Anthony D. Osborne

On the separability of the sine‐Gordon equation and similar quasilinear partial differential equations

The separability of the sine‐Gordon equation (SGE) is defined and studied in detail. We find a general class of dependent‐variable transformations under which the SGE is separable. This class may be reduced to a two‐parameter generalization of the usual transformation adopted, by requiring the transformations to reduce to the identity in the linear limit of the SGE (i.e., the Klein–Gordon equation). The method developed for studying the separability of the SGE is then applied to more general quasilinear equations and a discussion of the limitations of the method, and of separable solutions in general, is also given.

Separable solutions of the two-dimensional sine-Gordon equation

Abstract We discuss the reduction of the two-dimensional sine-Gordon equation to a pair of uncoupled, ordinary differential equations.

Separable solutions of the sine-Gordon equation in terms of a Painlevé transcendent

Abstract We show that, when expressed in characteristic coordinates, the two-dimensional sine-Gordon equation may be reduced to a separable form via the use of a fifth Painleve transcendent.

On the separability of the sine‐Gordon equation and similar quasilinear partial differential equations. II. Dependent‐ and independent‐variable transformations

In an earlier paper, we investigated the separability of the sine‐Gordon equation (SGE), and of similar quasilinear partial differential equations, under transformations of the dependent variable (i.e, of the codomain). We found, in particular, that there is a general class of dependent‐variable transformations which leads to separable forms of the SGE. In this paper, we extend our previous analysis to include independent‐ as well as dependent‐variable transformations (i.e., transformations of both the domain and codomain) and treat, in detail, constant coefficient equations of the first and second orders. We illustrate our method by applying it to the SGE and find combinations of domain and codomain transformations which reduce the equation to separable forms. Some of these transformations lead to known solutions of the SGE, but others give new solutions expressible in terms of a fifth Painleve transcendent. Our method can, in principle, be used to map out the space of separable solutions of the SGE and ...

A new approach to Special Relativity

We present a new commonsense approach to Special Relativity based on direct observation instead of from the usual rationale conceived in terms of light travelling invisibly in vacuo. This approach permits the construction of a model of the time dilation effect.

Separable solutions of nonlinear wave equations

Abstract It is shown how to obtain separable solutions, which include travelling-wave solutions as a special case, of a large class of nonlinear wave equations, including the Korteweg-de Vries-Burgers equation and Fishers equation.