Peter A. Clarkson

New similarity reductions of the Boussinesq equation

Some new similarity reductions of the Boussinesq equation, which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering, are presented. These new similarity reductions, including some new reductions to the first, second, and fourth Painleve equations, cannot be obtained using the standard Lie group method for finding group‐invariant solutions of partial differential equations; they are determined using a new and direct method that involves no group theoretical techniques.

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Abstract Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f (u) are considered. The method of differential Grobner 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 the nonlinear heat equation for cubic f (u) in terms of the roots of f (u) = 0.

The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation

Abstract In this paper it is shown that there exist exact solutions of the Fitzhugh-Nagumo equation which can be obtained using the nonclassical method for determining symmetry reductions of partial differential equations developed by Bluman and Cole [J. Math. Mech. 18 (1969) 1025], but which are not obtained using the direct method as developed by Clarkson and Kruskal [J. Math. Phys. 30 (1989) 2201].

Nonclassical symmetry reductions of the Boussinesq equation

Abstract In this paper we discuss symmetry reductions and exact solutions of the Boussinesq equation using the classical Lie method of infinitesimals, the direct method due to Clarkson and Kruskal and the nonclassical method due to Bluman and Cole. In particular, we compare and contrast the application of these three methods. We discuss the use of symbolic manipulation programs in the implementation of these methods and differential Grobner bases as a technique for solving the overdetermined systems of equations that arise. The relationship between the direct and nonclassical methods and other ansatz-based methods for deriving exact solutions of partial differential equations are also mentioned. To conclude we describe some of the important open problems in the field of symmetry analysis of differential equations.

Applications of analytic and geometric methods to nonlinear differential equations

Preface. I: Self-Dual Yang--Mills Equations. II: Completely Integrable Equations. III: Painleve Equations and Painleve Analysis. IV: Symmetries of Differential Equations. Author Index. Subject Index.

Algorithms for the nonclassical method of symmetry reductions

In this article the authors first present 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 differential 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 they demonstrate some algorithms which may be used to analyse, and often to solve, the resulting systems of overdetermined nonlinear PDEs. The authors take as their principal example a generalised Boussinesq equation, which arises in shallow water theory. Although the equation appears to be nonintegrable, the authors obtain an exact “two-soliton” solution from a nonclassical reduction.

Painlevé equations: nonlinear special functions

The six Painleve equations (PI-PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painleve equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.The Painleve equations may be thought of a nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations.In this paper, I discuss some of the remarkable properties which the Painleve equations possess including connection formulae, Backlund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painleve equation PII is used to illustrate these properties and some of the applications of PII are also discussed.

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

The second Painlev? 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 Painlev? equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev? analysis to ordinary differential equations. We extend these techniques in order to derive auto-B?cklund transformations for the second Painlev? hierarchy. We also derive a number of other B?cklund transformations, including a B?cklund transformation onto a hierarchy of equations, and a little known B?cklund transformation for itself. We then use our results on B?cklund transformations to obtain, for each member of the hierarchy, a sequence of special integrals.

Rogue waves, rational solutions, the patterns of their zeros and integral relations

The focusing nonlinear Schrodinger equation, which describes generic nonlinear phenomena, including waves in the deep ocean and light pulses in optical fibres, supports a whole hierarchy of recently discovered rational solutions. We present recurrence relations for the hierarchy, the pattern of zeros for each solution and a set of integral relations which characterizes them.

New similarity solutions for the modified Boussinesq equation

In this paper the author presents some new similarity solutions of the modified Boussinesq equation, which is a completely integrable soliton equation. These new similarity solutions include reductions to the second and fourth Painleve equations which are not obtainable using the standard Lie group method for finding group-invariant solutions of partial differential equations; they are determined using a new and direct method which involves no group theoretical techniques.