Existence of solitary waves in symmetric dimer lattices

Determine whether symmetric dimer lattices can possess solitary traveling-wave solutions, rather than only nanopteron solutions, and characterize whether such solutions occur only at special parameter values.

Background

The paper places its singularly perturbed fifth-order KdV equation in the context of nanopterons arising in heterogeneous FPUT lattices. Existing results establish nanopteron solutions for several dimer configurations and parameter limits, but the authors note that the corresponding nonexistence theory for solitary waves is incomplete. In particular, it is not known whether the oscillatory ripple in dimer nanopterons must always be nonzero, as it is for the scalar equation studied in the paper and for water waves with small surface tension.

Resolving this question would clarify whether dimer heterogeneity generically rules out solitary waves or whether isolated parameter values can support them, as occurs in related mass-in-mass lattices.

References

Despite this progress, numerous open questions about dimer nanopterons remain (to say nothing of how the KdV approximation might persist in lattices with more complicated material data). It is unclear if solitary waves in dimers definitely do not exist, as for eqn: the problem and the water wave problem with small surface tension.

eqn: the problem:

$\ep^2u^{(4)}+u''-u+u^2 = 0 $

— Phase-Shifted Nanopteron Solutions to a Singularly Perturbed Korteweg--de Vries Equation  (2608.19097 - Faver, 19 Aug 2026) in Section 1, Subsubsection “The lattice connection”