Extension of the integrable, (1+1) Gross-Pitaevskii equation to chaotic behaviour and arbitrary dimensions

The integrable, (1+1) Gross-Pitaevskii (GP-) equation with hermitian property is extended to chaotic behaviour as part of general complex fields within the sl(2,C) algebra for Lax pairs. Furthermore, we prove the involution property of conserved quantities in the case of GP-type equations with an arbitrary external potential. We generalize the approach of Lax pair matrices to arbitrary spacetime dimensions and conclude for the type of nonlinear equations from the structure constants of the underlying algebra. One can also calculate conserved quantities from loops within the (N-1) dimensional base space and the mapping to the manifold of the general SL(n,C) group or a sub-group, provided that the resulting fibre space is of nontrivial homotopic kind.

💡 Research Summary

The paper presents a comprehensive extension of the well‑known (1+1)‑dimensional integrable Gross‑Pitaevskii (GP) equation by embedding it into the complex Lie algebra sl(2,ℂ) and by allowing the associated Lax pair to take values in this non‑Hermitian algebra. This move replaces the traditional Hermitian Lax operators with complex‑valued matrices that depend on a spectral parameter λ living on the full complex plane. By permitting λ to trace non‑trivial paths in the complex plane, the authors show that the formerly integrable GP dynamics can be driven into chaotic regimes while still retaining a hierarchy of conserved quantities.

A central theoretical contribution is the proof that, even when the external potential V_ext(x,t) is an arbitrary, possibly time‑dependent function, the infinite set of conserved charges Q_k = ∮ Tr(L^k) dx (k ∈ ℕ) remain in involution. The proof proceeds by (i) demonstrating that the zero‑curvature condition ∂_tU − ∂_xV +