arXiv subjects

Peter A. Clarkson

Publications and source records attributed to Peter A. Clarkson.

27 records · Page 2Linked to original sources

arXiv2006

Painleve 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 $m=\tfrac12$, $n=-\tfrac12$, it is a real, generalized, pumped Maxwell-Bloch equation.

nlin.SI↗

arXiv1999

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(κu + γu^2\right)_{xx} + u u_{xxxx} +μu_{xxtt}+αu_x u_{xxx} + βu_{xx}^2, \ee where $α$, $β$, $γ$, $κ$ and $μ$ 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.

math.AP↗

arXiv1998

Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé analysis to ordinary differential equations. We extend these techniques in order to derive auto-Bäcklund transformations for the second Painlevé hierarchy. We also derive a number of other Bäcklund transformations, including a Bäcklund transformation onto a hierarchy of P34 equations, and a little known Bäcklund transformation for P2 itself. We then use our results on Bäcklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.

solv-int↗

arXiv1994

Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations

In this paper we describe Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation (PIV) $${\d^2 w\over\d z^2}={1\over2w}\left(\d w\over\d z\right)^2 + {3\over2}w^3 + 4zw^2 + 2(z^2-α)w+{β\over w},\eqno(1){\hbox to 16pt{\hfill}}$$ with $α$, $β$ constants. Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder functions as well as rational solutions will be derived. Included amongst these hierarchies are solutions of (1) for which $α=\pm\tfr12n$ and $β=-\tfr12n^2$, with $n$ an integer. These particular forms arise in quantum gravity and also satisfy a discrete analogue of the first Painlevé equation. We also obtain a number of exact solutions of the discrete fourth Painlevé equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(η-3δ^{-2}-z_n^2)x_n^2+μ^2\over (x_n+z_n+γ)(x_n+z_n-γ)},\eqno(2){\hbox to 16pt{\hfill}}$$}% {\narrower\noindent\baselineskip=12pt where $z_n=nδ$ and $η$, $δ$, $μ$ and $γ$ are constants, which, in an appropriate limit, reduces to PIV (1). A suitable factorisation of (2) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$ and the limits of these solutions yield rational solutions of (1).

solv-int↗

arXiv1994

Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations

In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + αu_x u_{xt} + βu_t u_{xx} - u_{xt} - u_{xx} = 0,\eqno(1)$$ where $α$ and $β$ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting $u_x=U$, have been discussed in the literature. The case $α=2β$ was discussed by Ablowitz, Kaup, Newell and Segur [{\it Stud.\ Appl.\ Math.}, {\bf53} (1974) 249], who showed that this case was solvable by inverse scattering through a second order linear problem. This case and the case $α=β$ were studied by Hirota and Satsuma [{\it J.\ Phys.\ Soc.\ Japan}, {\bf40} (1976) 611] using Hirota's bi-linear technique. Further the case $α=β$ is solvable by inverse scattering through a third order linear problem. In this paper a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole [{\it J.\ Math.\ Mech.\/}, {\bf 18} (1969) 1025]. The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth \p\ transcendents and Weierstrass elliptic functions. The nonclassical method yields a plethora of exact solutions of (1) with $α=β$ which possess a rich variety of qualitative behaviours. These solutions all like a two-soliton solution for $t<0$ but differ radically for $t>0$ and may be viewed as a nonlinear superposition of two solitons, one travelling to the left with arbitrary speed and the other to the right with equal and opposite speed.

solv-int↗

arXiv1994

New exact solutions for the discrete fourth Painlevé equation

In this paper we derive a number of exact solutions of the discrete equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(η-3δ^{-2}-z_n^2)x_n^2+μ^2\over (x_n+z_n+γ)(x_n+z_n-γ)},\eqno(1)$$ where $z_n=nδ$ and $η$, $δ$, $μ$ and $γ$ are constants. In an appropriate limit (1) reduces to the fourth \p\ (PIV) equation $${\d^2w\over\d z^2} = {1\over2w}\left({\d w\over\d z}\right)^2+\tfr32w^3 + 4zw^2 + 2(z^2-α)w +{β\over w},\eqno(2)$$ where $α$ and $β$ are constants and (1) is commonly referred to as the discretised fourth Painlevé equation. A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$. Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).

solv-int↗

arXiv1994

Algorithms for the Nonclassical Method of Symmetry Reductions

In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This algorithm requires significantly less computation time than that standardly used, and avoids many of the difficulties commonly encountered. The proof of correctness of the algorithm is a simple application of the theory of Grobner bases. In the second part we demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear PDEs. We take as our principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be non-integrable, we obtain an exact ``two-soliton'' solution from a nonclassical reduction.

solv-int↗

arXiv1994

On a Shallow Water Wave Equation

In this paper we study a shallow water equation derivable using the Boussinesq approximation, which includes as two special cases, one equation discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one by Hirota and Satsuma [J. Phys. Soc. Japan}, 40 (1976) 611--612]. A catalogue of classical and nonclassical symmetry reductions, and a Painleve analysis, are given. Of particular interest are families of solutions found containing a rich variety of qualitative behaviours. Indeed we exhibit and plot a wide variety of solutions all of which look like a two-soliton for t>0 but differ radically for t<0. These families arise as nonclassical symmetry reduction solutions and solutions found using the singular manifold method. This example shows that nonclassical symmetries and the singular manifold method do not, in general, yield the same solution set. We also obtain symmetry reductions of the shallow water equation solvable in terms of solutions of the first, third and fifth Painleve equations. We give evidence that the variety of solutions found which exhibit ``nonlinear superposition'' is not an artefact of the equation being linearisable since the equation is solvable by inverse scattering. These solutions have important implications with regard to the numerical analysis for the shallow water equation we study, which would not be able to distinguish the solutions in an initial value problem since an exponentially small change in the initial conditions can result in completely different qualitative behaviours.

solv-int↗

arXiv1993

Symmetry Reductions and Exact Solutions of a class of Nonlinear Heat Equations

Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gröbner bases is used both to find the conditions on $f(u)$ under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic $f(u)$ in terms of the roots of $f(u)=0$.

solv-int↗