In this work, we study a generalised high-order nonlinear Schrödinger equation with time-dependent coefficients, embracing a wide range of physical influences. By employing the Darboux transformation, we construct explicit breather and rogue wave solutions, illustrating how the spectral parameter governs waveform transitions. In these dynamics, dispersion determines stability and symmetry, nonlinearity influences the peak amplitude and width, and third-order dispersion introduces asymmetry and drift in the wave profile. We have demonstrated that stabilization, destabilization and shifting of the centre of the localization, or drifting towards the soliton in space or even temporal directions, can be possible by manoeuvring the spectral parameter relating dispersion and nonlinearity in optical fibre. Manoeuvring the spectral parameter relates the dispersion a1(t) and nonlinearity from 100 t to 0.1 t leads to the stabilization of the soliton by a notable decrease in the amplitude for two hundred folds. The results reveal that the inclusion of higher-order term functions as a control mechanism for managing instability and localisation in nonlinear optical fibre systems, offering promising prospects for future developments in nonlinear optics.

This study investigates soliton solutions and dynamic wave behaviors in the complex Ginzburg-Landau equation, a model that plays a central role in describing diverse physical phenomena such as superconductivity, nonlinear optical fibers, liquid crystals, second-order phase transitions, and field theory strings. To derive closed-form solutions, we employ two advanced analytical techniques: the new rational extended sinh-Gordon equation expansion method (ShGEEM) and the modified generalized exponential rational function method (mGERFM). These methods yield a wide range of solitonic structures, such as complex and singular solitons, oscillatory periodic waves, bright, dark, and multi-wave profiles. In this work, new families of exact solitary wave solutions with ShGEEM and several hyperbolic, trigonometric, and exponential solutions with mGERFM are presented. Further, the obtained solutions are checked for accuracy by substituting them back into Mathematica. For the dynamics of solutions, 2D plots, 3D surfaces, and contour graphs have been constructed for some values of parameters in the presence of the [Formula: see text]-fractional derivative to understand wave structures and their evolution. In general, the present study not only consolidates the aspects of nonlinear wave dynamics in the field of chemical and physical oceanography but also provides pathways for further research on nonlinear fractional-order models. The originality of the present study lies in the point that the complex Ginzburg-Landau equation has not been studied within the ShGEEM and mGERFM frameworks.

In this work, we develop the generalized Fourier transform and construct action-angle variables for the Lakshmanan-Porsezian-Daniel (LPD) equation. This framework mathematically characterizes ultrashort femtosecond soliton propagation in dispersive nonlinear optical fibers. By establishing completeness relations for squared eigenfunctions in the spectral problem, we prove that solutions of the Lax system span the Schwartz function space, thereby rigorously formulating the generalized Fourier transform. Furthermore, explicit Poisson brackets construction for scattering data generates canonical action-angle variables governed by spectral parameters. Crucially, the Hamiltonian formulation reveals a duality: conservation laws admit equivalent representations through both spectral parameters and potential functions, establishing a direct correspondence between scattering data and nonlinear potentials. This framework not only deepens the analytical understanding of the LPD equation’s integrability but also provides theoretical foundations for controlling optical solitons in defocusing media with higher-order nonlinear effects.

In this study, we investigate the coupled nonlinear integrable system known as the Akbota–Gudekli–Kairat–Zhaidary (AGKZ) equation. By employing a generalized extended simple equation method, we examine soliton and various solitary wave solutions with diverse physical structures. The nonlinear complex AGKZ equation is a newly introduced integrable model arising in the study of space curves and surfaces; therefore, its analytical exploration is essential for understanding its physical applications. The investigated solutions display distinct physical structures, including bright solitons, kink wave structures, dark solitons, peakon-type bright and dark waves, anti-kink wave structures, periodic waves with varying profiles, solitary waves through contour plots, two-dimensional plots, and three-dimensional visualizations using Mathematica tool. The novelty of this work lies in establishing enriched and distinct soliton solutions to the AGKZ equation and performing a comparative analysis of the proposed method, which has not been previously addressed in the literature. The derived solutions of the AGKZ equation may be applied to model ultrashort pulse propagation in nonlinear optical fibers, photonic crystals, waveguides, and solitary waves in shallow water. The results demonstrate that the proposed approach is practical, straightforward, and effective for generating a wide variety of soliton solutions applicable to other nonlinear equations.

Optical frequency conversion plays a key role in realizing large-scale quantum networks, including multi-qubit discrete-variable quantum computers and quantum communication links where photons serve as the fundamental qubits. However, achieving efficient conversion via nonlinear optical processes for specific target wavelengths remains a significant challenge, as precise dispersion control is essential to satisfy phase-matching conditions across specific frequency ranges. An intriguing approach to solve this challenge is leveraging the modal degree of freedom in spatially multimoded waveguides and realizing intermodal nonlinear interaction. Following this approach, we present the experimental demonstration of tunable, number-state-preserving frequency conversion of true single photons emitted from a quantum dot. The conversion is achieved in a multimode fiber and exhibits a peak internal efficiency of 85% while retaining single photon purity of 99% during conversion. Our results show that the intermodal platform presents a promising and versatile approach for overcoming phase-matching limitations in quantum frequency conversion, thus allowing the efficient interfacing of different optical quantum devices.

ABSTRACT In this study, we deal with the perturbed nonlinear Schrödinger equation (PNLSE). This equation is an important model for describing optical soliton propagation in nonlinear optical fibers with the Kerr law nonlinearity. Two efficient techniques are employed: the generalized Abel equation method (GAEM) with variable coefficients and the collective variable (CV) approach. As a consequence of GAEM, we derive a periodic‐type soliton solution and enhance its physical interpretation through graphical simulations. The CV method, carrying a fourth‐order Runge–Kutta scheme and the Gaussian ansatz, is used to model the dynamics of soliton parameters such as amplitude, width, chirp, and frequency, revealing periodic fluctuations influenced by propagation distance. Both methods provide a comprehensive understanding of soliton behaviors in optical fibers, offering valuable outcomes for advancing optical communication systems.

We analyzed the mechanism of alkali-ions migration taking part in obtaining the frozen ionic potential responsible for the creation of second-order nonlinearity in optical fibers made of silica and other oxide-based glasses. We confirmed that the presence of alkali ions in the oxide glass, together with a high χ(3) value, allows to obtain ionic potential, even though these ions are structural elements of glass rather than impurities. Creation of second-order susceptibility in glass is possible and, depending on alkali-ion concentration, χ(2) can be higher than for a silica-based fiber of the same geometry. Calculated χ(2) reached 0.58 pm/V, and the maximum value obtained for silica-based fiber for the same thermal poling conditions was only 0.33 pm/V. Full Text: PDF References U. Österberg and W. Margulis, "Experimental studies on efficient frequency doubling in glass optical fibers," Opt. Lett. 12(1), 57 (1987) CrossRef H. An and S. Fleming, "Creating second-order nonlinearity in pure synthetic silica optical fibers by thermal poling," Opt. Lett. 32(6), 832 (2007) CrossRef H. An and S. Fleming, "Second-order optical nonlinearity in thermally poled borosilicate glass," Appl. Phys. Lett. 89(18), 181111 (2006) CrossRef W. Margulis, O. Tarasenko, and N. Myrén, "Who needs a cathode? Creating a second-order nonlinearity by charging glass fiber with two anodes," Opt. Express 17(17), 15534 (2009) CrossRef F. De Lucia, D.W. Keefer, C. Corbari, and P.J.A. Sazio, "Thermal poling of silica optical fibers using liquid electrodes," Opt. Lett. 42(1), 69 (2017) CrossRef P. Kabaciński, T.M. Kardaś, Y. Stepanenko, and C. Radzewicz, "Nonlinear refractive index measurement by SPM-induced phase regression," Opt. Express 27(8), 11018 (2019) CrossRef A. Kudlinski, Y. Quiquempois, G. Martinelli, "Modeling of the χ(2) susceptibility time-evolution in thermally poled fused silica," Opt. Express 13(20), 8015 (2005) CrossRef A. Camara, O. Tarasenko, and W. Margulis, "Study of thermally poled fibers with a two-dimensional model," Opt. Express 22(15), 17700 (2014) CrossRef F. Bergmann et al., "Measuring the permittivity of fused silica with planar on-wafer structures up to 325 GHz," Appl. Phys. Lett. 124(7), 072902 (2024) CrossRef Corning Inc., "Datasheet for Corning Pyrex glass," Aug. 13, 2025. DirectLink B. Morova et al., "Fabrication and characterization of large numerical aperture, high-resolution optical fiber bundles based on high-contrast pairs of soft glasses for fluorescence imaging," Opt. Express 27(7), 9502 (2019) CrossRef R.W. Boyd, "The nonlinear optical susceptibility," in Nonlinear Optics, 3rd ed., Academic Press, 2008, 1-67. CrossRef

This paper investigates the resonant nonlinear Schrödinger equation (RNSE) with parabolic law nonlinearity, modeling optical pulse propagation in nonlinear optical fibers. By employing the Kumar–Malik approach, we have derived some analytical soliton solutions for the considered equation. These solutions are in the form of Jacobi elliptic, hyperbolic, trigonometric, exponential functions are obtained by this analytical approach. Dark, bright, singular, and periodic wave solitons are created by selecting proper values for the parameters. The new results are compared with previously obtained results. In addition, the physical properties of the presented solutions are represented by 2d, contour and 3d graphs created by selecting appropriate constant parameters. The findings of this study are novel. The acquired results highlight the simplicity, efficacy, and dependability of this method in the analysis of various nonlinear models encountered in the fields of mathematical physics and engineering.

This research investigates the paraxial nonlinear Schrödinger equation commonly used in quantum mechanics, plasma physics, and nonlinear fiber optics. Employing the extended modified auxiliary equation mapping method, we obtained different soliton solutions, which were tested via Hamiltonian method of stability analysis. The dynamic behavior of the solutions was realized by making use of Stream Density graphs, 3D slice contour graphs, Linear graphs, Density linear graphs, and 2D graphs. The results obtained were tabulated systematically to ensure accuracy; therefore, this research would be of practical use in soliton dynamics and nonlinear wave propagation and can be useful in furtherance of mathematics and bio-mathematics as well as industrial research. The given model and soliton solutions can be efficaciously used to model pulse propagation in optical fibers, investigate energy localization in plasmas, and examine wave packet dynamics in quantum systems. Such uses highlight the practical relevance of paraxial nonlinear Schrödinger equation to promote technologies in telecommunications, fusion science, and nanoscale materials science.

We describe in detail the optical Kerr nonlinearity in our photonic Ising machine (PIM), which employs ultrahigh-speed optical pulse propagation in a lossless fiber loop. Although the peak power of the pulses in the fiber loop is set as low as ∼1 mW, the present PIM requires ultralong-distance pulse propagation of the order of 100,000 km (e.g., ∼2,000 circulations in a 50 km loop) to calculate large-scale optimization problems. As a result, a nonlinear phase rotation of greater than π/2 is accumulated due to the Kerr effect. This nonlinear phase rotation makes it possible to couple between the real (I) and imaginary (Q) parts of the recirculating optical pulse. Thus, as the amplitude of the I-channel changes due to the nonlinear phase rotation, the Q-channel also varies accordingly, and vice versa. We show that this mutual coupling gives rise to a new phenomenon, which we name Kerr-resonanced bifurcation switching, where the accumulated nonlinear phase rotation results in a periodic dip in the cut value of a max-cut problem. This dip phenomenon can be understood as a consequence of optical power peaking in a nonlinear optical fiber loop resonator with Kerr phase rotation. Finally, we propose a method for preventing dip generation by combining a large core fiber and a chirped fiber Bragg grating (CFBG) over a short length, which can reduce the Kerr-induced nonlinear phase rotation.

Stability analysis of mode-locked pulses generated by nonlinear polarization rotation effect using nonlinear optical fiber Jones matrix

A vectorial envelope Maxwell formulation for electromagnetic waveguides with application to nonlinear fiber optics

This study, analyzed the explicit solitary wave soliton for the stochastic resonance nonlinear Schrödinger equation under the Brownian motion. The Schrödinger equations are mostly used to describe how light moves via planar wave guides and nonlinear optical fibres. Analytical technique is applied to gained the various solitary waves and soliton solutions for the resonance nonlinear Schrödinger equation namely, generalized exponential rational function method. This approach is used to find several new trigonometric, exponential, and hyperbolic solutions under the noise. This method is provided us the soliton solutions for nonlinear models that is a computed using an efficient, accurate, capable, and trustworthy method. Furthermore, by varying the parameters, a few graphs of the developed solutions are shown to illustrate the physical setup of stochastic solutions. We anticipate that the obtained results will have significant potential applications in quantum mechanics, magneto-electrodynamics, optical fibres, and heavy ion collisions. Moreover, using the Galilean transformation, the dynamical system of the governing equation is obtained, and the theory of the planar dynamical system is used to carry out its sensitivity, chaotic and bifurcation. By providing certain two- and three-dimensional phase pictures, the existence of chaotic behaviors of the resonance nonlinear Schrödinger equation is examined by taking into account a perturbed term in the resulting dynamical system.

In this study, we present a novel approach to the analytical analysis of the perturbed Gerdjikov–Ivanov model using the extended modified auxiliary equation mapping methodology. We found an even larger and more diverse collection of exact solutions using three parameters in our proposed method: half-dark, half-bright, half-periodic, dark, combined, semi-half-dark, doubly periodic, half-bright, and half-dark. This is the primary way in which our method differs from other currently employed techniques. Recent discoveries have had a significant impact on many natural and physical sciences, including theoretical fluid dynamics, nonlinear fiber optics, electromagnetism, mathematical physics, bio-mathematics, soliton dynamics, plasma physics, industrial research, quantum mechanics, and nuclear physics. Using Mathematica 14.0, we have shown the recently found solutions in graphs of different widths to give a more vivid image of the dynamic features of the solutions. We also verified the answers we had obtained for stability, and the results were displayed in Table 1.

In this paper, we investigate the time-fractional improved (2+1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity, group-velocity dispersion, and spatio-temporal dispersion in nonlinear optics. This equation models the propagation of optical pulses in nonlinear optical fibers. We derive novel optical soliton solutions expressed through exponential and hyperbolic functions, which include bright, bell-shaped, wave, and singular solitons. To illustrate the characteristics of these solutions, we provide two-dimensional, three-dimensional, and contour plots that visualize the magnitude of the conformable improved (2+1)-dimensional nonlinear Schrödinger equation. By selecting suitable values for physical parameters, we demonstrate the diversity of soliton structures and their behaviors. Furthermore, we investigated the influence of the temporal parameter and the conformable fractional-order derivative on the behavior of soliton solutions. The results highlighted the effectiveness and versatility of the modified Kudryashov method in addressing both integer- and fractional-order differential equations, providing analytical solutions that deepen our insight into the dynamics of complex optical systems. These results contribute to the advancement of soliton theory in nonlinear optics and mathematical physics.

Optical solitons have emerged as a highly active research domain in nonlinear fiber optics, driving significant advancements and enabling a wide range of practical applications. This study investigates the dynamics of dark solitons in systems governed by the resonant nonlinear Schrödinger equation (RNLSE). For the RNLSE with third-order (3OD) and fourth-order (4OD) dispersions, the dark soliton solution of the equation in the (1+1)-dimensional case is derived using the F-expansion method, and the analytical study is extended to the (2+1)-dimensional case via the self-similar method. Subsequently, the nonlinear equation incorporating perturbation terms is further studied, with particular attention given to the dark soliton solutions in both one and two dimensions. The soliton dynamics are illustrated through graphical representations to elucidate their propagation characteristics. Finally, modulation instability analysis is conducted to evaluate the stability of the nonlinear system. These theoretical findings provide a solid foundation for experimental investigations of dark solitons within the systems governed by the RNLSE model.

Soliton molecules can enhance information capacity through multi-dimensional coding and provide an efficient data transmission way compared with traditional single-hump solitons in nonlinear optical fiber communication. The multi-hump solitons and their asymptotic behaviors of the coupled cubic-quintic nonlinear Schrödinger equation are studied, which illustrates the influence of the quintic nonlinearity on the ultrashort optical pulse propagation in the medium. On the vanishing background, we deduce the N-soliton solutions. Based on those solutions, we present the proportional and disproportionate vector multi-hump solitons associated with the two, three, and four spectral parameters, and investigate their physical properties such as velocity and amplitude. Then, performing asymptotic analysis on soliton solutions of N = 3 and N = 4, we find that the exact solutions and asymptotic solutions show a clear consistency. When N = 3, we exhibit both asymptotically and graphically the transformation between the soliton molecules and single-, double-hump solitons before and after the interaction. When N = 4, we show the elastic or inelastic interactions among three different types of solitons: between double-hump solitons, between the soliton molecules and single solitons, the soliton molecules are controled by four spectral parameters. Moreover, we reveal that although the soliton's shape changes during the interaction, its intensity is not redistributed between the two components.

Weighted belief propagation (WBP) for the decoding of linear block codes is considered. In WBP, the Tanner graph of the code is unrolled with respect to the iterations of the belief propagation decoder. Then, weights are assigned to the edges of the resulting recurrent network and optimized offline using a training dataset. The main contribution of this paper is an adaptive WBP where the weights of the decoder are determined for each received word. Two variants of this decoder are investigated. In the parallel WBP decoders, the weights take values in a discrete set. A number of WBP decoders are run in parallel to search for the best sequence- of weights in real time. In the two-stage decoder, a small neural network is used to dynamically determine the weights of the WBP decoder for each received word. The proposed adaptive decoders demonstrate significant improvements over the static counterparts in two applications. In the first application, Bose–Chaudhuri–Hocquenghem, polar and quasi-cyclic low-density parity-check (QC-LDPC) codes are used over an additive white Gaussian noise channel. The results indicate that the adaptive WBP achieves bit error rates (BERs) up to an order of magnitude less than the BERs of the static WBP at about the same decoding complexity, depending on the code, its rate, and the signal-to-noise ratio. The second application is a concatenated code designed for a long-haul nonlinear optical fiber channel where the inner code is a QC-LDPC code and the outer code is a spatially coupled LDPC code. In this case, the inner code is decoded using an adaptive WBP, while the outer code is decoded using the sliding window decoder and static belief propagation. The results show that the adaptive WBP provides a coding gain of 0.8 dB compared to the neural normalized min-sum decoder, with about the same computational complexity and decoding latency.

In this study, novel solutions of the anti-cubic nonlinear fractional-order Biswas–Milovic equation are obtained for the first time using the Sardar sub-equation method, the new Kudryashov method, and the addendum to Kudryashov’s method. These methods have been successfully implemented, and several graphical representations have been provided to illustrate that higher-order nonlinear equations can be solved easily and efficiently. The obtained values and the comparisons made in the study provide a novel perspective on soliton solutions, giving new insights into the dynamics of the underlying physical systems, which are particularly relevant in fields such as nonlinear optics, quantum mechanics, and plasma physics. The results of this work show that the derived soliton solutions are directly applicable to modeling and enhancing pulse propagation in nonlinear optical fibers, enabling better control over signal dispersion and attenuation. In addition, these findings establish a mathematical framework for understanding nonlinear effects, which play a crucial role in energy transfer and stability in quantum mechanics and plasma physics.

The fractional nonlinear Schrödinger equation (FNLSE) describes the wave propagation in nonlinear optical fibers, ion-acoustic waves in plasmas, propagation of ultrashort laser pulses, and signal processing. In this study, we investigate the (3 + 1)-dimensional FNLSE with beta fractional derivative to explore soliton dynamics in diverse physical and engineering domains. A couple of analytical approaches, namely the extended sinh-Gordon expansion method and the two-variable (G′/G, 1/G)-expansion method, are used to determine the assorted soliton solutions expressed in trigonometric, hyperbolic, and rational forms. The obtained solutions include bright, bell-shaped, anti-peakon-shaped, anti-bell-shaped, periodic, singular, and singular periodic solitons. We examine the impact of the beta fractional parameter on soliton dynamics through graphical simulations and test the stability analysis using linear stability theory. We also conduct bifurcation analysis to investigate the qualitative behavior of the derived solutions, highlighting the emergence of static solitons via saddle–center bifurcation. The results indicate that the introduced methods generate a wide range of soliton structures and provide insights into their stability and dynamic properties. The results contribute to the theoretical understanding of soliton propagation in nonlinear optical fibers and other dispersive media.