Interplay between parity-time symmetry, supersymmetry, and nonlinearity: An analytically tractable case example

In the present work, we combine the notion of parity-time ($\mathcal{PT}$) symmetry with that of supersymmetry (SUSY) for a prototypical case example with a complex potential that is related by SUSY to the so-called Pöschl-Teller potential which is real. Not only are we able to identify and numerically confirm the eigenvalues of the relevant problem, but we also show that the corresponding nonlinear problem, in the presence of an arbitrary power-law nonlinearity, has an exact bright soliton solution that can be analytically identified and has intriguing stability properties, such as an oscillatory instability, which is absent for the corresponding solution of the regular nonlinear Schrödinger equation with arbitrary power-law nonlinearity. The spectral properties and dynamical implications of this instability are examined. We believe that these findings may pave the way toward initiating a fruitful interplay between the notions of $\mathcal{PT}$ symmetry, supersymmetric partner potentials, and nonlinear interactions.

Figure 1

The blue circles represent the numerically computed eigenvalues $E^{(+)}$ of the operator $H^{(+)}$ of Eq. (3), under the $\mathcal{PT}$-symmetric potential of Eq. (9). The solid lines represent the analytical predictions based on the potential's supersymmetric partner $V^{(-)}$ corresponding to an analytically tractable Pöschl-Teller potential.

Figure 2

Norm of the solutions of Eq. (17) as a function of $b_I$ and $\kappa$ when $m=1$.

Figure 3

Panel (a): Norm of the solutions with $\kappa =1$ as a function of $b_I$. The blue (full) line corresponds to the nodeless solution whereas the red (dashed) line corresponds to the “dipole” branch (see [23]) possessing a node. Panel (b) [(c)] showcases the modulus of the nodeless solution [solution with a node] as a function of $x$ for different values of $b_I$. It can clearly be seen that the amplitude of the solution goes to 0 as $b_I\to 3/2$ [$b_I\to 5/2$].

Figure 4

Panel (a): Norm of the solutions with $\kappa =3$ as a function of $b_I$. The blue (full) line corresponds to the nodeless solution whereas the red (dashed) line corresponds to the solution with a node (i.e., the “dipole”). Panel (b) [(c)] showcases the modulus of the nodeless solution [solution with a node, although the node itself disappears as $b_I$ increases] as a function of $x$ for different values of $b_I$. It can clearly be seen that the amplitude of the solution goes to 0 as $b_I\to 5/6$ [$b_I\to 11/6$], i.e., the corresponding linear limit value for $E=-1/9$.

Figure 5

Imaginary part [panels (a) and (c)] and real part [panels (b) and (d)] of the eigenvalues associated with the linearization around the solution of the nodeless [(a) and (b)] and the single-node (i.e., dipole) solution branches [panels (c) and (d)] for $\kappa =1$. It can be observed that the nodeless solutions become unstable in the interval $0.56<b_I<1.37$, whereas the solutions with a single node are unstable for all $b_I$ except for a very small interval $2.43<b_I<2.5$ close to the upper existence limit; in addition, for $b_I<0.48$ the latter waveform is also exponentially unstable.

Figure 6

Two case examples of the spectral plane $({\lambda }_r,{\lambda }_i)$ of eigenvalues $\lambda ={\lambda }_r+i{\lambda }_i$ of the solution for (a) $b_I=0$ and (b) $b_I=1$. The eigenvalue which is spectrally stable in the left panel but whose collision with the band edge of the continuous spectrum is responsible for the instability in the right panel is denoted by a red mark.

Figure 7

Imaginary part [(a) and (c)] and real part [(b) and (d)] of the eigenvalues associated with the linearization around the solution of the nodeless [(a) and (b)] and the single-node (i.e., dipole) solutions branches [(c) and (d)] for $\kappa =3$. It can be observed that all the solutions (for different values of $b_I$) are unstable; see the text for a detailed description of the eigenvalue variation over $b_I$.

Figure 8

Panel (a): Space-time contour plot of the evolution of the squared modulus (density) of the solution during its unstable dynamics for $b_I=0.65$ and $\kappa =1$; the inset shows the evolution of the (maximal) density of the solution occurring at $x=0$. Panel (b) [(c)] shows the evolution of the density for $b_I=1$ and $\kappa =1$ [$\kappa =3$]; the inset shows the evolution of the norm and displays its eventual indefinite growth. Panel (d) considers the evolution of the unstable soliton with a node for $b_I=0.2$ and $\kappa =1$, leading eventually to a split of the two humps into a stationary (at $x=0$) and a traveling one (at $x<0$).