Exponential Wave Research Articles

Exploring chaos and stability: dynamic insights into the stochastic Davey-Stewartson system through advanced sensitivity analysis

In this study, we investigate the stochastic Davey-Stewartson equation in the presence of noise. These two-dimensional integrable equations are higher-dimensional versions of the nonlinear Schrödinger equation. Davey-Stewartson equations are important in plasma physics, nonlinear optics, hydrodynamics, and other disciplines because the solutions they provide are valuable in understanding many complex physical phenomena. We employ a modified version of the G′G2 approach to handle variable-coefficient systems with imaginary components, such as nonlinear Schrödinger systems. We have discovered a wide range of precise traveling wave solutions, including solitons, kink, periodic, and rational solutions. These solutions may have a significant impact in the domains of engineering and plasma physics. The presented techniques effectively achieve a variety of exponential solutions, including bright, dark, single, rational, and periodic solitary wave solutions. We used MATLAB to simulate our findings and provide 3D, 2D, and counter graphs that illustrate the impact of noise on the precise solutions of the stochastic Davey-Stewartson problem. We apply the Galilean transformation to derive the planar dynamical system, which enhances our understanding of the system’s dynamical analysis. We also conduct a sensitivity analysis to observe the systematic response to various initial conditions. This analysis focuses on the symmetrical aspects of the system and includes the illustration of phase portraits with equilibrium points. We also investigate the chaotic behavior of the planer dynamical system by introducing an additional external force. We predict the system’s behavior to increase the value of intensity and frequency. Our analysis reveals periodic, quasi-periodic, and chaotic processes.

Comparative study of novel solitary wave solutions with unveiling bifurcation and chaotic structure modelled by stochastic dynamical system

Abstract In this study, we conduct a comprehensive investigation of the novel characteristics of the (2 + 1)-dimensional stochastic Hirota–Maccari System (SHMS), which is a prominent mathematical model with significant applications in the field of nonlinear science and applied mathematics. Specifically, SHMS plays a critical role in the study of soliton dynamics, nonlinear wave propagation, and stochastic effects in complex physical systems such as fluid dynamics, optics, and plasma physics. In order to account for the abrupt and significant fluctuation, the aforementioned system is investigated using a Wiener process with multiplicative noise in the Itô sense. The considered equation is studied by the new extended direct algebraic method (NEDAM) and the modified Sardar sub-equation (MSSE) method. By solving this equation, we systematically derived the novel soliton solutions in the form of dark, dark-bright, bright-dark, singular, periodic, exponential, and rational forms. Additionally, we also categorize and analyze the W-shape, M-shape, bell shape, exponential, and hyperbolic soliton wave solutions, which are not documented by researchers. The bifurcation, chaos and sensitivity analysis has been depicted which represent the applicability of the system in different dynamics. These findings greatly advance our knowledge of nonlinear wave events in higher-dimensional stochastic systems both theoretically and in terms of possible applications. These findings are poised to open new avenues for future research into the applicability of stochastic nonlinear models in various scientific and industrial domains.

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Propagation of wave insights to the Chiral Schrödinger equation along with bifurcation analysis and diverse optical soliton solutions

In this study, the modified Sardar sub-equation method is capitalised to secure soliton solutions to the (1+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional chiral nonlinear Schrödinger (NLS) equation. Chiral soliton propagation in nuclear physics is an extremely attractive field because of its wide applications in communications and ultrafast signal routing systems. Additionally, we perform bifurcation analysis to gain a deeper understanding of the dynamics of the chiral NLS equation. This highlights the complex behaviour of the system and exposes the conditions under which various types of bifurcations occur. Additionally, a sensitivity analysis is performed to assess how small changes in initial conditions and parameters influence the solutions, offering valuable perspectives on the stability and dependability of the acquired solutions. By employing the above-mentioned methodology, we derive a variety of exact solutions, including periodic, singular, dark, bright, mixed trigonometric, exponential, hyperbolic, and rational wave solutions. The study’s findings advance our theoretical knowledge of chiral NLS equations and have potential applications in optical communication and related fields.

Relativistic Correction from the Four-Body Nonadiabatic Exponential Wave Function.

We present a method for calculating the relativistic correction in hydrogen molecules that significantly exceeds the accuracy of all the previous literature results. This method utilizes the explicitly correlated nonadiabatic exponential wave function, and thus treats electrons and nuclei equivalently. The proposed method can be applied to any rovibrational state, including highly excited ones. The numerical precision of the relativistic correction reaches several kHz (∼10-7 cm-1), which is below the best experimental accuracy.

Privacy and Security Challenges in IoT Deployments

The IoT means the Internet of Things – the integration of things and systems in the digital world across numerous industries. As has been seen we can opened up an exponential wave of efficiency, automation and innovation in the domain of IoT while at the same time we have exposed an infinite number of privacy and security threats that could severely jeopardize individuals, organizations, or nations. This paper thus critically analyses the status and trends of IoT privacy and security threats focusing more on the most common threats as well as impacts arising from their exploitation. From the analysis of the most reported case studies in this paper, the kind of recent IoT security breaches and the efforts made to address these are revealed. Moreover, the paper provides guidelines on imposing improvements to IoT systems security by persuading the implementation of global and inclusive security that targets prior and emergent threats. At the same time, this work also synthesises the most current advances in the field of IoTs security research, as well as outline specific directions for future research, which is also evidence of the necessity for Iots stakeholders to contin ue the improvement and collaboration to protect the future of IoTs environments.

Error estimates of exponential wave integrators for the Dirac equation in the massless and nonrelativistic regime

AbstractWe present exponential wave integrator Fourier pseudospectral (EWI‐FP) methods and establish their error estimates of the fully discrete schemes for the Dirac equation in the massless and nonrelativistic regime. This regime involves a small dimensionless parameter where , and is inversely proportional to the speed of light. The solution exhibits highly oscillatory behavior in time and rapid wave propagation in space in this regime. Specifically, the time oscillations have a wavelength of , while the spatial oscillations have a wavelength of , with a wave speed of . We employ (symmetric) exponential wave integrators for temporal derivatives and Fourier spectral discretization for spatial derivatives. We rigorously derive the error bounds which explicitly depend on the mesh size , the time step and the small dimensionless parameter . The error estimates for the EWI‐FP methods demonstrate that their meshing strategy requirement (‐scalability) necessitates setting and when . Finally, some numerical examples are provided to validate the error bounds.

An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity

An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

Diverse exact solutions to Davey–Stewartson model using modified extended mapping method

In this study, we obtain solitary wave solutions and other exact wave solutions for Davey–Stewartson equation (DSE), which explains how waves move through water with a finite depth while being affected by gravity and surface tension. The study is conducted with the aid of the modified extended mapping method (MEMM). A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, and singular solitary wave solutions. Additionally, Jacobi elliptic function solutions, exponential wave solutions, singular periodic wave solutions, rational wave solutions, and periodic wave solutions are also offered. To help readers physically grasp the acquired solutions, graphical representations of some of the extracted solutions are provided.

Improved uniform error bounds of a Lawson-type exponential wave integrator method for the Klein-Gordon-Dirac equation

For the Klein-Gordon-Dirac equation (KGDE) with small coupling constant ε∈(0,1], we propose a Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method and establish the improved uniform error bounds in the time domain at O(1/ε). We first convert the KGDE to a coupled system and then consider LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric which is an important structure in numerical geometric integration. Through careful and rigorous convergence analysis, we establish the error bounds O(hm+ετ2+τ0m) for the full-discretization, where m is determined by the regularity conditions. If the solution is sufficiently smooth with m sufficiently large, we obtain the errors with improved uniform bounds at O(hm+ετ2) in the long-time domain up to O(1/ε). These error bounds are much better than the classical bounds O(hm+τ2) provided by the traditional analysis for the non-Lawson-type exponential wave integrators equipped with Fourier pseudo-spectral method. Combined with the classical analysis tools such as mathematical induction and energy method, we complete our error analysis by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation. By applying the LEWIFP method to some problems, we show the numerical results to support our error bounds. In addition, the numerical results also show that the discrete mass and energy are stable in the time domain which is long enough. Finally we extend our method to the oscillatory problem.

Derivation of optical solitons for perturbed highly dispersive conformable fractional nonlinear Schrödinger equation with sextic-power law refractive index

In this article, the modified extended direct algebraic method is applied for the perturbed highly dispersive nonlinear Schrödinger equation with conformable fractional derivative and sextic-power law refractive index. Various types of solutions are extracted such as bright solitons, dark solitons, combo bright-dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions and rational solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order.

Qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative

The focus of the article studies the qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative. Firstly, the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative is transformed into two-dimensional planar dynamic system through complex exponential wave transformation. Secondly, phase portrait and its orbit of the system are analyzed through the theory of dynamical system. By using Maple software the phase portrait, Poincaré section and sensitivity analysis are compared and displayed. Finally, parameter conditions for stable systems are provided through modulation instability.

Improved uniform error bounds on an exponential wave integrator method for the nonlinear Schrödinger equation with wave operator and weak nonlinearity

In this paper, we establish the improved uniform error bounds for the Lawson-type exponential wave integrator Fourier pseudo-spectral (LEWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW) and weak nonlinearity characterized by a small constant ε∈(0,1]. We first convert the NLSW to a coupled system and then consider the LEWIFP method for the coupled system. The LEWIFP method is proved to be time symmetric and mass-conservative which is very important from the point of view of numerical geometric integration. Through careful and rigorous convergence analysis, we establish improved uniform error bounds for the full-discretization at O(hm+ε2pτ2) in the long-time domain up to O(1/ε2p) where m is determined by the regularity conditions. These error bounds are much better than the classical error bounds O(hm+τ2) provided by the traditional error analysis. In error analysis, in addition to the classical tools such as mathematical induction and energy method, we adopt the regularity compensation oscillation (RCO) technique to analyze the accumulation of errors carefully. The numerical experiments support our error estimates and demonstrate the discrete mass-conservation. In addition, the numerical results show that the discrete energy is stable in the time domain up to O(1/ε2p).

Analysis of propagation of normal shocks in Van der Waals gas: Buongiorno model

This paper presents an analytical investigation into the propagation of a normal shock wave in Van der Waals gas flow, employing the Buongiorno model. It explores the dynamics of this flow type under a shock wave, considering modified Rankine–Hugoniot (RH) jump conditions within the Buongiorno model framework. By solving the RH conditions, this study provides a solution for gas flows with varying nanoparticle concentrations, facilitating an examination of parameter variations under discontinuity. The abstract further emphasizes the graphical representation of velocity coefficient and pressure in adiabatic gas flow during a shock wave, considering compressive and exponential waves. The analysis accounts for nanoparticle concentration and non-ideal parameters.

Augmenting sparse arrays in ocean acoustics: A Gaussian Process approach

Source localization and geoacoustic inversion performance depends on the location of receiving phones, their spacing, their number, and the array aperture. Performance suffers when practical design issues limit the capabilities of the array. In this work, sparse arrays are augmented leading to the generation of virtual arrays for better sampling of the ocean and, thus, improved estimation performance. To that effect, Gaussian Processes are employed, which are shown to create high-fidelity field “measurements” at virtual, densely spaced hydrophones. Kernel functions are key building blocks in the implementation of Gaussian Processes, as they quantify the field coherence at neighboring spatial points. Functions of interest are the squared exponential, Matern, and plane wave kernels. We validate our method with application to synthetic data as well as data collected during the Seabed Characterization Experiment conducted in 2022. [Work supported by ONR.]

Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation

In this paper, we derive a new exponential wave integrator sine pseudo-spectral (EWI-SP) method for the higher-order Boussinesq equation involving the higher-order effects of dispersion. The method is fully-explicit and it has fourth order accuracy in time and spectral accuracy in space. We rigorously carry out error analysis and establish error bounds in the Sobolev spaces. The performance of the EWI-SP method is illustrated by examining the long-time evolution of the single solitary wave, single wave splitting, and head-on collision of solitary waves. Numerical experiments confirm the theoretical results.

Optimal Error Bounds on the Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity

Optimal Error Bounds on the Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity

Investigation of solitons in magneto-optic waveguides with Kudryashov’s law nonlinear refractive index for coupled system of generalized nonlinear Schrödinger’s equations using modified extended mapping method

In this work, we investigate the optical solitons and other waves through magneto-optic waveguides with Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation factors using the modified extended mapping approach. Many classifications of solutions are established like bright solitons, dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions, rational wave, solutions, Weierstrass elliptic doubly periodic solutions, and Jacobi elliptic function solutions. Some of the extracted solutions are described graphically to provide their physical understanding of the acquired solutions.

Exploring analytical results for (2+1) dimensional breaking soliton equation and stochastic fractional Broer-Kaup system

<abstract> <p>This paper introduces a pioneering exploration of the stochastic (2+1) dimensional breaking soliton equation (SBSE) and the stochastic fractional Broer-Kaup system (SFBK), employing the first integral method to uncover explicit solutions, including trigonometric, exponential, hyperbolic, and solitary wave solutions. Despite the extensive application of the Broer-Kaup model in tsunami wave analysis and plasma physics, existing literature has largely overlooked the complexity introduced by stochastic elements and fractional dimensions. Our study fills this critical gap by extending the traditional Broer-Kaup equations through the lens of stochastic forces, thereby offering a more comprehensive framework for analyzing hydrodynamic wave models. The novelty of our approach lies in the detailed investigation of the SBSE and SFBK equations, providing new insights into the behavior of shallow water waves under the influence of randomness. This work not only advances theoretical understanding but also enhances practical analysis capabilities by illustrating the effects of noise on wave propagation. Utilizing MATLAB for visual representation, we demonstrate the efficiency and flexibility of our method in addressing these sophisticated physical processes. The analytical solutions derived here mark a significant departure from previous findings, contributing novel perspectives to the field and paving the way for future research into complex wave dynamics.</p> </abstract>