Stationary modes and integrals of motion in nonlinear lattices with a \(\mathcal{PT}\)-symmetric linear part (Q2861465)

Nonlinear systems of the form NEWLINE\[NEWLINE i\dot{\mathbf{q}}=-H(\gamma)\mathbf{q}-F(\mathbf{q})\mathbf{q}NEWLINE\]NEWLINE are studied, where \(\mathbf{q}=\mathbf{q}(t)\) is a column-vector of \(N\) elements. The linear part of the finite lattice is described by an \(N\times N\) symmetric matrix \(H(\gamma)\). \(\mathcal{PT}\)-symmetric Hamiltonians \(H(\gamma)\) are considered, that is there exist a parity, \(\mathcal{P}\), and time-reversal, \(\mathcal{T}\), operators such that \(\mathcal{P}^2=\mathcal{T}^2=I\), \([\mathcal{P},\mathcal{T}]=0\) and \([\mathcal{PT}, H]=0\). (\(I\) is the identity operator). The authors investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of \(\mathcal{PT}\)-symmetric nonlinearities allowing the existence of families of nonlinear modes. The particular attention is paid to situations when the underlying linear \(\mathcal{PT}\)-symmetric operator is characterized by the presence of degenerate eigenvalues or an exceptional-point singularity.