Rigorous dynamics of the full time-dependent managed system

Establish a rigorous description of the solutions of the full two-component Gross–Pitaevskii system with time-dependent inter-species interaction coefficient under nonlinearity management, including their collapse or global-existence behavior for specified management profiles and initial data.

Background

The paper rigorously establishes the absence of blow-up only for the averaged two-component system. The nonlinear quantum-pressure term generated by rapid management is positive and scales as L{-6} under norm-preserving concentration, whereas the kinetic and mean-field contributions scale as L{-2}; consequently, the averaged Hamiltonian is bounded from below at fixed component norms.

The original Gross–Pitaevskii equations retain explicitly time-dependent inter-species coupling and need not share the averaged system’s global behavior. Numerical simulations indicate that the managed full system can remain localized for long times, but the authors explicitly state that a rigorous description of its behavior, including dependence on the management form and initial data, remains unresolved.

References

We stress that this statement concerns the averaged model. The behavior of the solutions of the full vector NLSE with time-dependent coefficients depends on the form of the management and on the initial data, and its rigorous description remains an open problem.

— Nonlinearity management of matter-wave vector solitons of Bose-Einstein condensates in two dimensions  (2610.11536 - Abdullaev et al., 8 Oct 2026) in Section 2, subsection “Suppression of collapse in the averaged system” (immediately before Section 3)