Classification of conservation laws of compressible isentropic fluid flow in n>1 spatial dimensions

For the Euler equations governing compressible isentropic fluid flow with a barotropic equation of state (where pressure is a function only of the density), local conservation laws in $n>1$ spatial dimensions are fully classified in two primary cases of physical and analytical interest: (1) kinematic conserved densities that depend only on the fluid density and velocity, in addition to the time and space coordinates; (2) vorticity conserved densities that have an essential dependence on the curl of the fluid velocity. A main result of the classification in the kinematic case is that the only equation of state found to be distinguished by admitting extra $n$-dimensional conserved integrals, apart from mass, momentum, energy, angular momentum and Galilean momentum (which are admitted for all equations of state), is the well-known polytropic equation of state with dimension-dependent exponent $\gamma=1+2/n$. In the vorticity case, no distinguished equations of state are found to arise, and here the main result of the classification is that, in all even dimensions $n\geq 2$, a generalized version of Kelvin's two-dimensional circulation theorem is obtained for a general equation of state.Comment: 24 pages; published version with misprints correcte

Group classification of the Sachs equations for a radiating axisymmetric, non-rotating, vacuum space-time

We carry out a Lie group analysis of the Sachs equations for a time-dependent axisymmetric non-rotating space-time in which the Ricci tensor vanishes. These equations, which are the first two members of the set of Newman-Penrose equations, define the characteristic initial-value problem for the space-time. We find a particular form for the initial data such that these equations admit a Lie symmetry, and so defines a geometrically special class of such spacetimes. These should additionally be of particular physical interest because of this special geometric feature.Comment: 18 Pages. Submitted to Classical and Quantum Gravit

Supersymmetric Reciprocal Transformation and Its Applications

The supersymmetric analog of the reciprocal transformation is introduced. This is used to establish a transformation between one of the supersymmetric Harry Dym equations and the supersymmetric modified Korteweg-de Vries equation. The reciprocal transformation, as a B\"{a}cklund-type transformation between these two equations, is adopted to construct a recursion operator of the supersymmetric Harry Dym equation. By proper factorization of the recursion operator, a bi-Hamiltonian structure is found for the supersymmetric Harry Dym equation. Furthermore, a supersymmetric Kawamoto equation is proposed and is associated to the supersymmetric Sawada-Kotera equation. The recursion operator and odd bi-Hamiltonian structure of the supersymmetric Kawamoto equation are also constructed.Comment: 31 pages, expande

A Necessary Condition for existence of Lie Symmetries in Quasihomogeneous Systems of Ordinary Differential Equations

Lie symmetries for ordinary differential equations are studied. In systems of ordinary differential equations, there do not always exist non-trivial Lie symmetries around equilibrium points. We present a necessary condition for existence of Lie symmetries analytic in the neighbourhood of an equilibrium point. In addition, this result can be applied to a necessary condition for existence of a Lie symmetry in quasihomogeneous systems of ordinary differential equations. With the help of our main theorem, it is proved that several systems do not possess any analytic Lie symmetries.Comment: 15 pages, no figures, AMSLaTe

Ordinary differential equations which linearize on differentiation

In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.Comment: 9 page

Integrability of Lie systems through Riccati equations

Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.Comment: Corrected typo

Pluripolarity of Graphs of Denjoy Quasianalytic Functions of Several Variables

In this paper we prove pluripolarity of graphs of Denjoy quasianalytic functions of several variables on the spanning se

Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid

Using group theoretical methods, we analyze the generalization of a one-dimensional sixth-order thin film equation which arises in considering the motion of a thin film of viscous fluid driven by an overlying elastic plate. The most general Lie group classification of point symmetries, its Lie algebra, and the equivalence group are obtained. Similar reductions are performed and invariant solutions are constructed. It is found that some similarity solutions are of great physical interest such as sink and source solutions, travelling-wave solutions, waiting-time solutions, and blow-up solutions.Comment: 8 page

Nonlocal aspects of λ\lambda-symmetries and ODEs reduction

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $\lambda$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $\lambda$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In this framework, by embedding $\mathcal{Y}$ into a suitable system $\mathcal{Y}^{\prime}$ determined by the function $\lambda$, any $\lambda$-symmetry of $\mathcal{Y}$ can be recovered by a local symmetry of $\mathcal{Y}^{\prime}$. As a consequence, the reduction method of Muriel and Romero follows from the standard method of reduction by differential invariants applied to $\mathcal{Y}^{\prime}$.Comment: 13 page

Extreme value statistics and return intervals in long-range correlated uniform deviates

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider e.g., maximum relative to minimum are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself i.e., a Weibull distribution, reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form