satisfy the following self-similar Burgers’ equation $$-\frac{1}{2i}  {\overline}W_1 + \left( \frac{3}{2}x + {\overline}W_1 \right) \partial_x  {\overline}W_1=0\,.$$ In addition to ${\overline}{W}_1$ defined above, the Burgers’ equation admits a countable family of smooth self-similar profiles [@Fontelos]. For each $i\in\mathbb N$, there exists a unique non-trivial analytic profile $W_i$ satisfying the ODE $$-\frac 1{2i}  {\overline}{W}_i + \left( \frac{(2i+1)x}{2i} + {\overline}{W}_i \right) \partial_x  {\overline}{W}_i = 0\,.$$ such that $$w_i(x,t)=(-t)^{\frac1{2i}}{\overline}{W}_i(x(-t)^{\frac{2i+1}{2i}})\,,$$ defines a self-similar solution to the Burgers’ equation. Unlike ${\overline}{W}_1$, the solutions ${\overline}{W}_{i}$ for $i>1$ are unstable: generic small perturbations of initial data $w_i(\cdot,0)$ lead to singularities described by the stable self-similar profile ${\overline}W_1$. Indeed a generic smooth perturbation of $w_i(x,0)$ leads to initial data with a global minimum at a point where the third derivative is positive, which by the discussion above leads to a shock with asymptotic profile ${\overline}W_1$.

The profiles ${\overline}{W}_{i}$ for $i>1$ are nevertheless stable modulo a finite co-dimension of initial data: Suppose we are given initial data $w_0$ with a global minimum, as a consequence of the invariances of Burgers’ equation, we may further assume $w(0)=0$ and $w_0'(0)=-1$. If we further assume that $w_0^{(n)}=0$ for $n=2,\dots,2i $ and that $w_0^{(2i+1)}=\nu>0$, then $$\begin{aligned}
 \label{limit:2}
\lim_{t\rightarrow 0} (-t)^{-\frac1{2i}}w(x(-t)^{-\frac{2i+1}{2i}},t)= \left(\frac{\nu}{(2i+1)!}\right)^{-\frac1{2i}}{\overline}{W}_i\left(\left(\frac{\nu}{(2i+1)!}\right)^{\frac1{2i}}x\right)\,,\end{aligned}$$ for all $x\in\mathbb 