 Odd pair of Gaussian profiles under diffusion

For $D,\gamma ,a>0$, apply the superposition principle to two translated solutions for Gaussian initial data for the heat equation. The displayed solution of $u_t=D{u}_{xx}$ has odd initial profile $A(e^{-\gamma (x-a{)}^2}-e^{-\gamma (x+a{)}^2})$. Its total spatial integral is zero. For $A>0$, it is positive on $x>0$ and negative on $x<0$ at every $t\ge 0$.

Writing it as $2A{s}^{-1/2}{e}^{-\gamma (x^2+a^2)/s}\sinh (2\gamma ax/s)$ shows how cancellation affects its late-time shape. On the diffusive scale $x=O(\sqrt{s})$, its leading form is $4A\gamma ax{s}^{-3/2}{e}^{-\gamma x^2/s}$. Thus the two lobe maxima asymptotically lie at $x=\pm \sqrt{s/(2\gamma )}$, while their absolute heights decay as $s^{-1}$. Their centers are not fixed at the initial Gaussian centers. The first spatial moment remains ${\int }_{\mathbb{R}}xu(x,t)dx=2Aa\sqrt{\pi /\gamma }$, as follows directly from the Gaussian integral and translation.