A Reversible System.

The fourth-order differential equation

$$u'''' + P u'' + u -u^3 =0$$

arises in a number of contexts, e.g., as the travelling-wave equation for a nonlinear-Schrödinger equation with fourth-order dissipation [#!BuAk:95!#] and as a model of a strut on a symmetric nonlinear elastic foundation [#!HuBoTh:89!#]. It may be expressed as a system

$$\left \{ \begin{array}{rcl} \dot{u_1} & = & u_2 \\ \dot{u_2} & = ... ...\dot{u_3} & = & u_4 \\ \dot{u_4} & = & -P u_3 - u_1 + u_1^3 \end{array} \right.$$ (26.1)

Note that (26.1) is invariant under two separate reversibilities

$$R_1: (u_1,u_2,u_3,u_4,t) \mapsto (u_1,-u_2,u_3,-u_4,-t)$$ (26.2)

and

$$R_2: (u_1,u_2,u_3,u_4,t) \mapsto (-u_1,u_2,-u_3,u_4,-t)$$ (26.3)

First, we copy the demo into a new directory

@dm rev

For this example, we shall make two separate starts from data stored in equation and data files rev.c.1, rev.dat.1 and rev.c.3, rev.dat.3 respectively. The first of these contains initial data for a solution that is reversible under $R_1$ and the second for data that is reversible under $R_2$.