Equation for the background [in my PDEs]

If u[t, x] is independent of x, as it is sufficiently far away from the main pattern, then the partial differential equation on page 165 reduces to the ordinary differential equation

u''[t] == (1 - u[t]^^{2})(1 + a u[t]) u[0] == u'[0] == 0

For a = 0, the solution to this equation can be written in terms of Jacobi elliptic functions as

(Sqrt[3] JacobiSN[t/3^(1/4), 1/2]^^{2}) / (1 + JacobiCN[t/3^(1/4), 1/2]^^{2})

In general the solution is

(b d JacobiSN[r t, s]^^{2})/(b - d JacobiCN[r t, s]^^{2})

where

r = -Sqrt[a c (b - d)/8] s = (d (c - b))/(c (d - b))

and b, c, d are determined by the equation

(x-b)(x-c)(x-d) == -(12 + 6 a x - 4 x^^{2} - 3 a x^^{3})/(3a)

In all cases (except when -8/3 < a < -1/Sqrt[6]), the solution is periodic and non-singular. For a = 0, the period is 2 3^^{1/4} EllipticK[1/2] ≅ 4.88. For a = 1, the period is about 4.01; for a = 2, it is about 3.62; while for a = 4, it is about 3.18. For a = 8/3, the solution can be written without Jacobi elliptic functions, and is given by

3 Sin[Sqrt[5/6] t]^^{2}/(2 + 3 Cos[Sqrt[5/6] t]^^{2})