Related papers: The Gross-Pitaevskii equation: B\"acklund transfor…
A method for the study of steady-state nonlinear modes for Gross-Pitaevskii equation (GPE) is described. It is based on exact statement about coding of the steady-state solutions of GPE which vanish as $x\to+\infty$ by reals. This allows to…
We study the Bose-Einstein condensate trapped in a three-dimensional spherically symmetrical potential. Exact solutions to the stationary Gross-Pitaevskii equation are obtained for properly modulated radial nonlinearity. The solutions…
Extensions to higher-dimensions are given for a convolution estimate used by Klainerman and Machedon in their study of uniqueness of solutions for the Gross-Pitaevskii hierarchy. Such estimates determine more general forms of Stein-Weiss…
In this paper, we have studied the integrability nature of a system of three coupled Gross-Pitaevskii type nonlinear evolution equations arising in the context of spinor Bose-Einstein condensates by applying the Painlev\'e singularity…
We derive a generalized Gross-Pitaevski (GP) equation immersed on a electromagnetic and a weak gravitational field starting from the covariant Complex Klein-Gordon field in a curved space-time. We compare it with the GP equation where the…
The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrodinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an…
We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze…
We introduce certain B\"acklund transformations for rational solutions of the Painlev\'e VI equation. These transformations act ona family of Painlev\'e VI tau functions. They are obtained from reducing the Hirota bilinear equations that…
We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, B\"acklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we…
The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case…
In this paper we present the analytic solution to the problem of bound states of the Gross-Pitaevskii (GP) equation in 1D and its properties, in the presence of external potentials in the form of finite square wells or attractive Dirac…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the von Mises transformation, with the unknown function taken as the new independent variable and an appropriate partial…
A variational technique is established to deal with the Schrodinger equation with parity-time(PT) symmetric Gaussian complex potential. The method is extended to the linear and self-focusing and defocusing nonlinear cases. Some unusual…
Wide classes of nonlinear mathematical physics equations are described that admit order reduction through the use of the Crocco transformation, with a first-order partial derivative taken as a new independent variable and a second-order…
In this paper we have solved the Bohmian equations of quantum gravity, perturbatively. Solutions up to second order are derived explicitly, but in principle the method can be used in any order. Some consequences of the solution are…
We show that integro-differential generalized Langevin and non-Markovian master equations can be transformed into larger sets of ordinary differential equations. .On the basis of this transformation we develop a numerical method for solving…
We present a generalized Gross-Pitaevskii equation that describes also the dissipative dynamics of a trapped partially Bose condensed gas. It takes the form of a complex nonlinear Schr\"odinger equation with noise. We consider an…
We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures…
Solutions of the classical and nonlocal Gross-Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second order nonisospectral Ablowitz-Kaup-Newell-Segur system and reduction technique of double…
The B\"acklund transformation and its nonlinear superposition principle are presented for the Krichever-Novikov equation $u_t= u_{xxx} - {3/(2u_x)} (u^2_{xx} - r(u)) + cu_x, r^{(5)}=0$.