Nonlinear Schrodinger Equation Research Articles

Explicit Synchronized Solitary Waves for Some Models for the Interaction of Long and Short Waves in Dispersive Media

Abstract A system coupling the cubic, nonlinear Schrodinger equation (NLS) and the Korteweg–de Vries equation (KdV) commonly known as the NLS–KdV system was previously analysed to be inconsistently derived and a few alternative systems were proposed as more suitable models for describing the interaction of long and short waves in dispersive media (Deconinck et al. in J Phys A: Math Theor 49:415501, 2016. https://doi.org/10.1088/1751-8113/49/41/415501; Nguyen and Liu in Water Waves 2:327–359, 2020. https://doi.org/10.1007/s42286-020-00038-6; Commun Math Sci 21(3):641–669, 2023. https://doi.org/10.4310/CMS.2023.v21.n3.a3). In this manuscript, these alternative systems are shown to possess synchronized solitary waves via the method of constrained minimization, as well as by direct computation of the associated systems of ODEs. These synchronized solitary waves have the hyperbolic $$\textrm{sech}^2$$ sech 2 -profile typical of dispersive equations.

Nonlinear SchröDinger Equation for Broad Bandwidth Interfacial Waves With a Current Jump

ABSTRACTEmploying multiscale expansion, a modified nonlinear Schrödinger equation (NSE) of interfacial gravity waves for broad bandwidth in a two‐layer fluid of infinite depths with uniform velocity of the upper fluid has been developed. This NSE is widened by slackening the narrow bandwidth constraint with the purpose that it will be more appropriate for utilization to a realistic sea‐wave spectrum. The main results of this work are (a) to find the effect of wind velocity on the wave profiles and the modulational instability regions and (b) to discuss how the broad wave bandwidth affects on the stability results. On the basis of broad and narrow bandwidth equations, the instability regions in the wave numbers plane and the growth rate of instability curves against wave number for Stokes waves are drawn for several values of the dimensionless speed of the upper layer and the wave steepness. The key feature of this problem is that the new equation due to broad bandwidth is observed to predict the instability region in an excellent agreement with the exact results and due to sufficient bandwidth, this broad‐banded equation is expected to be more appropriate for realistic ocean wave problems.

New insights into the diversity of stochastic solutions and dynamical analysis for the complex cubic NLSE with δ-potential through Brownian process

Abstract The nonlinear Schrodinger equation is a key tool for modeling wave propagation in nonlinear and dispersive
media. This study focuses on the complex cubic nonlinear Schrodinger equation with δ-potential,
explored through the Brownian process. The investigation begins with the derivation of stochastic solitary
wave solutions using the modified exp(−Ψ(ξ)) expansion method. To illustrate the noise effects, 3D and 2D
visualizations are displayed for different non-negative values of noise parameter under suitable parameter
values. Additionally, qualitative analysis of both perturbed and unperturbed dynamical systems is conducted
using bifurcation and chaos theory. In bifurcation analysis, we analyze the detailed parameter analysis near
fixed points of the unperturbed system. An external periodic force is applied to perturb the system, leading
to an investigation of its chaotic behavior. Chaos detection tools are employed to predict the behavior of the
perturbed dynamical system, with results validated through visual representations. Multistability analysis
is conducted under varying initial conditions to identify multiple stable states in the perturbed dynamical
system, contributing to chaotic behavior. Also, sensitivity analysis of the Hamiltonian system is performed
for different initial conditions. The novelty of this work lies in the significance of the obtained results, which
have not been previously explored for the considered equation. These findings offer noteworthy insights into
the behavior of the complex cubic nonlinear Schrodinger equation with δ-potential and its applications in
fields such as nonlinear optics, quantum mechanics and Bose-Einstein condensates.

New perspective to the coupled fractional nonlinear Schrodinger equations in dual-core optical fibers

New perspective to the coupled fractional nonlinear Schrodinger equations in dual-core optical fibers

Novel insights into high-order dispersion and soliton dynamics in optical fibers via the perturbed Schrödinger–Hirota equation

This study offers a comprehensive analysis of the Perturbed Schrödinger -Hirota Equation (PSHE), crucial for understanding soliton dynamics in modern optical communication systems. We extended the traditional Nonlinear Schrödinger Equation (NLSE) to include higher-order nonlinearities and spatiotemporal dispersion, capturing the complexities of light pulse propagation. Employing the modified auxiliary equation method and Adomian Decomposition Method (ADM), we derived a spectrum of exact traveling wave solutions, encompassing exponential, rational, trigonometric, and hyperbolic functions. These solutions provide insights into soliton behaviors across diverse parameters, essential for optimizing fiber optic systems. The precision of our analytical solutions was validated through numerical solutions, and we explored modulation instability, revealing conditions for soliton formation and evolution. The findings have significant implications for the design and optimization of next-generation optical communication technologies.

OAM mode propagation and supercontinuum generation in a nested photonic crystal fiber

Abstract The proposed work presents a nested Photonic Crystal Fiber(PCF) with
a hexagonal lattice of air holes, capable of supporting Orbital Angular Momentum
modes(OAM) and generating a wide-ranging supercontinuum spectrum, further
enhancing its utility in various optical applications. The Photonic Crystal Fiber
consists of a dense flint SF6 and a very light flint LLF1, fabricated from Schott. Finite
Element Method is used for the analysis of Nested Photonic Crystal Fiber characteris-
tics using COMSOL Multiphysics. A total of 120 Orbital Angular Momentum modes
modes have been obtained in the fiber, with 86 in the outer core and 34 in the inner
core. The generalised Nonlinear Schrodinger Equation provides a numerical description
of the pulse that is propagating through the PCF. In this article, the proposed nested
PCF can be used to generate supercontinuum by pumping with a hyperbolic secant
pulse, with red an average power of 71 mW. The resulting supercontinuum spans an
impressive bandwidth, extending from approximately 1122 to 2072 nm.

A fast normal splitting preconditioner for attractive coupled nonlinear Schroedinger equations with fractional Laplacian

A fast normal splitting preconditioner for attractive coupled nonlinear Schroedinger equations with fractional Laplacian

Nonplanar wave modulation and rogue wave formation in lunar dusty plasma

Abstract Motivated by the coming lunar missions, namely Luna-25 and Luna-27, scheduled for landing in the coming years, the investigation of the properties and characteristics of lunar dust, which, in contrast to terrestrial dust, adversely affects technology and creates significant risks to humans. As the Moon orbits Earth, plasma from its surroundings impacts its surface and dust grains above and below; thus, about one-third of the moon's orbit is occupied by the tail magnetosphere, which is thicker and more energetic than the tail regions' very thin plasma. Therefore, investigating the plasma properties in this region is of interest to avoid any unfavorable waves that may affect on the space vessels electronics. For this purpose, we use a three-component plasma fluid model study non-planar nonlinear electrostatic wave modulation on the lunar dark side. A non-planar, nonlinear Schrödinger equation (NLSE) is derived to examine the nonlinear wave envelope, which is produced from the dynamics of lunar negatively charged dust particles, positively charged ions, and Maxwellian electrons. Slow and fast dust-ion acoustic modes exist. The instability region allows the occurrence of nonlinear wave envelopes in the form of rogue waves. The latter are proposed as a physical, temporal, and spatial concentration of energy within a localized wave packet, which is sufficient to establish accumulation particles and influence the overall stability of the environment. The relative ion-to-electron temperature ratio enhances the fast mode's maximum growth rate. The growth rate of modulational instability for the spherical mode is lower than that for the cylindrical mode.

Introducing and analyzing a new combined version of the unstable Schrödinger equations with strong and weak stability effects

In the literature, two types of unstable nonlinear Schrodinger equations have been independently developed and studied. Each was derived by incorporating either a self-effect term or a time-space dispersion term into the standard nonlinear Schrodinger equation. Both models describe the time evolution of disturbances in unstable media. The primary contribution of this work is the combination of these two types into a single, new unstable version of the nonlinear Schrodinger equation. This new model is analyzed using two effective methods: the rational sine-cosine and the rational sinh-cosh functions. Additionally, a comparison test of the embedded unstable terms is conducted to assess their respective impacts on the stability of the Schrodinger model. Finally, graphical analyses, including 2D and 3D plots, are performed to validate the study’s findings.

A higher-order quadratic NLS equation on the half-line

The well-posedness of the initial-boundary value problem for higher-order quadratic nonlinear Schrödinger equations on the half-line is studied by utilizing the Fokas solution formula for the corresponding linear problem. Using this formula, linear estimates are derived in Bourgain spaces for initial data in spatial Sobolev spaces on the half-line and boundary data in temporal Sobolev spaces suggested by the time regularity of the linear initial value problem. Then, the needed bilinear estimates are derived and used for showing that the iteration map defined via the Fokas solution formula is a contraction in appropriate solution spaces. Finally, well-posedness is established for optimal Sobolev exponents in a way analogous to the case of the initial value problem on the whole line with solutions in classical Bourgain spaces.

Fokas-Lenells Derivative nonlinear Schrödinger equation its associated fundamental forms and Gaussian curvature

Abstract One of the most important tasks in mathematics and physics is to connect differential geometry and nonlinear differential equations. In the study of nonlinear optics, integrable nonlinear differential equations such as the nonlinear Schrödinger equation (NLSE) and higher-order NLSE (HNLSE) play crucial roles. Because of the medium’s balance between dispersion and nonlinearity, all of these systems display soliton solutions. The soliton surfaces, or manifolds, connected to these integrable systems hold significance in numerous areas of mathematics and physics. We examine the use of soliton theory in differential geometry in this paper. We build the two-dimensional soliton surface in the three-dimensional Euclidean space by taking into account the Fokas-Lenells Derivative nonlinear Schrödinger equation (also known as the gauged Fokas-Lenells equation). The same is constructed by us using the Sym-Tafel formula. The first and second fundamental forms, surface area, and Gaussian curvature are obtained using a lax representation of the gauged FLE.

Stable and Periodic Soliton Molecules in the Fiber Ring Laser

ABSTRACTStable and periodic soliton molecules (SMs) have been studied numerically by coupled nonlinear Schrodinger equations (CNLSEs). Split and formation processes of different‐order SMs are observed by changing saturation power and filtering bandwidth. Bimodal pulses of SMs are in a transition state toward higher‐order SMs. Short‐period and long‐period SMs are observed by adjusting saturation power. The research contributes to revealing SM dynamics, which has potential applications in encoding as well as optical storage.

On Action Ground States of Defocusing Nonlinear Schrodinger Equations

On Action Ground States of Defocusing Nonlinear Schrodinger Equations

Dynamic description of soliton solutions of nonlinear schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Dynamical structures of optical solitons for highly dispersive perturbed NLSE with β-fractional derivatives and a sextic power-law refractive index using a novel approach

In this paper, we investigate the highly dispersive perturbed nonlinear Schrödinger equation (NLSE) with β-fractional derivatives, generalized nonlocal laws and sextic-power law refractive index. This equation is crucial for modeling complex phenomena in nonlinear optics, such as soliton formation, light pulse propagation in optical fibers, and light wave control, with potential applications in designing efficient optical communication devices. Furthermore, it provides a framework for understanding the intricate interactions between high dispersion, nonlocality, and complex nonlinearity, contributing to the development of new theories in wave physics. To accomplish this, we use the modified extended direct algebraic method. A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, combo bright-dark and singular soliton solutions. Additionally, singular periodic solutions, rational and exponential solutions. Furthermore, graphical simulations are presented that highlight the distinctive characteristics of these solutions. Compared to Nofal et al. (Optik 228:166120, 2021), the proposed technique produced novel and diversified results. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. The extracted solutions confirmed the efficacy and strength of the current approach. The parameter constraints ensure the existence of the obtained soliton solutions. It is worth noting that the proposed method, being effective, consistent, and influential, can be applied to solve various other physical models and related disciplines.

Radial solutions of initial boundary value problems of nonlinear Schrödinger equations in ℝn

The article studies an initial boundary valueproblem (ibvp) for the radial solutions of the nonlinear Schrödinger (NLS) equation in a radially symmetric region $\Omega\in \mathbb R^n$ with boundaries. All such regions can be classified into three types: a ball Ω0 centred at origin, a region Ω1 outside a ball, and an n-dimensional annulus Ω2. To study the well-posedness of those ibvps, the function spaces for the boundary data must be specified in terms of the solutions in appropriate Sobolev spaces. It is shown that when $\Omega = \Omega_1$ , the ibvp for the NLS equation is locally well-posed in $ C( [0, T^*]; H^s(\Omega_1))$ if the initial data is in $H^s(\Omega_1)$ and boundary data is in $ H^{\frac{2s+1}{4}}(0, T)$ with $s \geq 0$ . This is the optimal regularity for the boundary data and cannot be improved. When $\Omega = \Omega_2$ , the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_2))$ if the initial data is in $ H^s(\Omega_2)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with $s \geq 0$ . In this case, the boundary data requires $1/4$ more derivative compared to the case when $\Omega = \Omega_1$ . When $\Omega = \Omega_0$ with n = 2 (the case with n > 2 can be discussed similarly), the ibvp is locally well-posed in $ C( [0, T^*]; H^s(\Omega_0))$ if the initial data is in $ H^s(\Omega_0)$ and boundary data is in $ H^{\frac{s+1}{2}}(0, T)$ with s > 1 (or $s \gt n/2$ ). Due to the lack of Strichartz estimates for the corresponding boundary integral operator with $ 0 \leq s \leq 1$ , the local well-posedness can only be achieved for s > 1. It is noted that the well-posedness results on Ω0 and Ω2 are the first ones for the ibvp of NLS equations in bounded regions of higher dimension.

Unveiling of Highly Dispersive Dual-Solitons and Modulation Instability Analysis for Dual-Mode Extension of a Non-Linear Schrödinger Equation

The two-mode equations are nonlinear models that describe the behavior of two-way waves moving simultaneously while being affected by confined phase velocity. This article expands a non-linear Schrödinger equation (NLSE) by constructing it as a dual-mode structure. Applying the modified extended direct algebraic method (MEDAM) yields exact and explicit solutions. The results of this investigation have significant implications for the propagation of solitons in nonlinear optics. There are multiple resulted solutions that comprise singular periodic solutions, Weierstrass elliptic doubly periodic solutions, Jacobi elliptic function (JEF), singular soliton, bright soliton, dark soliton, and rational solutions, moreover, hyperbolic wave solutions. We show our acquired traveling wave solutions’ uniqueness and significant addition to current research by contrasting them with the body of existing literature. The method’s effectiveness shows that it may be used to address a wide variety of nonlinear problems across multiple disciplines, particularly in the theory of soliton, as the studied model appears in many applications. Additionally, we display the outlines of some of these discovered solution behaviors in 3D and 2D graphs to help with comprehension. Finally, we analyze modulation instability to examine the stability of the discovered solutions.

Nonlinear Interaction and Modulational Instability of Dust Acoustic Disturbances in a Magnetized Cometary Plasma

ABSTRACTThe interplay of dust acoustic waves (DAWs) and it's amplitude modulation is investigated in a magnetized five‐component cometary plasma comprising positively charged hydrogen and oxygen ions, negatively charged dust grains, along with superthermal electrons (solar and cometary). The interplay dynamics of the dust acoustic waves (DAWs) is described by the Kortweg–de Vries (KdV) equation. The appearance of multi‐soliton profiles are derived using Hirota's method. The amplitude modulation of the DAWs is investigated through nonlinear Schrodinger equation. In the numerical study, it appears that the dark‐envelope soliton may exist in such cometary plasmas, but the emergence of bright‐envelope soliton is not possible at any condition. Superthermality, magnetic field strength, ion and electron concentration, and other physical factors have all been explored for their impacts. The findings of this research may contribute to our understanding of the nonlinear characteristics and wave propagation stability of electrostatic wave excitations in cometary coma environments.

New dynamical behaviors and soliton solutions of the coupled nonlinear Schrodinger equation

New dynamical behaviors and soliton solutions of the coupled nonlinear Schrodinger equation

On the radius of spatial analyticity for the quintic fourth-order nonlinear Schrodinger equation on ℝ2

On the radius of spatial analyticity for the quintic fourth-order nonlinear Schrodinger equation on ℝ²