Unbreakable $\mathcal{PT}$ symmetry of exact solitons supported by transversally modulated nonlinearity acting as a psuedopotential

We demonstrate analytically and numerically the existence of exact solitons in the form of double-kink and fractional-transform, supported by a symmetric transversally modulated defocusing nonlinearity acting as a psuedopotential combined with an antisymmetric gain-loss profile. We explicate here that the $\mathcal{PT}$ symmetry is never broken in the dynamical system under study, even in the absence of any symmetric modulation of linear refractive-index, and the transversally modulated defocusing nonlinearity comes in the way as a requirement to establish the ensuing $\mathcal{PT}$ symmetry.

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