Dipole and quadrupole nonparaxial solitary waves.

The cubic nonlinear Helmholtz equation with third and fourth order dispersion and non-Kerr nonlinearity, such as the self steepening and the self frequency shift, is considered. This model describes nonparaxial ultrashort pulse propagation in an optical medium in the presence of spatial dispersion originating from the failure of slowly varying envelope approximation. We show that this system admits periodic (elliptic) solitary waves with a dipole structure within a period and also a transition from a dipole to quadrupole structure within a period depending on the value of the modulus parameter of a Jacobi elliptic function.

Coupled Helmholtz equations: Chirped solitary waves.

We investigate the existence and stability properties of chirped gray and anti-dark solitary waves within the framework of a coupled cubic nonlinear Helmholtz equation in the presence of self-steepening and a self-frequency shift. We show that for a particular combination of self-steepening and a self-frequency shift, there is not only chirping but also chirp reversal. Specifically, the associated nontrivial phase has two intensity dependent terms: one varies as the reciprocal of the intensity, while the other, which depends on non-Kerr nonlinearities, is directly proportional to the intensity.

Bright-dark and dark-dark solitons in coupled nonlinear Schrödinger equation with PT-symmetric potentials.

We explore different nonlinear coherent structures, namely, bright-dark (BD) and dark-dark (DD) solitons in a coupled nonlinear Schrödinger/Gross-Pitaevskii equation with defocusing/repulsive nonlinearity coefficients featuring parity-time ( PT)-symmetric potentials. Especially, for two choices of PT-symmetric potentials, we obtain the exact solutions for BD and DD solitons. We perform the linear stability analysis of the obtained coherent structures.