The WWB cylindrical pulse solution and the equivalent G_2 solution are analysed particularly as the wave is reflected off the axis. Apparent phase shifts are revealed that are relevant to the discussion of whether or not phase shifts occur in gravitational soliton interactions.

In this paper we discuss the concepts of short-range and long-range correlated stochastic processes and we investigate the presence of such variables in two model complex systems. The selected model systems are DNA sequences of complete genomes and financial time series of equities traded in a stock market. Specifically, by starting from our research results, we discuss the statistical properties of (i) coding and non-coding regions of DNA and (ii) equity returns and volatility in financial markets. The stylized facts about these variables are presented and discussed with a focus on the statistical tools already used and/or still needed to better characterize these model complex systems.

We consider solutions of the Sine-Gordon equation which are given in terms of elliptic functions. In the first part of the paper we devote our attention to solutions obtained through a separation ansatz: $\phi=\alpha/\beta$. As a result, the equation decouples into two non-linear equations for $\alpha$ and $\beta$, the solutions of which turn out to be Jacobi elliptic functions. The explicit form of the functions is determined by the zeroes of some quartic polynomials. According to the nature of these, we shall distinguish several cases and provide a full classification of the solutions. Jacobi elliptic functions also appear in the solutions which arise by considering a Bäcklund transformation to a seed which describes a steady progressing wave. This particular choice of the seed considerably simplifies the two Riccati equations associated to the Bäcklund transformation. As a result these equations can be solved exactly. As before, several cases are considered. Computer generated 3-D plots of typical solutions are shown.

Soliton solutions of Einstein's field equations [1-2] for space-times with two Killing vectors are exact solutions obtained using the solution-generating-techniques that resemble the well known Inverse Scattering Methods that have been widely used in the solution of certain nonlinear equations (i.e. Kotewrg-de Vries, non-linear Schrodinger, *etc*). In these contexts, the solitons tetain their shape and velocity on interaction but experience a characteristic time shift. It is not clear whether or not gravitational solitons also experiencea similar behaviour [3-4].

By considering the Weber-Wheeler-Bonnor solution, we have recently shown [5] that a kind of "time shift" occurs as gravitational wave pulses interact near the axis for both cylindrical waves and their cosmological equivalent - even though they are solution of a linear equation. In order to analyse genuinely non-linear effects we must:

To do this we will study a 4-soliton non-diagonal solution propagating in Kasner background.

**References**

Last modified **14 February 2001**