= Odd pair of Gaussian profiles under diffusion
{title2=$u(x,t)=A s^{-1/2}\{e^{-\gamma(x-a)^2/s}-e^{-\gamma(x+a)^2/s}\},\quad s=1+4\gamma Dt$}
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 $u_t=Du_{xx}$ 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 $t\geq0$.
Writing it as $2As^{-1/2}e^{-\gamma(x^2+a^2)/s}\sinh(2\gamma ax/s)$ 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.
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