The Gross–Pitaevskii equation: Bäcklund transformations and admitted solutions

Abstract

In honour of Tommaso Ruggeri on his 70th Birthday. Abstract Bäcklund transformations are applied to study the Gross-Pitaevskii equation. Supported by previous results, a class of Bäcklund transformations admitted by this equation are constructed. Schwarzian derivative as well as its invariance properties turn out to represent a key tool in the present investigation. Examples and explicit solutions of the Gross-Pitaevskii equation are obtained.

Schwarzian derivative, Painlevé XXV–Ermakov equation, and Bäcklund transformations

“…By comparison with Proposition 3.1, the ratio on the right-hand side of the previous equation must be equal to 𝑘∕𝑢 3 , that is,…”

Schwarzian derivative, Painlevé XXV-Ermakov equation and Bäcklund transformations

“…In the previous work [11] (see also [12]) another family of Bäcklund transformations for equation ( 27) has been presented. These transformations depend on a certain function f (z) that solve a suitable functional equation that is closely related with the properties of the Schwarzian derivative.…”

Ermakov-Pinney and Emden-Fowler equations: New solutions from novel Bäcklund transformations

“…где G(z) -произвольная функция, удовлетворяющая уравнению (30). Тогда w(z) является решением уравнения…”

“…принадлежат и другие физически интересные системы, возникающие, например, как редукции уравнений в частных производных; в нашей работе мы сосредоточились только на некоторых физически важных примерах. Другие результаты, касающиеся уравнения Гросса-Питаевского, можно найти в [30]. Кроме того, ведутся исследования, касающиеся структуры уравнения (1), а также его решений в комплексной области.…”