We consider the multicomponent Yajima-Oikawa (YO) system and show that the two-component YO system can be derived in a physical setting of a three-coupled nonlinear Schrödinger (3-CNLS) type system by the asymptotic reduction method. The derivation is further generalized to the multicomponent case. This set of equations describes the dynamics of nonlinear resonant interaction between a one-dimensional long wave and multiple short waves. The Painlevé analysis of the general multicomponent YO system shows that the underlying set of evolution equations is integrable for arbitrary nonlinearity coefficients which will result in three different sets of equations corresponding to positive, negative, and mixed nonlinearity coefficients. We obtain the general bright N -soliton solution of the multicomponent YO system in the Gram determinant form by using Hirotas bilinearization method and explicitly analyze the one-and two-soliton solutions of the multicomponent YO system for the above mentioned three choices of nonlinearity coefficients.We also point out that the 3-CNLS system admits special asymptotic solitons of bright, dark, anti-dark, and gray types, when the long-wave-short-wave resonance takes place. The short-wave component solitons undergo two types of energy-sharing collisions. Specifically, in the two-component YO system, we demonstrate that two types of energy-sharing collisions-(i) energy switching with opposite nature for a particular soliton in two components and (ii) similar kind of energy switching for a given soliton in both components-result for two different choices of nonlinearity coefficients. The solitons appearing in the long-wave component always exhibit elastic collision whereas those of short-wave components exhibit standard elastic collisions only for a specific choice of parameters. We have also investigated the collision dynamics of asymptotic solitons in the original 3-CNLS system. For completeness, we explore the three-soliton interaction and demonstrate the pairwise nature of collisions and unravel the fascinating state restoration property.

We study the formation and propagation of chirped elliptic and solitary waves in cubic-quintic nonlinear Helmholtz (CQNLH) equation. This system describes nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities along with spatial dispersion originating from the nonparaxial effect that becomes dominant when the conventional slowly varying envelope approximation (SVEA) fails. We first carry out the modulational instability (MI) analysis of a plane wave in this system by employing the linear stability analysis and investigate the influence of different physical parameters on the MI gain spectra. In particular, we show the nonparaxial parameter suppresses the conventional MI gain spectrum and also leads to a nontrivial monotonic increase in the gain spectrum near the tails of the conventional MI band, a qualitatively distinct behaviour from the standard nonlinear Schrödinger (NLS) system. We then study the MI dynamics by direct numerical simulations which demonstrate production of ultra-short nonparaxial pulse trains with internal oscillations and slight distortions at the wings. Following the MI dynamics, we obtain exact elliptic and solitary wave solutions using the integration method by considering physically interesting chirped traveling wave ansatz. In particular, we show that the system features intriguing chirped anti-dark, bright, gray and dark solitary waves depending upon the nature of nonlinearities. We also show that the chirping is inversely proportional to the intensity of the optical wave. Especially, the bright and dark solitary waves exhibit unusual chirping behaviour which will have applications in nonlinear pulse compression process.Nonlinear Helmholtz (NLH) equation can play a significant role in modeling a progressive miniaturization of photonic devices and plasmonics. By obeying the intrinsic higher dimensions of spatial symmetry of uniform planar waveguides, many experimental configurations could be modeled that are otherwise inaccessible in the standard SVEA. In the present work, we consider the NLH equation with cubic-quintic (CQ) nonlinearities. The implication of non-Kerr nonlinearity in the NLH system leads to a more stabilized nonparaxial pulse propagation and it can easily be realized in available materials including semiconductors and doped fibers. We study the phenomenon of MI in the CQNLH system, where we address some peculiar features of nonparaxiality. Also, we numerically demonstrate the generation of train of ultra-short nonparaxial solitary pulses. In addition to different types of Helmholtz solitons, we present the first ever study of formation and propagation of chirped elliptic and solitary waves, which are not reported for any NLH system before. In particular, we obtain various interesting chirped solitary profiles, which include bright, dark, gray and anti-dark solitary waves depending upon the nature of nonlinearities. Among them, we find chirped profiles of bright and dark solitary waves exhibit some unusual nonlinear dynamics of compr...

We obtain a class of elliptic wave solutions of coupled nonlinear Helmholtz (CNLH) equations describing nonparaxial ultra-broad beam propagation in nonlinear Kerr-like media, in terms of the Jacobi elliptic functions and also discuss their limiting forms (hyperbolic solutions). Especially, we show the existence of non-trivial solitary wave profiles in the CNLH system. The effect of nonparaxiality on the speed, pulse width and amplitude of the nonlinear waves is analysed in detail. Particularly a mechanism for tuning the speed by altering the nonparaxial parameter is proposed. We also identify a novel phase-unlocking behaviour due to the presence of nonparaxial parameter.

We consider (2+1) and (1+1) dimensional long-wave short-wave resonance interaction systems. We construct an extensive set of exact periodic solutions of these systems in terms of Lamé polynomials of order one and two. The periodic solutions are classified into three categories as similar, mixed, superposed elliptic solutions. We also discuss the hyperbolic solutions as limiting cases.

We systematically construct a distinct class of complex potentials including parity-time (PT ) symmetric potentials for the stationary Schrödinger equation by using the soliton and periodic solutions of the four integrable real nonlinear evolution equations (NLEEs) namely the sine-Gordon (sG) equation, the modified Korteweg-de Vries (mKdV) equation, combined mKdV-sG equation and the Gardner equation. These potentials comprise of kink, breather, bion, elliptic bion, periodic and soliton potentials which are explicitly constructed from the various respective solutions of the NLEEs. We demonstrate the relevance between the identified complex potentials and the potential of the graphene model from an application point of view.