• Corpus ID: 233324224

Reductions of nonlocal nonlinear Schr\"odinger equations to Painlev\'e type functions

@inproceedings{Liu2021ReductionsON,
  title={Reductions of nonlocal nonlinear Schr\"odinger equations to Painlev\'e type functions},
  author={Jonathon Liu},
  year={2021}
}
In this paper, we take ODE reductions of the general nonlinear Schrödinger equation (NLS) AKNS system, and reduce them to Painlevé type equations. Specifically, the stationary solution is solved in terms of elliptic functions, and the similarity solution is solved in terms of the Painlevé IV transcendent. Since a number of newly proposed integrable ‘nonlocal’ NLS variants (the PT -symmetric nonlocal NLS, the reverse time NLS, and the reverse spacetime NLS) are derivable as specific cases of… 
View PDF on arXiv

References

SHOWING 1-10 OF 32 REFERENCES
Integrable Nonlocal Nonlinear Equations
A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the…
Solitons
General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system.
TLDR
It is shown that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent PT-symmetric potentials, and the symmetry breaking does not occur in the PT-Symmetric linear system with the associated potentials.
Non-Hermitian physics and PT symmetry
In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss…
The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability
Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation
It is shown that the nonlocal nonlinear Schr\"odinger equation recently proposed by Ablowitz and Musslimani [Phys. Rev. Lett. 110, 064105 (2013)] is gauge equivalent to the unconventional system of…
On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions
Soliton solutions for the nonlocal nonlinear Schrödinger equation
Abstract.The Darboux transformation (DT) for the integrable nonlocal nonlinear Schrödinger equation (nNLSE) is constructed with the aid of loop group method. Based on the DT, we derive several types…
On the relation between nonlinear Schrödinger equation and Painlevé IV equation
  • M. Can
  • Mathematics
    Il Nuovo Cimento B
  • 1991
SummaryIn this short communication it is shown that the similarity equation for the nonlinear Schrödinger equation is equivalent to the σ form of the Painlevé IV equation given by M. Jimbo and T.…
Chazy Classes IX–XI Of Third‐Order Differential Equations
In this article, we study Classes IX–XI of the 13 classes introduced by Chazy (1911) in his classification of third‐order differential equations in the polynomial class having the Painlevé property.…
...
...