Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation

We show how quasi-continuum methods can be used to construct approximate solutions of nonlinear differential delay equations derived from symmetry reductions of the discrete nonlinear telegraph equation. Travelling wave solutions are proven to exist and the existence of solutions to two other symmetry reductions are studied. Two of the less familiar reductions are studied; the first supports both a one-parameter family of single-pulse solitary-wave-type solutions and a one-parameter family of periodic waves. The size and shape of these waves are examined using the quasi-continuum technique; this approximates the differential-difference equation with a higher-order differential equation, which is integrated and analysed using phase plane techniques. In the large-amplitude limit, the shape of the pulse approaches a limiting form which has a corner at its peak. The manner of this approach is elucidated using matched asymptotic expansions. The second reduction, though differing only by the addition of a single term, appears not to support the solitary-wave type of solution-even in the limit where the additional term is premultiplied by an asymptotically small constant.