scholarly journals Exact Solutions and Symmetry Operators for the Nonlocal Gross-Pitaevskii Equation with Quadratic Potential

2005 ◽  
Author(s):  
Alexander Shapovalov
Keyword(s):  
Exact Solutions ◽  
Pitaevskii Equation ◽  
Quadratic Potential ◽  
Symmetry Operators

Symmetry operators and exact solutions of a type of time-fractional Burgers–KdV equation

2019 ◽  
Vol 16 (02) ◽  
pp. 1950032 ◽  
Author(s):  
Azadeh Naderifard ◽  
S. Reza Hejazi ◽  
Elham Dastranj ◽  
Ahmad Motamednezhad
Keyword(s):  
Exact Solutions ◽  
Fourth Order ◽  
Vector Fields ◽  
Kdv Equation ◽  
Group Analysis ◽  
Invariant Subspaces ◽  
Similarity Solutions ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Symmetry Operators

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

scholarly journals The Gross–Pitaevskii Equation with a Nonlocal Interaction in a Semiclassical Approximation on a Curve

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 201
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin ◽  
Andrey Yu. Trifonov
Keyword(s):  
Linear Equation ◽  
Semiclassical Approximation ◽  
Nonlocal Interaction ◽  
Pitaevskii Equation ◽  
Dimensional Manifold ◽  
Complex Germ ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Symmetry Operators ◽  
Interaction Term

We propose an approach to constructing semiclassical solutions for the generalized multidimensional Gross–Pitaevskii equation with a nonlocal interaction term. The key property of the solutions is that they are concentrated on a one-dimensional manifold (curve) that evolves over time. The approach reduces the Cauchy problem for the nonlocal Gross–Pitaevskii equation to a similar problem for the associated linear equation. The geometric properties of the resulting solutions are related to Maslov’s complex germ, and the symmetry operators of the associated linear equation lead to the approximation of the symmetry operators for the nonlocal Gross–Pitaevskii equation.

Symmetry operators of a Hartree-type equation with quadratic potential

2005 ◽  
Vol 46 (1) ◽  
pp. 119-132 ◽  
Author(s):  
A. L. Lisok ◽  
A. Yu. Trifonov ◽  
A. V. Shapovalov
Keyword(s):  
Type Equation ◽  
Quadratic Potential ◽  
Symmetry Operators

Exact solutions of the Gross-Pitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates

2010 ◽  
Vol 81 (6) ◽  
Author(s):  
Lei Wu ◽  
Lu Li ◽  
Jie-Fang Zhang ◽  
Dumitru Mihalache ◽  
Boris A. Malomed ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Pitaevskii Equation ◽  
Two Dimensional ◽  
Bose Einstein ◽  
Stable Vortex

scholarly journals Berry phases for the nonlocal Gross–Pitaevskii equation with a quadratic potential

2006 ◽  
Vol 39 (5) ◽  
pp. 1191-1206 ◽  
Author(s):  
F N Litvinets ◽  
A V Shapovalov ◽  
A Yu Trifonov
Keyword(s):  
Pitaevskii Equation ◽  
Quadratic Potential ◽  
Berry Phases

Mathematical and physical aspects of controlling the exact solutions of the 3D Gross–Pitaevskii equation

2010 ◽  
Vol 374 (5) ◽  
pp. 788-795 ◽  
Author(s):  
Renato Fedele ◽  
Dušan Jovanović ◽  
Sergio De Nicola ◽  
Bengt Eliasson ◽  
Padma K. Shukla
Keyword(s):  
Exact Solutions ◽  
Pitaevskii Equation
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