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arXiv:2605.30704

Nonlinear Schrodinger Equations for Dense Bose Fluid and He4 Film at Low Temperatures

Vladimir I. Kruglov

Published May 29, 2026• 0 citations
arXivHTMLPDF

Abstract

We have derived the nonlinear Schrodinger equation generalizing the Gross-Pitaevskii (GP) equa tion for dilute Bose gas. The derivation is based on the Hartree-Fock time-dependent mean-field theory with an arbitrary intermolecular interaction potential. It is shown that obtained nonlinear Schrodinger equation with appropriate redefinition of coefficients can be used for description of dense Bose fluid at low temperatures. We also present the Schrodinger type equation describing the superfluid component of helium in two fluid hydrodynamics. This approach leads to quantum correction for superfluid component of velocity in two fluid hydrodynamics. We also have derived the nonlinear Schrodinger equation for superfluid He4 film at low temperatures. It is shown that this Schrodinger type equation for superfluid He4 film leads to phonon-roton dispersion relation for elementary excitation at low temperatures.

DOI
10.48550/arxiv.2605.30704

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We have derived the nonlinear Schrodinger equation generalizing the Gross-Pitaevskii (GP) equa tion for dilute Bose gas. The derivation is based on the Hartree-Fock time-dependent mean-field theory with an arbitrary intermolecular interaction potential. It is shown that obtained nonlinear Schrodinger equation with appropriate redefinition of coefficients can be used for description of dense Bose fluid at low temperatures. We also present the Schrodinger type equation describing the superfluid component of helium in two fluid hydrodynamics. This approach leads to quantum correction for superfluid component of velocity in two fluid hydrodynamics. We also have derived the nonlinear Schrodinger equation for superfluid He4 film at low temperatures. It is shown that this Schrodinger type equation for superfluid He4 film leads to phonon-roton dispersion relation for elementary excitation at low temperatures.
Abstract Chapter 2 describes the basic knowledge of Bose-condensed systems in order to study the quantum hydrodynamics for both the superfluid helium and the atomic-gas BECs in the following sections. Starting from the field theoretical formulation, we will first give a self-contained explanation of the derivation of the most fundamental GP equation for weakly interacting Bose gases at zero temperature. We will also highlight the theoretical treatment for incorporating the concept of elementary excitations to describe low-energy excitations of Bose systems, the definition of normal and superfluid components, and eventually the celebrated model of two-fluid hydrodynamics. Finally, the microscopic structure and dynamics of topological excitations obtained from the GP equation are discussed by exemplifying solitons, vortex lines, and vortex rings.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.
The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.