Nonzero phase shift in the phase-selected nanopteron construction

Determine whether the phase shift selected when a nonzero exponentially small ripple amplitude is prescribed is necessarily nonzero for the singularly perturbed Korteweg–de Vries equation, by resolving whether the selection equation can admit \(\Theta_{q,b}^{\epsilon,A}(\eta)=0\).

Background

The paper constructs nanopterons by prescribing a small nonzero amplitude and selecting a phase shift through an intermediate-value argument. The resulting phase shift is shown to be small, but the argument does not exclude the value zero.

A nonzero phase shift is obtained in Lombardi’s related construction because an additional phase shift in the solvability condition forces it. The authors’ formulation lacks that mechanism, leaving open whether the selected phase shift for the present scalar equation can vanish.

References

We do not rule out here the possibility that \Theta_{q,b}{\ep,A}(\eta) = 0$. In this case, eqn: sel mech PS 2 becomes

eqn: sel mech PS 2:

$\sin((\ep\omega_{\ep})\theta) + \M_{q,b}^{\ep}(\eta,\alpha,\theta) = 0, $

— Phase-Shifted Nanopteron Solutions to a Singularly Perturbed Korteweg--de Vries Equation  (2608.19097 - Faver, 19 Aug 2026) in Section 7, Subsection “The problem for \(\theta\)”