Complex sine-Gordon-2: a new algorithm for multivortex solutions on the plane

We present a new vorticity-raising transformation for the second integrable complexification of the sine-Gordon equation on the plane. The new transformation is a product of four Schlesinger maps of the Painlev\'{e}-V to itself, and allows a more efficient construction of the $n$-vortex solution than the previously reported transformation comprising a product of $2n$ maps.Comment: Part of a talk given at a conference on "Nonlinear Physics. Theory and Experiment", Gallipoli (Lecce), June-July 2004. To appear in a topical issue of "Theoretical and Mathematical Physics". 7 pages, 1 figur

Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation

We construct exact localised solutions of the PT-symmetric Gross-Pitaevskii equation with an attractive cubic nonlinearity. The trapping potential has the form of two $\delta$-function wells, where one well loses particles while the other one is fed with atoms at an equal rate. The parameters of the constructed solutions are expressible in terms of the roots of a system of two transcendental algebraic equations. We also furnish a simple analytical treatment of the linear Schr\"odinger equation with the PT-symmetric double-$\delta$ potential.Comment: 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

PT\mathcal{PT}-symmetry breaking in a necklace of coupled optical waveguides

We consider parity-time ($\mathcal{PT}$) symmetric arrays formed by $N$ optical waveguides with gain and $N$ waveguides with loss. When the gain-loss coefficient exceeds a critical value $\gamma_c$, the $\mathcal{PT}$-symmetry becomes spontaneously broken. We calculate $\gamma_c(N)$ and prove that $\gamma_c \to 0$ as $N \to \infty$. In the symmetric phase, the periodic array is shown to support $2N$ solitons with different frequencies and polarisations.Comment: 6 pages, 4 figure

Dimer with gain and loss: Integrability and PT\mathcal{PT}-symmetry restoration

A $\mathcal{PT}$-symmetric nonlinear Schr\"odinger dimer is a two-site discrete nonlinear Schr\"odinger equation with one site losing and the other one gaining energy at the same rate. In this paper, two four-parameter families of cubic $\mathcal{PT}$-symmetric dimers are constructed as gain-loss extensions of their conservative, Hamiltonian, counterparts. We prove that all these damped-driven equations define completely integrable Hamiltonian systems. The second aim of our study is to identify nonlinearities that give rise to the spontaneous $\mathcal{PT}$-symmetry restoration. When the symmetry of the underlying linear dimer is broken and an unstable small perturbation starts to grow, the nonlinear coupling of the required type diverts progressively large amounts of energy from the gaining to the losing site. As a result, the exponential growth is saturated and all trajectories remain trapped in a finite part of the phase space regardless of the value of the gain-loss coefficient.Comment: Update presented at 13th Workshop on Pseudo-Hermitian Hamiltonians (Israel Institute for Advanced Studies, Jerusalem 12-16 July, 2015

Traveling solitons in the damped driven nonlinear Schr\"odinger equation

The well known effect of the linear damping on the moving nonlinear Schr\"odinger soliton (even when there is a supply of energy via the spatially homogeneous driving) is to quench its momentum to zero. Surprisingly, the zero momentum does not necessarily mean zero velocity. We show that two or more parametrically driven damped solitons can form a complex traveling with zero momentum at a nonzero constant speed. All traveling complexes we have found so far, turned out to be unstable. Thus, the parametric driving is capable of sustaining the uniform motion of damped solitons, but some additional agent is required to stabilize it.Comment: 13 pages, 9 figures; to appear in SIAM Journal of Applied Mathematic