It is known that imaginary potentials are a source of particles when included into the Gross-Pitaevskii equation. As far as the dynamics of a Bose gas is concerned, is it possible that these potentials could be a source for chaos? Did anyone investigate this before?
Not necessarily. There do exist non-Hermitian Hamiltonians that do describe integrable motion.
Among all dynamical systems, there are those not derived from an Hamiltonian, that are chaotic (i.e. with mixing and exponential asymptotic divergence of nearby trajectories).
The set of all dynamical systems, the solutions of dx/dt = F(x(t)), where x is a n-dimensional real vector, can be sorted in subsets according to the property of asymptotic phase space volumes:
1) always contracting volumes (dissipative systems),
2) always expanding volumes (anti-dissipative),
3) expanding or contracting volumes including conserving volumes for some solutions,
4) conserving all volumes,
5) with indefinite asymptotic behaviour.
Among the 4th) category, one finds the subset of Hamiltonian systems, which have the additional restriction that phase space coordinates appear as pair of conjugate coordinates, which individually conserve phase space area.
There exists a Hilbert space approach that enables to reduce arbitrary classical nonlinear dynamical systems with analytic nonlinearities to the Schrodinger-like equation with non-Hermitian boson Hamiltonian (see K. Kowalski Methods of
Hilbert Spaces in the Theory of Nonlinear Dynamical Systems (World Scientific, Singapore, 1994)). A particular case of the formalism is the Carleman linearization also referred to as the Carleman embedding technique described in the monograph: K. Kowalski and W.-H. Steeb, Nonlinear Dynamical Systems and Carleman Linearization (World Scientific, Singapore, 1991).
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