Dubrovin's universality conjecture for Hamiltonian perturbations

Let $u_0=u_0(x,t)$ be a smooth solution to the unperturbed equation

$$u_t+a(u)u_x=0,$$

defined for all $x\in\mathbb R$ and $0\leq t<t_0$, and monotone in $x$ for any $t$. Consider a Hamiltonian perturbation

$$u_t+a(u)u_x+\epsilon\left[b_1(u)u_{xx}+b_2(u)u_x^2\right]+\epsilon^2\left[b_3(u)u_{xxx}+b_4(u)u_xu_{xx}+b_5(u)u_x^3\right]+\cdots=0,$$

where the $b_j$ are smooth functions. A gradient catastrophe occurs at a point $(x_0,t_0)$ of the unperturbed solution.

Dubrovin's universality conjecture. There exists a solution $u=u(x,t;\epsilon)$ to the perturbed equation on the same domain with

$$u(x,t;\epsilon)=u_0(x,t)+\epsilon^2u_1(x,t)+\epsilon^4u_2(x,t)+o(\epsilon^4),$$

where $u_1$ and $u_2$ can be written down explicitly. The equation $P_I^2$ has a unique solution $y=y(x,T)$ smooth for all real $x$ and all values of $T$. The generic solution can be extended up to $t=t_0+\delta$ for sufficiently small positive $\delta=\delta(\epsilon)$ and, near $(x_0,t_0)$, satisfies

$$u(x,t;\epsilon)=u_0(x,t)+a\epsilon^{2/7}y\left(b\epsilon^{-6/7}(x-c(t-t_0)-x_0),d\epsilon^{-4/7}(t-t_0)\right)+O(\epsilon^{4/7}),$$

for constants $a,b,c,d$ depending on the hyperbolic equation, the solution $u$, and the perturbation. Here $y$ is the unique smooth solution specified above.

This conjecture predicts universal behavior near gradient catastrophe for generic Hamiltonian perturbations and connects the small-dispersion limit with the special pole-free solution of the fourth-order analogue of the Painlevé I equation. The source attributes the conjecture to Dubrovin; its resolution status is not specified here.