Related papers: Negative Volterra Flows and Mixed Volterra Flows a…
Taking the standard zero curvature approach we derive an infinite set of integrable equations, which taken together form the negative Volterra hierarchy. The resulting equations turn out to be nonlocal, which is usual for the negative…
In this article, by means of using discrete zero curvature representation and constructing opportune time evolution problems, two new discrete integrable lattice hierarchies with n-dependent coefficients are proposed, which related to a new…
In this article, several 2+1 dimensional lattice hierarchies proposed by Blaszak and Szum [J. Math. Phys. {\bf 42}, 225(2001)] are further investigated. We first describe their discrete zero curvature representations. Then, by means of…
This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the "meaningful objects" are the fluxes, evaluated across domain…
We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of…
Modified Volterra lattice admits two vector generalizations. One of them is studied for the first time. The zero curvature representations, B\"acklund transformations, nonlinear superposition principle and the simplest explicit solutions of…
Time-dependent models of fluid motion in thin layers, subject to signed source terms, represent important sub-problems within climate dynamics. Examples include ice sheets, sea ice, and even shallow oceans and lakes. We address these…
We consider the problem of existence of entropy weak solutions to scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect both to the space variable x and the unknown quantity u. The problem…
The initial boundary value problem for a class of scalar non autonomous conservation laws in one space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity…
The negative integrable hierarchies of shallow water waves and dispersionless Toda lattice equations are considered. The integrability is shown by explicit construction of an infinite set of conservation laws.
Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational…
We prove the well-posedness of entropy weak solutions for a class of space-discontinuous scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem adding a viscosity term and we provide $L^\infty$…
We study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we…
We introduce a simple model of the time evolution of a binary mixture of compressible fluids including the thermal effects. Despite its apparent simplicity, the model is thermodynamically consistent admitting an entropy balance equation. We…
We consider a new nonlocal formulation of the water-wave problem for a free surface with an irrotational flow based on the work of Ablowitz, Fokas, and Musslimani and presented in the recent work of Oliveras. The main focus of the short…
The existing paradox between theory and computational experiment for weak solutions of systems of conservation laws in higher space dimensions is arguably resolved. Apparently successful computations are identified with underlying…
We build a minimal model of dissipative vortex dynamics in two spatial dimensions, subject to a kinematic constraint: dipole conservation. The additional conservation law implies anomalously slow decay rates for vortices. We argue that this…
We show that the simplest universality classes of fracton hydrodynamics in more than one spatial dimension, including isotropic theories of charge and dipole conservation, can exhibit hidden "quasiconservation laws", in which certain higher…
We study pathwise entropy solutions for scalar conservation laws with inhomogeneous fluxes and quasilinear multiplicative rough path dependence. This extends the previous work of Lions, Perthame and Souganidis who considered spatially…
We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the…