Let varphi be a highest-wave profile for a Whitham-type equation, with crest at x=0 and crest regularity varphi∈ C^s for some s∈(0,1). The conjecture asserts that, for every integer n≥ 0, the derivative partial_x^n varphi(x) has the complete asymptotic expansion at the crest predicted by the leading-order highest-wave expansion, as x→0. The supplied sources do not state the explicit asymptotic scale or coefficients of this expansion.

Whitham highest-wave asymptotic-expansion conjecture

The conjecture predicts a complete asymptotic expansion for derivatives of highest Whitham waves near their singular crests. Earlier work established the leading cusp and partial correction estimates; the new paper claims the entire expansion in a substantial parameter range.

Known results

2016: Existence of a highest cusped periodic Whitham wave and its leading crest asymptotic were proved.
2018: Convexity and a sharper local remainder estimate were established.
Earlier work: Leading-order constants and first-derivative asymptotics were determined for Whitham-type dispersive orders.

September 2026 higher-order proof

Ehrnström, Mæhlen, and Varholm report that strong induction proves all higher-order asymptotic limits for $s\in(0,1)$ , while interval arithmetic establishes the zeroth-order estimate only for $s\in[0.35,1)$ . Thus the full conjectured expansion is claimed in that latter range; extending the base estimate to all $s\in(0,1)$ remains open.

Current status (as of September 2026): A paper claims the conjecture for $s\in[0.35,1)$ , while the extension to all $s\in(0,1)$ and independent verification remain open.

Whitham highest-wave asymptotic-expansion conjecture

Primary source

Additional references

Progress summary

Solutions 0