We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.

The deformed supersymmetric sine-Gordon model, obtained through known deformation of the corresponding potential, is found to be quasi-integrable, like its non-supersymmetric counterpart, which was observed earlier. The system expectedly possesses finite number of conserved quantities, leaving-out an infinite number of non-conserved anomalous charges. The quasiintegrability of this supersymmetric model heavily rely on the boundary conditions of the potential, otherwise rendered to be completely non-integrable. Moreover, interesting additional algebraic structures appear, absent in the non-supersymmetric counterparts.

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The hierarchy of equations belonging to two different but related integrable systems, the Nonlinear Schrödinger and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not preserve the integrability, only asymptotically integrable, and non-holonomic deformation (N HD) that does. QID is carried out generically for the NLS hierarchy while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No QI anomaly is observed at the level of EOMs which suggests that at that level the QID may be identified as some integrable deformation. NHD is applied to the NLS hierarchy generally as well as with the specific focus on the NLS equation itself and the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN) and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently, different aspects of the results are discussed. *

For one-dimensional PT -symmetric systems, it is observed that the non-local product ψ * (−x, t)ψ(x, t), obtained from the continuity equation can be interpreted as a conserved correlation function. This leads to physical conclusions, regarding both discrete and continuum states of such systems. Asymptotic states are shown to have necessarily broken PT -symmetry, leading to modified scattering and transfer matrices. This yields restricted boundary conditions, e.g., incidence from both sides, analogous to that of the proposed PT CPA laser [1]. The interpretation of 'left' and 'right' states leads to a Hermitian S-matrix, resulting in the non-conservation of the 'flux'. This further satisfies a 'duality' condition, identical to the optical analogues [2]. However, the non-local conserved scalar implements alternate boundary conditions in terms of 'in' and 'out' states, leading to the pseudo-Hermiticity condition in terms of the scattering matrix. Interestingly, when PT -symmetry is preserved, it leads to stationary states with real energy, naturally interpretable as bound states. The broken PT -symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.