A systematic construction of parity-time (\(\mathcal{PT}\))-symmetric and non-\(\mathcal{PT}\)-symmetric complex potentials from the solutions of various real nonlinear evolution equations. (English) Zbl 1380.37124
The authors construct parity-time (PT)-symmetric as well as non-PT-symmetric complex potentials of the linear stationary Schrödinger equation by means of known solitons and periodic solutions of some integrable real nonlinear evolution equations (NLEEs). These NLEEs are made up of the sine-Gordon (sG), modified Korteweg de Vries (mKdV), combined mKdV-sG, and the Gardner equations, and the complex potentials are constructed using a method suggested in [M. Wadati, J. Phys. Soc. Japan 77, 074005 (2008)].
The interest in the PT-symmetry of the potentials is justified by the fact that the PT-invariance of the Hamiltonian is a more relaxed condition than Hermiticity, provided that this PT-symmetry is not spontaneously broken. For each of these NLEEs, a small number of their solitons is considered and each such a solution is used to construct a complex potential of the linear Schrödinger equation.
Complex potentials constructed from solutions obtained by the inverse scattering transform (IST) method are PT-symmetric, and it is easy to verify if this PT-symmetry is spontaneously broken or not. For all the cases considered in the paper, it turns out that symmetries are not broken.
For complex potentials constructed from solutions which are not obtained by the IST method such as the periodic solitons, there is no general formalism to determine whether the corresponding energy eigenvalues are real or not. Thus for such complex potentials, the authors make use of the numerical evaluation of the corresponding (isolated) low-energy eigenvalues. It turns out that in all such cases considered in the paper the numerically computed energy eigenvalues are real when the complex potential satisfies PT-symmetry, and complex when it does not.
Possible applications to the graphene model of the PT-symmetric potentials obtained from the sG equation and the combined mKdV-sG equation are also discussed. These discussions are carried out along the lines of similar connections established in [C. L. Ho et al., Europhys. Lett. 112, 47004 (2015)] between the graphene model and the PT-invariant potentials obtained from the mKdV and the Gardner equations.
The interest in the PT-symmetry of the potentials is justified by the fact that the PT-invariance of the Hamiltonian is a more relaxed condition than Hermiticity, provided that this PT-symmetry is not spontaneously broken. For each of these NLEEs, a small number of their solitons is considered and each such a solution is used to construct a complex potential of the linear Schrödinger equation.
Complex potentials constructed from solutions obtained by the inverse scattering transform (IST) method are PT-symmetric, and it is easy to verify if this PT-symmetry is spontaneously broken or not. For all the cases considered in the paper, it turns out that symmetries are not broken.
For complex potentials constructed from solutions which are not obtained by the IST method such as the periodic solitons, there is no general formalism to determine whether the corresponding energy eigenvalues are real or not. Thus for such complex potentials, the authors make use of the numerical evaluation of the corresponding (isolated) low-energy eigenvalues. It turns out that in all such cases considered in the paper the numerically computed energy eigenvalues are real when the complex potential satisfies PT-symmetry, and complex when it does not.
Possible applications to the graphene model of the PT-symmetric potentials obtained from the sG equation and the combined mKdV-sG equation are also discussed. These discussions are carried out along the lines of similar connections established in [C. L. Ho et al., Europhys. Lett. 112, 47004 (2015)] between the graphene model and the PT-invariant potentials obtained from the mKdV and the Gardner equations.
Reviewer: Jean-Claude Ndogmo (Thohoyandou)
MSC:
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
| 35C08 | Soliton solutions |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
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