Universality of the two-soliton bound state attractor in the Schrödinger–Helmholtz equation

Establish whether the two-soliton bound state observed in numerical simulations of the Schrödinger–Helmholtz equation with nonlocality parameter β>0 is the universal statistical attractor, i.e., prove that broad classes of initial conditions evolve towards a solitary double-soliton bound state surrounded by weakly nonlinear waves.

Background

The paper numerically studies the nonintegrable Schrödinger–Helmholtz equation (SHE) and repeatedly observes a robust two-soliton bound state that self-organizes from diverse initial conditions, including random wave spectra and injected solitons. The authors propose that this bound state is the preferred statistical attractor for nonlocal media, contrasting earlier literature that suggested a single-soliton attractor in related contexts.

To elevate this proposal from empirical evidence to a rigorous result, the authors explicitly note the need to show that the bound state is indeed a universal attractor across generic initial data, in the sense envisioned in prior theoretical works on soliton turbulence and statistical mechanics of nonlinear Schrödinger-type systems.