Painlevé analysis, bilinear form, Bäcklund transformation, solitons, periodic waves and asymptotic properties for a generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system in a fluid or plasma

2021 ◽  
Vol 136 (9) ◽  
Author(s):  
Shao-Hua Liu ◽  
Bo Tian ◽  
Meng Wang
Keyword(s):  
Bilinear Form ◽  
Bäcklund Transformation ◽  
Asymptotic Properties ◽  
Painlevé Analysis ◽  
Periodic Waves ◽  
Backlund Transformation ◽  
Painleve Analysis

scholarly journals Bäcklund Transformation and New Exact Solutions of the Sharma-Tasso-Olver Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Lin Jianming ◽  
Ding Jie ◽  
Yuan Wenjun
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Analytic Study ◽  
Bäcklund Transformations ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Painleve Analysis

The Sharma-Tasso-Olver (STO) equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.

scholarly journals On Two Aspects of the Painlevé Analysis

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Sergei Sakovich
Keyword(s):  
Nonlinear Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Nonlinear Pdes ◽  
Painlevé Analysis ◽  
Painlevé Test ◽  
Backlund Transformation ◽  
Independent Variables ◽  
Painleve Analysis

We use the Calogero equation to illustrate the following two aspects of the Painlevé analysis of nonlinear PDEs. First, if a nonlinear equation passes the Painlevé test for integrability, the singular expansions of its solutions around characteristic hypersurfaces can be neither single-valued functions of independent variables nor single-valued functionals of data. Second, if the truncation of singular expansions of solutions is consistent, the truncation not necessarily leads to the simplest, or elementary, auto-Bäcklund transformation related to the Lax pair.

PAINLEVÉ ANALYSIS, LAX PAIR AND BÄCKLUND TRANSFORMATION FOR THE GROSS–PITAEVSKII EQUATION IN THE BOSE–EINSTEIN CONDENSATES

2011 ◽  
Vol 25 (08) ◽  
pp. 1037-1047
Author(s):  
FENG-HUA QI ◽  
BO TIAN ◽  
TAO XU ◽  
HAI-QIANG ZHANG ◽  
LI-LI LI ◽  
...  
Keyword(s):  
Bäcklund Transformation ◽  
Scattering Length ◽  
Lax Pair ◽  
Time Dependent ◽  
Painlevé Analysis ◽  
Pitaevskii Equation ◽  
Backlund Transformation ◽  
Ansatz Method ◽  
Bose Einstein ◽  
Painleve Analysis

Due to their relevance to physics and technology, the Bose–Einstein condensates (BECs) are of current interest. Certain dynamics of the BECs, such as the cigar-shaped condensate confined in a cylindrically symmetric parabolic trap, can be described by the Gross–Pitaevskii (GP) equation with a time-dependent trap. In this paper, by virtue of the Painlevé analysis and symbolic computation, we derive the integrable condition for the GP equation with the time-dependent scattering length in the presence of a confining or expulsive time-dependent trap. Lax pair for this equation is directly obtained via the Ablowitz–Kaup–Newell–Segur scheme under the integrable condition. Bright one-soliton-like solution of the GP equation is presented via the Bäcklund transformation and some analytic solutions with variable amplitudes are obtained by the ansatz method. In addition, an infinite number of conservation laws are also derived. Those results could be of some value for the studies on the lower-dimensional condensates.

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