Symmetries of a class of nonlinear fourth order partial differential equations

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations \be u_{tt} = \left(\kappa u + \gamma u^2\right)_{xx} + u u_{xxxx} +\mu u_{xxtt}+\alpha u_x u_{xxx} + \beta u_{xx}^2, \ee where $\alpha$, $\beta$, $\gamma$, $\kappa$ and $\mu$ are constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further equation (1) is a ``Boussinesq-type'' equation which arises as a model of vibrations of an anharmonic mass-spring chain and admits both ``compacton'' and conventional solitons. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole. In particular we obtain several reductions using the nonclassical method which are no} obtainable through the classical method

Painlevé Analysis and Similarity Reductions for the Magma Equation

In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE test on their other symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions and these are completely integrable. The case when m = 0, n = −1 is related to the Hirota-Satsuma equation and for m = ½, n = −½, it is a real, generalized, pumped Maxwell-Bloch equation

Classical and nonclassical symmetries of a generalized Boussinesq equation

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions $f(u)$ for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be derived

The non-linear Schr\"odinger equation and the conformal properties of non-relativistic space-time

The cubic non-linear Schr\"odinger equation where the coefficient of the nonlinear term is a function $F(t,x)$ only passes the Painlev\'e test of Weiss, Tabor, and Carnevale only for $F=(a+bt)^{-1}$, where $a$ and $b$ are constants. This is explained by transforming the time-dependent system into the constant-coefficient NLS by means of a time-dependent non-linear transformation, related to the conformal properties of non-relativistic space-time. A similar argument explains the integrability of the NLS in a uniform force field or in an oscillator background.Comment: Thoroughly revised version, in the light of new interest in non-relativistic conformal tranformation, with a new reference list. 8 pages, LaTex, no figures. To be published in Int. J. Theor. Phy

On Orthogonal and Symplectic Matrix Ensembles

The focus of this paper is on the probability, $E_\beta(0;J)$, that a set $J$ consisting of a finite union of intervals contains no eigenvalues for the finite $N$ Gaussian Orthogonal ($\beta=1$) and Gaussian Symplectic ($\beta=4$) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary ($\beta=2$) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlev\'e II function.Comment: 34 pages. LaTeX file with one figure. To appear in Commun. Math. Physic

Experimental and modeling study of the autoignition of 1-hexene/iso-octane mixtures at low temperatures

Autoignition delay times have been measured in a rapid compression machine at Lille at temperatures after compression from 630 to 840 K, pressures from 8 to 14 bar, \Phi = 1 and for a iso octane/1 hexene mixture containing 82% iso-octane and 18% 1 hexene. Results have shown that this mixture is strongly more reactive than pure iso-octane, but less reactive than pure 1 hexene. It exhibits a classical low temperature behaviour, with the appearance of cool flame and a negative temperature coefficient region. The composition of the reactive mixture obtained after the cool flame has also been determined. A detailed kinetic model has been obtained by using the system EXGAS, developed in Nancy for the automatic generation of kinetic mechanisms, and an acceptable agreement with the experimental results has been obtained both for autoignition delay times and for the distribution of products. A flow rate analysis reveals that the crossed reactions between species coming from both reactants (like H-abstractions or combinations) are negligible in the main flow consumption of the studied hydrocarbons. The ways of formation of the main primary products observed and the most sensitive rate constants have been identified

Fredholm Determinants, Differential Equations and Matrix Models

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only the abstract and decreases length of typeset versio

Point vortices and polynomials of the Sawada-Kotera and Kaup-Kupershmidt equations

Rational solutions and special polynomials associated with the generalized K_2 hierarchy are studied. This hierarchy is related to the Sawada-Kotera and Kaup-Kupershmidt equations and some other integrable partial differential equations including the Fordy-Gibbons equation. Differential-difference relations and differential equations satisfied by the polynomials are derived. The relationship between these special polynomials and stationary configurations of point vortices with circulations Gamma and -2Gamma is established. Properties of the polynomials are studied. Differential-difference relations enabling one to construct these polynomials explicitly are derived. Algebraic relations satisfied by the roots of the polynomials are found.Comment: 23 pages, 8 figure

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation

We extend the formalism of integrable operators a' la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi-infinite interval and to matrix integral operators with a kernel of the form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painleve' II (recently introduced by Retakh and Rubtsov), a related noncommutative equation of Painleve' type. We construct a particular family of solutions of the noncommutative Painleve' II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painleve' II. Such a solution plays the same role as its commutative counterpart relative to the Tracy-Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.Comment: 46 pages, no figures (oddly