ are constants and $A+B+C\neq 0$, using a nonisospectral Lax pair. As a byproduct, we…
We review recent developments of soliton theories and integrable systems on noncommutative spaces. The former part is a brief review of noncommutative gauge theories focusing on ADHM construction of noncommutative instantons. The latter…
This paper is devoted to the study of propagation dynamics for a large class of non-monotone evolution systems. In two directions of the spatial variable, such a system has two limiting systems admitting the spatial translation invariance.…
In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive…
The Hamiltonian formulation of superfluids based on noncanonical Poisson brackets is studied in detail. The assumption that the momentum density is proportional to the flow of the conserved energy is shown to lead to the covariant…
An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete…
Through a reciprocal transformation $\mathcal{T}_0$ induced by the conservation law $\partial_t(u_x^2) = \partial_x(2uu_x^2)$, the Hunter-Saxton (HS) equation $u_{xt} = 2uu_{2x} + u_x^2$ is shown to possess conserved densities involving…
Constructal Law states that a finite-size flow system that persists in time evolves its configuration so as to provide progressively easier access to the currents that flow through it. Classical Constructal theory derives hierarchical flow…
We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like…
The aim of this work is twofold. From a mathematical point of view, we show the existence of a hyperbolic system of equations that is not symmetrizable in the sense of Friedrichs. Such system appears in the theory of compressible fluid…
The paper considers the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The inviscid, thermally non-conducting medium is modeled by a polytropic gas. The equations are examined for symmetries and…
Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike…
We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using…
We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function $f$ is strictly convex and show that, for every $…