Narrow Results

This work studies the development of windows of harmonious instability in negative refractive material. After taking into account the fundamental need to monitor instability, such as self-phase modulation and group velocity spreading, we first examine the effects on the gain spectrum of fourth and third-order dispersals, cubic–quintic nonlinearity, higher-order nonlinear dispersions, self-steepening, and detuning parameter. Furthermore, in metamaterials with the aforementioned nonlinear effects, we investigate the influence of an adjustable nonlinear saturation effect over the modulational instability. Tunable modulation instability (MI) gain spectrum formation results from the tunable material parameters made possible by the engineering freedom offered by metamaterials. As a structure is operated at high incident power, generally exceeding the medium’s saturation inception, the SNL becomes a robust case. The nonlinear saturation window with negative index material can also produce adjustable instability gain spectrum formation. More methods for forming solitons and ultrashort pulses with desired parameters can be achieved with this tunable gain spectrum. The numerical approach validates the analytical prediction that came from the linear stability study.

The symplectic integrator of the Gauss–Legendre type is tested on the nonlinear Schrödinger equation. Preservation of high integrals (up to 10 or more) and quasiperiodic motion have been detected for dynamics on both stable soliton and homoclinic manifolds, which indicate applicability of symplectic integrators for adequate simulation of integrable equation. The tested integrator is applied to the problem of long-time stability of the solitons in higher-derivative nonlinear Schrödinger equation. The slow logarithmic-type depletion of the soliton amplitude with time has been detected.

The modulational instability of the coupled Gross–Pitaevskii equation (alias nonlinear Schrödinger equation), which describes two Bose–Einstein condensates trapped in an asymmetric double-well potential, is investigated. The nonlinear dispersion relation that relates the frequency and wave number of the modulating perturbations is found and its analysis shows several possibilities for the modulational stability region. Exact soliton and periodic solutions are constructed via elliptic ordinary differential equations.

From the discrete nonlinear Schrödinger equation and the linear stability analysis, the modulational instability (MI) of dipolar Bose–Einstein Condensates (BECs) in an optical lattice is studied. The MI relation of the dipolar BECs in the optical lattice with the on-site interaction and the inter-site interaction is obtained. The results show that there is a great influence of inter-site interaction on MI of dipolar BEC in the optical lattice. This gives us some useful information for manipulating dipolar BECs in practice.

Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.

The modulational instability (MI) of binary condensates with cubic-quintic nonlinearities is investigated. Using a linear stability analysis, a gain of instability is derived then, effects of the quintic nonlinearities on the instability gain are identified. To be precise, attractive intraspecie quintic nonlinearities enhance the instability, while repulsive quintic intraspecie nonlinearities soften the instability. Besides, small attractive and large repulsive quintic inter-species nonlinearities increase the instability. Numerical experiments quite well corroborate the analytical predictions. Further numerical results show effects of the cubic and the quintic nonlinearities on the propagation of trains of bright solitons generated.

We investigate the generation of soliton-like pulses along a DNA chain which takes into account both torsional and solvent interaction effects. Interactions between neighboring base pairs are described by a twist angle. Twisting is essential in the model to capture the importance of nonlinear effects for the thermodynamical properties. The nonlinear dynamics of the DNA is then modeled in the Hamiltonian approach by the generalized Dauxois–Peyrard–Bishop model (combination of several models). We introduce the generalized discrete nonlinear Schrödinger equation describing the dynamics of modulated wave through the twisted DNA with solvent interaction. The modulational instability is studied and we present an analytical expression for the MI gain to show the effects of twist angle on MI gain spectra as well as on stability diagram. With the increase of the twist angle the MI gain decreases then increases. Some interesting MI phenomena appear with an additional new MI region as the twist angle increases. The instability and stability diagrams are also affected. Numerical simulations are carried out to show the validity of the analytical approach. The result is that the initial wave breaks into a train of ultrashort pulses with repetition rates, which are trapped in some sites. The impact of the twist angle is investigated and we obtain that the twist angle affects the dynamics of stable patterns generated through the molecule. Thereafter, we study energy localization in the framework of twisted DNA with solvent interaction. While the twist angle leads to a stronger localization of energy, the solvent interaction delocalizes energy along the molecule.

In this paper, we study the modulational instability (MI) in a biexciton molecular chain taking into account the saturable nonlinearity effects (SNE). Under the adiabatic approximation, the biexciton system is reduced to two coupled nonlinear Schrödinger equations. We perform the linear stability analysis of continuous wave solutions of the coupled system. This analysis reveals that the MI gain is deeply influenced by the SNE. Indeed, the gain spectrum decreases when increasing the saturable nonlinearity parameters. The numerical simulations reveal that the system exhibits incoherent periodic array of patterns and we also observe train of pulses due to the SNE.

We exclusively analyze the onset and condition of formation of modulated waves in a diffusive FitzHugh–Nagumo model for myocardial cell excitations. The cells are connected through gap junction coupling. An additive magnetic flux variable is used to describe the effect of electromagnetic induction, while electromagnetic radiation is imposed on the magnetic flux variable as a periodic forcing. We used the discrete multiple scale expansion and obtained, from the model equations, a single differential-difference amplitude nonlinear equation. We performed the linear stability analysis of this equation and found that instability features are importantly influenced by the induced electromagnetic gain. We present the unstable and stable regions of modulational instability (MI). The resulting analytic predictions are confirmed by numerical experiments of the generic equations. The results reveal that due to MI, an initial steady state that consisted of a plane wave with low amplitude evolves into a modulated localized wave patterns, soliton-like in shape, with features of synchronization. Furthermore, the formation of periodic pulse train with breathing motion presents a disappearing pattern in the presence of electromagnetic radiation. This could provide guidance and better understanding of sudden heart failure exposed to heavily electromagnetic radiation.

To analytically investigate the matter-wave solitons of Bose–Einstein condensates (BECs) in time-dependent complex potential, we consider a cubic-quintic Gross–Pitaevskii (GP) equation with distributed coefficients and a dissipative term. By introducing a suitable ansatz, we establish the criterion of the modulational instability (MI) of the system and present an explicit expression for the growth rate of a purely growing MI. Effects of the parabolic background potential, as well as of the linear potential, the gain/loss parameter, and the two- and three-body interatomic interactions on the MI are investigated. We show how the feeding/loss parameter can be well used to control the instability of the system. The analytical resolution of the considered GP equation leads to exact bright, dark and kink solitary wave solutions which are used to investigate analytically the dynamics of matter-wave solitons in BECs under consideration. These analytical investigations show that the amplitude and the motion of bright, dark and kink solitary waves depend on the strengths of the two- and three-body interatomic interactions, as well as on the strengths of the external trapping potential and the parameter of the gain/loss of atoms in the condensate.

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies.

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose–Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross–Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent structures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review.

We consider a generalized fourth-order nonlinear Schrödinger (NLS) equation. Based on the ansatz method, its bright, dark single-soliton is constructed under some constraint conditions. Furthermore, combining the Riccati equation extension approach, we also derive some exact singular solutions. With several parameters to play with, we display the dynamic behaviors of bright, dark single-soliton. Finally, the condition for the modulational instability (MI) of continuous wave solutions for the equation is generated. It is hoped that our results can help enrich the nonlinear dynamics of the NLS equations.

Under investigation is the (2+1)-dimensional KMN equation, which is a new extension of the well-known nonlinear Schrödinger equation. Firstly, we investigate the modulational instability (MI) of the plane wave states for this equation. Secondly, we explore several types of interesting mixed breather–lump solutions in orders $N=2$ and $N=3$ cases by using the generalized $(2,N-2)$-fold Darboux transformation (DT). Moreover, we also provide the graphical analysis of such mixed interaction solutions to better understand their dynamical behavior. Finally, we sum up a few mathematical features to obtain special mixed interaction structures through the generalized $(n,N-n)$-fold DT. The results obtained in this paper might excite the interest in nonlinear optics and relevant fields.

Apply style for article title, author, affiliation and email as per stylesheet. Several decades ago, antennas had simple shapes that were described in Euclidean geometry. Nowadays, scientists try to make the structure of fractal geometry for applications in the field of electromagnetism, which has led to the development of new innovative antenna devices. Non-integer dimensional space (NDS) is useful to describe the concept of fractional space in fractal structure for real phenomenon of electromagnetic wave propagation. In this work, we investigate effects of NDS and normalized frequency on modulational instability (MI) gain in lossless left-handed metamaterials (LHM). We derive the nonlinear Schrödindiger equation (NLSE) with non-integer transverse laplacian. By means of linear stability analysis method, MI gain expression is also determined. Different forms of figures are obtained due to the signs of group velocity dispersion (GVD) and defocusing/focusing nonlinearity. We show how the increasing value of the normalized frequency enhances the amplitude as well as the bandwidth of MI gain, and waves are more unstable due to non-integer dimension. The obtained results are new and have a relatively newer application in telecommunication by constructing the fractal-shaped antennas operating in multi-frequency bands.

This paper aims to theoretically examine the phenomenon of modulation instability (MI) in the presence of a considerable nonlinear refractive index movement in a triad-core nonlinear coupler, including a bi-negative refractive material transmission medium. The formula for the instability gain is deduced with the help of the linear stability analysis. The significant role of cubic, quintic, septic, nonic, and stability of the front-to-reverse-transmitting wave nonlinearities in the phenomenon of MI in the nonlinear triad-core coupler is critically confirmed via MI gain spectra analysis. It is found that the cubic, quintic, septic, and nonic nonlinearities improve the MI in both dispersion regimes (normal and anomalous) by boosting the gain and width of the instability profile by taking account of reverse-to-front ratio power (f). On the other hand, quintic nonlinearity improves the MI in the cubic case by enhancing its width and gain in the normal and anomalous dispersion areas. Using 3D pictures, we analyze the dynamic behavior of the MI under an anomalous dispersion area and find that, in both normal and anomalous dispersion scenarios, the instability shape develops with septic and nonic nonlinearity. Furthermore, as the figures clearly show, neither the normal nor the anomalous dispersion scenario significantly changes due to septic and nonic nonlinearity alone. We aim to shed light on the role of contamination in saturation nonlinearity by examining soliton and MI, their production, and control in triad-core couplers, focusing on a Negative Index Meta Material (NIMM) transmission medium.

Optical propagation in nematic liquid crystals is characterized by a large and highly-nonlocal Kerr-like nonlinearity. We investigate the fundamental role played by spatial nonlocality in nonlinear optical propagation, and develop a model able to predict the main features of spatial solitons and modulational instability in nematic liquid crystals. The model unifies solitons in physical systems exhibiting different degrees of nonlocality, disclosing a connection between nonlocal solitons and parametric solitons in quadratic media. Finally, soliton breathing as well as other characteristics of nonlocal propagation are experimentally demonstrated in a specifically-engineered liquid crystal cell.

We study the modulation instability (MI) for nonlinear Schrödinger equation phase locked to an external source. We analytically compute the growth rate and analyze the possibility of existence of MI for self-focusing and self-defocusing nonlinearities for positive and negative values of the homogeneous source term coefficient. We further investigate the impact of variation of nonlinear coefficient and variation of source term coefficient on the existence of MI. Phase-locking is the key characteristic that enables us to conduct analytical investigations of MI in this dynamical system, that appertains to pulse propagation through asymmetric twin-core fibers.

We study the modulation instability (MI) for optical wave propagation in the presence of slowly varying saturable nonlinearity, group velocity dispersion and a PT-symmetric external potential over the length of tapered graded-index waveguide. First we study the MI with space-dependent dispersion and saturated nonlinearity and then with constant dispersion and saturated nonlinearity. We analytically compute the growth rate and analyze the possibility of existence of MI for focussing and defocussing nonlinearities for different choices of group velocity dispersion terms and saturated nonlinearities. We observe that MI is independent of PT symmetric potential term and typically depends upon the choice of group velocity dispersion term, nonlinearity, saturated nonlinearity and wave amplitude. MI gain decreases with increase in cw amplitude however it increases with increase in saturated nonlinearity both for focussing and defocussing nonlinearities.

The evolution of crossing sea states and the emergence of rogue waves in such systems are studied via numerical simulations performed using a higher order spectral (HOS) method to solve the free surface Euler equations with a flat bottom. Two classes of crossing sea states are analyzed: one using directional spectra from the Draupner wave crossing at different angles, another considering a Draupner-like spectra crossed with a narrowband JONSWAP state to model spectral growth between wind sea and swell. These two classes of crossing sea states are constructed using the spectral output of a WAVEWATCH III hindcast on the Draupner rogue wave event. We measure ensemble statistical moments as functions of time, finding that although the crossing angle influences the statistical evolution to some degree, there are no significant third-order effects present. Additionally, we pay particular attention to the mean sea level measured beneath extreme crest heights, the elevation of which (set up or set down) is shown to be related to the spectral content in the low wavenumber region of the corresponding spectrum.