Completeness of the squared eigenfunction connection for the DP Lax pair

Determine whether all solutions and, in particular, all L^2 eigenfunctions of the linearized spectral problem A[u_0]v=λv about a smooth solitary wave u_0 of the Degasperis–Procesi equation can be constructed via quadratic combinations of eigenfunctions of its Lax pair and adjoint (a squared eigenfunction connection); that is, prove completeness of the squared eigenfunction representation for the DP equation.

Background

The paper derives a new squared eigenfunction connection for the DP equation that maps products of Lax pair eigenfunctions to solutions of the linearized equation. For many integrable PDEs (e.g., Camassa–Holm), such connections are known to be complete, enabling full spectral resolution.

For the DP equation, however, completeness is not established. The authors can show completeness on the imaginary axis for genus-1 (traveling wave) settings to rule out embedded point spectrum, but a general completeness theorem remains unavailable.

References

Unfortunately, however, it is not known if all eigenfunctions can be constructed in this way: indeed, while the “completeness” of the squared eigenfunction connection is known in many cases, such a completeness result is not available for the DP equation.

— Linear Asymptotic Stability of the Smooth 1-Solitons for the Degasperis-Procesi Equation  (2604.03060 - Deng et al., 3 Apr 2026) in Section 3 (Analysis of the Point Spectrum), discussion preceding Section 4