Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation

Abstract

This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+i{W}_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.

To appear in Proceedings of the 15 Conference on Pseudo-Hermitian Hamiltonians in Quantum Physics, May 18-23 2015, Palermo, Italy (Springer Proceedings in Physics, 2016)

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• Stationary through-flows in a Bose-Einstein condensate with a PT-symmetric impurity
• Continuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach
• Effects of the third-order dispersion on continuous waves in complex potentials
• Krein signature for instability of $\mathcal{P}\mathcal{T}$-symmetric states
• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• A systematic construction of parity-time ($\mathcal{P}\mathcal{T}$)-symmetric and non-$\mathcal{P}\mathcal{T}$-symmetric complex potentials from the solutions of various real nonlinear evolution equations
• Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential