Related papers: Painlev{\'e} singularity structure analysis of thr…
The Gross-Pitaevskii equation has been extremely successful in the theory of weakly-interacting Bose-Einstein condensates. However, present-day experiments reach beyond the regime of its validity due to the significant role of correlations.…
The magnetically induced Richtmyer-Meshkov instability in a two-component Bose-Einstein condensate is investigated. We construct and study analytical models describing the development of the instability at both the linear and nonlinear…
We test the $\mathbb{C}P^{N-1}$ sigma models for the Painlev\'e property. While the construction of finite action solutions ensures their meromorphicity, the general case requires testing. The test is performed for the equations in the…
We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multi-component nonlinear wave systems of nonlinear Schr{\"o}dinger type. While our computations relate to…
We consider dynamics of simple waves in a two-component Bose-Einstein condensates. The evolution of the condensate is described by the Gross-Pitaevskii equations which can be reduced for simple wave solutions to a system of ordinary…
We investigate formation of Bose-Einstein condensates under non-equilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the…
We study the problem of dimension reduction for the three dimensional Gross-Pitaevskii equation (GPE) describing a Bose-Einstein condensate confined in a strongly anisotropic harmonic trap. Since the gas is assumed to be in a strong…
We investigate dynamics in two-component Bose-Einstein condensates in the context of coupled Gross-Pitaevskii equations and derive results for the evolution of the total density fluctuations. Using these results, we show how, in many cases…
We investigate the bifurcation structure of stationary states in a dipolar Bose-Einstein condensate located in an external PT-symmetric potential. The imaginary part of this external potential allows for the effective description of in- and…
We develop a simple and general variational method to estimate the solutions of the Gross-Pitaevskii equations and obtain the corresponding density profiles for all spin states of a trapped spin-1 Bose-Einstein condensate. We further employ…
An analysis of possible extension of the Painlev\'e test, to encompass the one-dimensional Vlasov equation, is performed. The extending requires a nontrivial generalization of the test. The proposed singularity analysis provides…
We solve the Gross-Pitaevskii equation to model the behaviour of a weakly-interacting Bose condensate. Solutions are presented for the eigenstates in one- and two-dimensional harmonic traps. We include the effect of gravity and coupling to…
A nonuniform condensate is usually described by the Gross-Pitaevskii (GP) equation, which is derived with the help of the c-number ansatz $\hat{ \Psi}(\mathbf{r},t)=\Psi (\mathbf{r},t)$. Proceeding from a more accurate operator ansatz…
We study density isolines in quantum turbulence under the Schramm-Loewner framework using direct numerical simulations of the truncated Gross-Pitaevskii equation, in both spherical and cylindrical traps with three-dimensional dynamics.…
A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved \textit{via} a method initiated by Painleve'…
A phenomenological model is proposed to describe the behavior of spinor Bose-Einstein condensates. In the absence of hyperfine spin-spin interactions, Bose-Einstein condensation leads to a spontaneous magnetization at the same transition…
The ``Painlev\'e analysis'' is quite often perceived as a collection of tricks reserved to experts. The aim of this course is to demonstrate the contrary and to unveil the simplicity and the beauty of a subject which is in fact the theory…
In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional…
A mixture of two kinds of identical bosons, species $1$ and species $2$, held in a harmonic potential and interacting by harmonic intra-species and inter-species particle-particle interactions is discussed. To prove Bose-Einstein…
The Gross-Pitaevskii equation, or more generally the nonlinear Schr\"odinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the…