Is the Schrödinger-Newton equation time-reversal symmetric? What about PT-symmetry or similar symmetries?
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I tried to figure it out myself. If you take the integro-differential form of the equation, a complex square of the time-dependent wavefunction appears.. It seems to me that this means the equation cannot be time symmetric but I'm not sure about this reasoning. The same goes for PT-symmetry, since the Laplacian is the same even under a P reversal.
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HERE IS THE ANSWER ✌
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Stability of soliton families in one-dimensional nonlinear Schrödinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically.
It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition.
In addition, a pseudo-Hamiltonian–Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian–Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian.
The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets.(λ,−λ,λ⁎,−λ⁎), similar to conservative systems and PT-symmetric systems.
This quartet eigenvalue symmetry is very surprising for non-PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.
✅✅✅
_______________
HOPE IT WILL HELP YOU ☺☺☺
HERE IS THE ANSWER ✌
_____________
⬇⬇⬇
Stability of soliton families in one-dimensional nonlinear Schrödinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically.
It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition.
In addition, a pseudo-Hamiltonian–Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian–Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian.
The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets.(λ,−λ,λ⁎,−λ⁎), similar to conservative systems and PT-symmetric systems.
This quartet eigenvalue symmetry is very surprising for non-PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.
✅✅✅
_______________
HOPE IT WILL HELP YOU ☺☺☺
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