Painlevé singularity structure analysis of three component Gross–Pitaevskii type equations

Kanna, T.; Sakkaravarthi, K.; Kumar, C. Senthil; Lakshmanan, M.; Wadati, Miki

doi:10.1063/1.3263936

Cited by 20 publications

(22 citation statements)

References 34 publications

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“…the matrix NLS equation with ǫ = −1 describes a special case of a spin-1 Bose-Einstein condensate [102][103][104][105][106][107][108][109][110][111][112][113][114][115]. (6.3) is then equivalent to the system…”

Section: The Complex Conjugation Reductionmentioning

confidence: 99%

“…A brief summary of the physical relevance can be found in [3]. In particular, a (symmetric) 2 × 2 matrix NLS equation turned out to be of relevance for the description of a special Bose-Einstein condensate (with atoms in a spin 1 state) [102][103][104][105][106][107][108][109][110][111][112][113][114][115]. Semi-discrete matrix NLS equations appeared in [3,[116][117][118][119][120][121][122][123][124][125][126][127][128][129], a full discretisation has been elaborated in [130] and a dispersionless limit studied in [127].…”

Section: Introductionmentioning

confidence: 99%

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Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

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Using a bidifferential graded algebra approach to "integrable" partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework, by specifying a representation of the bidifferential graded algebra and imposing a reduction. By application of a general result, corresponding families of exact solutions are obtained that in particular comprise the matrix soliton solutions in the focusing NLS case. The solutions are parametrised in terms of constant matrix data subject to a Sylvester equation (which previously appeared as a rank condition in the integrable systems literature). These data exhibit a certain redundancy, which we diminish to a large extent.More precisely, we first consider more general AKNS-type systems from which two different matrix NLS systems emerge via reductions. In the continuous case, the familiar Hermitian conjugation reduction leads to a continuous matrix (including vector) NLS equation, but it is well-known that this does not work as well in the discrete cases. On the other hand there is a complex conjugation reduction, which apparently has not been studied previously. It leads to square matrix NLS systems, but works in all three cases (continuous, semi-and fully-discrete). A large part of this work is devoted to an exploration of the corresponding solutions, in particular regularity and asymptotic behaviour of matrix soliton solutions.

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Section: The Complex Conjugation Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solutions of matrix NLS systems and their discretizations: a unified treatment

Dimakis¹,

Müller‐Hoissen²

2010

Inverse Problems

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“…(9)- (11) in Eq. (6) we get the following set of known integrable three-component GP equations 33,36 : iq 2ρ + q 2,χχ + 2(|q 1 | 2 + |q 2 | 2 + |q 3 | 2 )q 2 + 2q 1 q 3 q * 2 = 0 ,…”

Section: Reduction Of Nonautonomous Three Coupled Gp Equation Tomentioning

confidence: 99%

“…The corresponding autonomous model of Eq. (1) in the absence of external trap has been investigated thoroughly for integrability aspect, 36 for obtaining multi soliton solutions, 33 for rogue wave solutions, 15 and for studying the interactions between polar and ferromagnetic solitons. 37 Additionally, the soliton dynamics have been analyzed in the BEC of alkali atoms in F = 1 hyperfine state such as 7 Li, 87 Rb and in 23 Na, confined in the 1D space by purely optical means.…”

Section: Introductionmentioning

confidence: 99%

The management of matter rogue waves in F = 1 spinor Bose–Einstein condensates

Loomba

2015

Int. J. Mod. Phys. B

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We have reported the matter rogue wave solutions for the nonautonomous three coupled Gross-Pitaevskii (GP) equation which governs the pulse propagation in spin-1 Bose-Einstein condensates (BECs) with time modulated nonlinearities. The system under consideration has attractive mean-field interactions and ferromagnetic spin-exchange interactions. The exact solutions have been worked out by using similarity and scaling transformations. We have depicted the controllable center characteristics of rogue waves for the kink-type spin-exchange interactions. Additionally, we have also discussed the management of rogue waves for the hyperbolic form of spin-exchange interactions by invoking isospectral Hamiltonian deformation technique. Keywords: Rogue wave; similarity transformation; spinor Bose-Einstein condensate. PACS numbers: 05.45.Yv, 52.35.Sb, 42.65.Tg 1550125-1 Int. J. Mod. Phys. B 2015.29. Downloaded from www.worldscientific.com by PRINCETON UNIVERSITY on 08/11/15. For personal use only. 1550125-2 Int. J. Mod. Phys. B 2015.29. Downloaded from www.worldscientific.com by PRINCETON UNIVERSITY on 08/11/15. For personal use only.

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“…The corresponding equations are non-linear second-order ordinary differential equations, used by physicists and mathematicians since their discovery to describe a growing variety of systems. Some examples involve the description of the asymptotic behavior of non-linear equations [2], statistical mechanics [3], correlation functions of the XY model [4], bidimensional ising model [5], superconductivity [6], Bose-Einstein condensation [6], stimulated Raman dispersion [7], quantum gravity and quantum field theory [8], aleatory matrix models [9], topologic field theory (e.g., the so-called WittenDijkgraaf-Verlinde-Verlinde equations) [10], general relativity [11], solutions of Einstein axialsymmetric equations [11], negative curvature surfaces [12], plasma physics [6], Hele-Shaw problems [13] and non-linear optics [14]. During the last years, more and more researchers are interested in these equations and they have found interesting analytic, geometric, and algebraic properties.…”

Section: Introductionmentioning

confidence: 99%

Sierra-Porta

Núñez²

2017

IJAMR

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The Painlevé equations and their solutions arises in pure, applied mathematics and theoretical physics. In this manuscript we apply the Optimal Homotopy Asymptotic Method (OHAM) for solving the first Painlevé equation. Our approximation technique is based on the use of polynomial solutions, which are shown to be accurate when compared to the computed numerical solutions, thus providing a very close description of the evolution of the system.

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Painlevé singularity structure analysis of three component Gross–Pitaevskii type equations

Cited by 20 publications

References 34 publications

Solutions of matrix NLS systems and their discretizations: a unified treatment

Solutions of matrix NLS systems and their discretizations: a unified treatment

The management of matter rogue waves in F = 1 spinor Bose–Einstein condensates

On the polynomial solution of the first PainlevÃ© equation

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