Fig. 6.

Comparison between numerical, multiple scales (MMS), and long-wave solutions for (a) $\beta =1$, $\tau =1.15$, $a=1\times 10^{-5}$ (numerical), $c_1=93$ (MMS), and $P\left(0\right)=0.87\times 10^{-4}$ (long-wave); (b) $\beta =1$, $\tau =1.25$, $a=1\times 10^{-5}$ (numerical), $c_1=66$ (MMS), and $P\left(0\right)=0.78\times 10^{-4}$ (long-wave). The initial condition used for integrating Eq. (18) is $P\left(t\right)=a\left(1+\frac{t}{1+\tau }\right),t\le 0$. The initial condition $P\left(0\right)$ is used to integrate the long-wave equation (59). The coefficient $c_1$ is used in the multiple scales solution (see Eq. (47)).