Uniqueness of the π-phase configuration in avoiding binary soliton merger

Prove that, for two equal-amplitude Schrödinger–Helmholtz solitons launched with equal and opposite velocities in a periodic domain, an initial phase difference Δφ_i=π uniquely avoids a merger for arbitrarily long times in the limit of infinite numerical precision.

Background

The authors systematically paper collisions of two identical Schrödinger–Helmholtz solitons with varying phase differences, observing prompt merger for in-phase collisions (Δφ_i=0), delayed merger for other phase differences, and an apparent non-merging case for Δφ_i=π under increasing numerical precision.

They conjecture that only the anti-phase case (Δφ_i=π) remains non-merging indefinitely at infinite precision, highlighting a potential selection rule for merger dynamics and phase-synchrony effects in nonintegrable soliton collisions.

References

We expect that if we were to realise infinite precision, the case with $\Delta\phi_{\rm i}=\pi$ would never result in a merger. We conjecture that this would be the only case in which the binary soliton merger could be avoided up to arbitrary time.

— A bound state attractor in optical turbulence (2410.12507 - Colleaux et al., 16 Oct 2024) in Section 5 (Collisions of SHE solitons)