Nonlinear Schrödinger equation for a PT symmetric delta-functions double well

Abstract

The time-independent nonlinear Schrödinger equation is solved for two attractive delta-function shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the other. We show that for vanishing nonlinearity the model captures all the features known from studies of PT symmetric optical wave guides, e.g., the coalescence of modes in an exceptional point at a critical value of the loss/gain parameter, and the breaking of PT symmetry beyond. With the nonlinearity present, the equation is a model for a Bose-Einstein condensate with loss and gain in a double well potential. We find that the nonlinear Hamiltonian picks as stationary eigenstates exactly such solutions which render the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and bifurcation scenarios different from those known from linear PT symmetric Hamiltonians.

16 pages, 9 figures, to be published in Journal of Physics A

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• Dissipative quadratic solitons supported by a localized gain
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• Stationary states in the many-particle description of Bose-Einstein condensates with balanced gain and loss
• Pinned modes in two-dimensional lossy lattices with local gain and nonlinearity
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• Transmission singularities in resonant electron tunneling through double complex potential barrier
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