Local universality of singularity formation at multiple discontinuities

Establish whether, for Burgers’ equation with piecewise-continuous initial conditions having multiple discontinuities, the local complex-plane singularity structure near each discontinuity is governed by the corresponding step-down or step-up Heaviside solution according to the direction of the jump.

Background

The paper analyzes Heaviside step-down and step-up initial conditions and derives asymptotic descriptions for the infinitely many simple poles that emerge from a single discontinuity as time approaches zero. The authors then consider initial conditions with multiple discontinuities, including a top-hat profile and a saw-tooth profile, and observe that the pole patterns near each individual discontinuity resemble those of the corresponding single-jump problems.

The unresolved issue is whether this observed correspondence is a general principle: namely, whether the local behavior near every discontinuity can be determined independently from the local orientation of the jump, even when singularity trajectories from distinct discontinuities later interact or collide.

References

For other examples with more than one discontinuity, our conjecture is that the solutions in section~\ref{Heaviside_initial_conditions} will be relevant close to each individual discontinuity, depending on whether the real-valued initial condition steps down or up at the discontinuity.

— Complex singularities for Burgers' equation with piecewise-continuous initial conditions  (2609.04691 - Gentner et al., 4 Sep 2026) in Section 4, “Other initial conditions,” opening paragraph

Our conjecture is that for any initial condition in $Cn$ but not in $C{n+1}$, for $n\geq 0$, an infinite number of simple-pole singularities emerge from the points at which the $(n+1)$th derivative is discontinuous in a convex-shaped configuration that extends out to rays at angles $\pi/4$ and $3\pi/4$.

— Complex singularities for Burgers' equation with piecewise-continuous initial conditions  (2609.04691 - Gentner et al., 4 Sep 2026) in Section 5, “Discussion,” second paragraph