Nonexistence of accelerating shape-preserving wave packets for quartic dispersion

Prove that the fourth-order Schrödinger equation with quartic dispersion admits no rigidly accelerating, shape-preserving wave packets beyond the hyper-Airy ansatz, including profiles other than the fourth-order Airy function.

Background

The paper first analyzes a fourth-order Schrödinger equation and shows that a fourth-order Airy profile cannot undergo nonconstant translation while preserving its shape under a physical multiplicative potential. The coefficient conditions force the translation to be constant, so the specific hyper-Airy construction does not yield an accelerating nonspreading packet.

The authors then consider a more general translated profile with an arbitrary shape function. Their formal operator calculation likewise forces the phase gradient to vanish and consequently forces the translation speed to be constant. Nevertheless, the passage explicitly presents the broader nonexistence claim as a conjecture, making the complete exclusion of all possible fourth-order accelerating shape-preserving packets an explicitly stated unresolved claim.

References

While the above analysis shows that the $\mathrm{Ai}_{4}$ wavepacket does not accelerate while preserving its shape under eq:4th_order_schrodinger, we conjecture that there are no such wavepackets in the fourth order Schrödinger equation.

— Is there an accelerating nonspreading wave packet in the Schrödinger equation with higher even-order dispersions?  (2609.25799 - Adriano et al., 22 Sep 2026) in Section 2, subsection “Nonexistence of Rigidly Accelerating Shape-Preserving Solutions”