Dark solitons and vortices in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>phase transitions

Abstract

We consider nonlinear analogs of parity-time- ($\mathcal{PT}$-) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and odd excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter $\ensuremath{\varepsilon}$ controlling the strength of the $\mathcal{PT}$-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as $\ensuremath{\varepsilon}$ is further increased, the ground state and first excited state, as well as branches of higher multisoliton (multivortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear $\mathcal{PT}$ phase transition---thus termed the nonlinear $\mathcal{PT}$ phase transition. Past this critical point, initialization of, e.g., the former ground state, leads to spontaneously emerging solitons and vortices.