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

(√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[1/8 a c (b - d)]

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 x² - 3 a x³)/(3a)

In all cases (except when -8/3 < a < -1/√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)