arXiv subjects

P. A. Clarkson

Publications and source records attributed to P. A. Clarkson.

4 recordsLinked to original sources

arXiv1999

Discrete equations and the singular manifold method

The Painleve expansion for the second Painleve equation (PII) and fourth Painleve equation (PIV) have two branches. The singular manifold method therefore requires two singular manifolds. The double singular manifold method is used to derive Miura transformations from PII and PIV to modified Painleve type equations for which auto-Backlund transformations are obtained. These auto-Backlund transformations can be used to obtain discrete equations.

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arXiv1996

Symmetries of a class of Nonlinear Third Order Partial Differential Equations

In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations $u_t -εu_{xxt} +2κu_x= u u_{xxx} +αu u_x +βu_x u_{xx}$ where $ε$, $κ$, $α$ and $β$ are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave solutions: the Fornberg-Whitham equation, for the parameters $ε=1$, $α=-1$, $β=3$ and $κ=\tfr12$, admits a wave of greatest height, as a peaked limiting form of the travelling wave solution; the Rosenau-Hyman equation, for the parameters $ε=0$, $α=1$, $β=3$ and $κ=0$, admits a ``compacton'' solitary wave solution; and the Fuchssteiner-Fokas-Camassa-Holm equation, for the parameters $ε=1$, $α=-3$ and $β=2$, has a ``peakon'' solitary wave solution. A catalogue of symmetry reductions for equation (1) is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole.

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arXiv1996

Application of Uniform Asymptotics to the Second Painlev{é} Transcendent

In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{é} equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort. Our method is reliant on finding uniform approximations of differential equations of the generic form ${d^2ϕ}/{dη^2} = - ξ^2F(η,ξ)ϕ$ as the complex-valued parameter $ξ\to \infty.$ The details of the treatment rely heavily on the locations of the zeros of the function $F$ in this limit. If they are isolated then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of $F$ coalesce as $|ξ| \to \infty$ then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the ``classical'' connection problem associated with the second Painlev{é} transcendent. Future papers will show how the technique can be applied with very little change to the other Painlev{é} equations, and to the wider problem of the asymptotic behaviour of the general solution to any of these equations.

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arXiv1994

Symmetries and Exact Solutions of a 2+1-dimensional Sine-Gordon System

We investigate the classical and nonclassical reductions of the $2+1$-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalisation of the sine-Gordon equation. A family of solutions obtained as a nonclassical reduction involves a decoupled sum of solutions of a generalised, real, pumped Maxwell-Bloch system. This implies the existence of families of solutions, all occurring as a decoupled sum, expressible in terms of the second, third and fifth Painlevé transcendents, and the sine-Gordon equation. Indeed, hierarchies of such solutions are found, and explicit transformations connecting members of each hierarchy are given. By applying a known Bäcklund transformation for the system to the new solutions found, we obtain further families of exact solutions, including some which are expressed as the argument and modulus of sums of products of Bessel functions with arbitrary coefficients. Finally, we prove the sine-Gordon system has the Painlevé property, which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalised, real, pumped Maxwell-Bloch system.

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