Quasiparticle solutions for the nonlocal NLSE with an anti-Hermitian term in semiclassical approximation

Abstract

We deal with the $n$-dimensional nonlinear Schrödinger equation (NLSE) with a cubic nonlocal nonlinearity and an anti-Hermitian term, which is widely used model for the study of open quantum system. We construct asymptotic solutions to the Cauchy problem for such equation within the formalism of semiclassical approximation based on the Maslov complex germ method. Our solutions are localized in a neighbourhood of few points for every given time, i.e. form some spatial pattern. The localization points move over trajectories that are associated with the dynamics of semiclassical quasiparticles. The Cauchy problem for the original NLSE is reduced to the system of ODEs and auxiliary linear equations. The semiclassical nonlinear evolution operator is derived for the NLSE. The general formalism is applied to the specific one-dimensional NLSE with a periodic trap potential, dipole-dipole interaction, and phenomenological damping. It is shown that the long-range interactions in such model, which are considered through the interaction of quasiparticles in our approach, can lead to drastic changes in the behaviour of our asymptotic solutions.

arXiv comment38 pages, 6 figures; version prior to peer review

References in corpus (21)

• Theory of Bose-Einstein condensation in trapped gases
• Non-Hermitian Physics
• Quantum filaments in dipolar Bose-Einstein condensates
• Phenomenological damping in trapped atomic Bose-Einstein condensates
• Solitons in quasi one-dimensional Bose-Einstein condensates with competing dipolar and local interactions
• Quantitative Derivation of the Gross-Pitaevskii Equation
• Observation of vortices and vortex stripes in a dipolar Bose-Einstein condensate
• Droplet formation in a Bose-Einstein condensate with strong dipole-dipole interaction
• Quantum many-body fluctuations around nonlinear Schrödinger dynamics
• The Trajectory-Coherent Approximation and the System of Moments for the Hartree-Type Equation
• Algebras of integrals of motion for the Hamilton-Jacobi and Klein-Gordon-Fock equations in spacetime with a four-parameter groups of motions in the presence of an external electromagnetic field
• Optical Spatial Shock Waves in Nonlocal Nonlinear Media
• Dissipative Dynamics of an Open Bose Einstein Condensate
• A generalized Haus master equation model for mode-locked class-B lasers
• Semiclassical Propagation of Coherent States for the Hartree equation
• Semiclassical approach to the nonlocal nonlinear Schrödinger equation with a non-Hermitian term
• Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates
• On Nonlocal Gross-Pitaevskii Equations with Periodic Potentials
• Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation
• Particle and wave dynamics of nonlocal solitons in external potentials
• Focusing dynamics of 2D Bose gases in the instability regime

Cited by in corpus (1)

• Semiclassical states localized on a one-dimensional manifold and governed by the nonlocal NLSE with an anti-Hermitian term