We numerically investigate how self-steepening and quintic nonlinearity influence the bifurcation structure and stability of solitary waves in the cubic–quintic nonlinear Schrödinger equation with self-steepening and a symmetric double-well potential. In the regime with self-focusing cubic and self-defocusing quintic nonlinearities, the interplay between competing nonlinearities gives rise to intricate bifurcation patterns, including double pitchfork and saddle–node bifurcations. Increasing the self-steepening strength modifies the bifurcation topology by eliminating certain branches and reducing multistability through the suppression of saddle–node bifurcations. As the defocusing strength of the quintic term increases, symmetry breaking is progressively suppressed, and the bifurcation structure reduces to a continuous symmetric branch that folds at a saddle–node bifurcation, where both the lower and upper segments are stable. In contrast, when both nonlinearities are self-focusing, the bifurcation structure exhibits a single supercritical pitchfork bifurcation accompanied by stable asymmetric branches and remains qualitatively unchanged under variations in either the quintic or self-steepening parameters. These results provide valuable insight into how higher-order nonlinearities shape the existence and stability of localized states, with potential applications in ultrafast optics and nonlinear wave phenomena.

We numerically investigate the existence and stability dynamics of self-steepening optical solitons in a periodic PT-symmetric potential. We show that self-steepening solitons of the modified nonlinear Schrödinger (MNLS) equation undergo a position shift and amplitude increase during their evolution in the MNLS equation. The stabilization of solitons by an external potential is a challenging issue. This study demonstrates that the suppression of both the amplitude increase and the position shift of self-steepening solitons can be achieved by adding a periodic PT-symmetric potential to the MNLS equation.

We numerically demonstrate the existence of parabolic and rectangular self-similar propagations of optical beams in saturable media. Rectangular selfsimilar evolution is achieved by imposing an external optical potential (lattice) that acts as a pulse shaping mechanism and it is shown that a rectangular shaped profile can be obtained by providing a delicate balance between the nonlinearity and saturation of the medium.

The fundamental lattice solitons are explored in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity. The band-gap boundaries, soliton profiles, and stability domains of fundamental solitons are investigated comprehensively by the linear stability spectra and nonlinear evolution of the solitons. It is demonstrated that fundamental lattice solitons can stay stable for a wide range of parameters with the weak self-focusing and self-defocusing quintic nonlinearity, while strong self-focusing and self-defocusing quintic nonlinearities are shortened the propagation distance of evolved solitons. Furthermore, it is observed that when the instability emerges from strong quintic nonlinearity, increasing anisotropy of the medium and modification of lattice depth can be considered as a collapse arrest mechanism.

Stability dynamics of dipole solitons have been numerically investigated in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity by the squared-operator method. It has been demonstrated that solitons can stay nonlinearly stable for a wide range of each parameter, and two nonlinearly stable regions have been found for dipole solitons in the gap domain. Moreover, it has been observed that instability of dipole solitons can be improved or suppressed by modification of the potential depth and strong anisotropy coefficient.

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water.

The self-similar propagation of optical beams in a broad class of nonlocal, nonlinear optical media is studied utilizing a generic system of coupled equations with linear gain. This system describes, for instance, beam propagation in nematic liquid crystals and optical thermal media. It is found, both numerically and analytically, that the nonlocal response has a focusing effect on the beam, concentrating its power around its center during propagation. In particular, the beam narrows in width and grows in amplitude faster than in local media, with the resulting beam shape being parabolic. Finally, a general initial localized beam evolves to a common shape.

With this book, we aim to capture different perspectives of researchers on nonlinear optics and optical devices and we intend to cover the latest developments in optics from theoretical, numerical, and experimental aspects. The eleven selected chapters cover a variety of topics related to nonlinear optics including bright, dark, kink solitary waves in various media, magnetic solitons, lattice solitons, rogue-waves, solid-state lasers, laser cladding, optical sensors, optical vortices, and molecular switches. The book is intended to draw the attention of scientists in academia, as well as researchers and engineers in industry, since the field has a significant potential for the production and design of novel optical devices and other technological applications.

We numerically demonstrate controlled self-similar evolution of optical pulses in fibers. In so doing, we utilize the nonlinear Schrödinger equation with constant gain to which we add a linear forcing term, which we call an optical potential in analogy to other optical media, which acts as the shape forming mechanism; this term, in earlier studies is added in the form of a periodically placed filter. Here, we show that a distributed equation not only makes the modelling and analysis of the system simpler but also allows for initial Gaussian shaped pulses to grown self-similarly, under evolution, to rectangular or triangular shaped localized structures.

We investigate the existence and stability of two dimensional nonlinear localized modes in parity-time symmetric optical media with self-focusing/self-defocusing cubic and quintic nonlinearities. The exact analytical form of the localized modes is found for all values of competing parameters.

In this paper, we discuss the existence of lattice solitons supported by cubic-saturable nonlinearity in the framework of nonlinear Schrödinger (NLS) equation with external potentials such as parity-time-symmetric lattices with and without defects by using the pseudo-spectral renormalization method. Linear and nonlinear stability properties of the lattice solitons centered on the maximum of the PT-symmetric potential are investigated in detail.

A pulse shaping mechanism applied to mode-locked lasers is proposed. By adding a linear (forcing) term in the power energy saturation model, we are able to control the resulting pulses in both energy and shape. In fact, this term also provides a focusing effect keeping most of the pulse's energy confined within the width of the forcing. The appropriate condition for which mode-locking occurs is also derived and links the physical parameters of the system (gain, loss, filtering) to those of the pulse (amplitude, width, energy). Thus, given the desired pulse one only needs to fix the laser's parameters accordingly, so as to obey this condition, and mode-locking will occur.

Wave collapse is arrested in the self-focusing nonlinear Schrödinger equation with coupling to a mean term (NLSM) by adding an external potential (lattice) to the governing equation. It is numerically demonstrated that collapse will eventually occur in a lattice-free system and it can be suppressed by adding an external periodic lattice to the governing system. It is numerically shown that lattice depth provides great controllability on soliton stability and more robust solitons can be obtained.

In this paper, the existence and stability properties of optical solitons on parity-time (PT) symmetric lattices are investigated. The governing equation for the physical model is the (1+1)D cubic-quintic nonlinear Schrödinger equation (CQNLS) with a PT-symmetric potential. The solution to this equation is obtained both analytically and numerically by spectral methods. The numerical existence of fundamental solitons on PT-symmetric lattices is shown for various medium and potential depths.

Using computational methods, the numerical existence and nonlinear stability of fundamental solitons in saturable me dia on crystal and certain type of quasicrystal lattices are inves tigated. In a certain parameter regime of the lattice depth a nd the propagation constant, the first nonlinear band-gap structures are obtai ned and the effect of the DC bias field (external electric field ) and the lattice depth to the gap width are analyzed.

We put forward a mechanism for delaying the collapse of the fundamental solitons in nonlinear media whose dynamics is governed by two-dimensional nonlinear Schrödinger (NLS) equation with parity-time symmetric (PT-symmetric) periodic potentials with/without a vacancy defect. We observed that strengthened gain–loss component (imaginary part of the potential) in the periodic lattice impoverish the stability properties of the solitons, on the other hand adding a vacancy defect to the periodic PT-symmetric lattice acts as a delaying mechanism for collapsing solitons.

We report the numerical existence of dipole and vortex solitons for the two-dimensional nonlinear Schrödinger (NLS) equation with external potentials that possess strong irregularities, i.e., edge dislocations and a vacancy defects. Multi-humped solitons are computed by employing a spectral fixed-point computational scheme. The nonlinear stability of these solitons is investigated using direct simulations of the NLS equation and it is observed that these multi-humped modes in the defect lattices can be stable or unstable.

We modified the spectral renormalization method as the pseudospectral renormalization method in order to find the localized solutions. The pseudospectral renormalization method can be applied to a large class of problems including different homogeneities. Using this computational method, we demonstrate the existence of two different solitons in optical media described by the self-focusing cubic and the self-defocusing quintic nonlinear Schrödinger equation with quasicrystal lattice. It is shown that there are two different lattice solitons corresponding to the first and the second renormalization factors for the self-focusing cubic and the self-defocusing quintic model. However, the self-focusing quintic nonlinearity without optical lattice does not support two different solitons. We showed that the lattice solitons corresponding to the first and the second renormalization factors have the same powers and amplitudes. We also demonstrate that quintic nonlinearity supports bistable solitons by adding the optical lattice such as a quasicrystal lattice. The linear and nonlinear stabilities of these solitons are investigated using direct simulation of the nonlinear Schrödinger equation with the cubic-quintic nonlinearity and its linearized equation.

In this paper, exact solutions of two-dimensional nonlinear Schrödinger equation with kerr, saturable and quintic type of nonlinearities are studied by means of the Homotopy analysis method (HAM). Linear stability properties of these solutions are investigated by the linearized eigenvalue problem. We also investigate nonlinear stability properties of the exact solutions obtained by HAM by direct simulations.