We discuss the problem of consistent description of higher spin massive fields coupled to external gravity. As an example we consider massive field of spin 2 in arbitrary gravitational field. Consistency requires the theory to have the same number of degrees of freedom as in flat spacetime and to describe causal propagation. By careful analysis of lagrangian structure of the theory and its cons...

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...

For the initial-boundary value problem (IBVP) if isentropic gas dynamics written in Lagrangian coordinates written in terms of Riemann invariants we show how to obtain necessary conditions for existence of global smooth solution using the Lax technique. Under these conditions we formulate the existence theorem in the class of piecewise-smooth functions. A priori estimates with respect to the in...

Approximations of the Sedov self-similar solution for a strong point explosion in a medium with the power-law density distribution ρ ∝ r−m are reviewed and their accuracies are analyzed. The Taylor approximation is extended to cases m / = 0 and spherical, cylindrical and plane geometry. Two approximations of the solution are presented in the Lagrangian coordinates for all types of geometry. The...

In order to investigate corrections to the common KdV approximation for surface water waves in a canal, we derive modulation equations for the evolution of long wavelength initial data. We work in Lagrangian coordinates. The equations which govern corrections to the KdV approximation consist of linearized and inhomogeneous KdV equations plus an inhomogeneous wave equation. These equations are e...

We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa– Holm system in Lagrangian variable...

Approximations of the Sedov self-similar solution for a strong point explosion in a medium with the power-law density distribution ρo ∝ r−m are reviewed and their accuracy are analyzed. Taylor approximation is extended to cases m 6= 0. Two approximations of the solution are presented in the Lagrangian coordinates for spherical, cylindrical and plane geometry. These approximations may be used fo...

We analyse the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an...

We present a topological Lagrangian field theory that is geometrically similar to Yang-Mills(-Higgs) theory, and study the Bogomol'nyi solitons contained within this theory. The topological field theory may provide an example of a dual field theory. The existence of a dual field theory to Yang-Mills(-Higgs) theory was conjectured by Montonen and Olive.