Solitary Waves Research Articles

Lump Kink interactional and breather-type waves solutions of (3+1)-dimensional shallow water wave dynamical model and its stability with applications

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient.

Dynamics of propagation patterns: An analytical investigation into fractional systems

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ([Formula: see text]) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ([Formula: see text]) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics.

Effects of Brownian motion on solitary wave structures for 1-D stochastic Poisson Nernst Plank system in electrobiochemical

Effects of Brownian motion on solitary wave structures for 1-D stochastic Poisson Nernst Plank system in electrobiochemical

An overview of ion-acoustic solitary and shock waves in a magnetized nonthermal plasma: influence of trapped positrons and electrons

A magnetized nonthermal electron–positron-ion (e-p-i) plasma is considered to study the propagation properties of ion-acoustic solitary and shock waves in the presence of trapped positrons and electrons for the first time. The Schamel-κ (kappa) distribution function that describes plasma nonthermality and particle trapping is assumed to consider electrons and positrons. The diffusive effect of ion plasma fluid, which is responsible for shock dynamics, is taken into account. A nonlinear Schamel-Korteweg–de Vries-Burgers’ (SKdVB) equation is derived by employing the reductive perturbation approach, and the solitary and shock wave solutions of the SKdVB equation have also been derived for different limiting cases. It is found that only positive potential nonlinear structures (for both solitary and shock waves) are formed in the proposed plasma system. The condition for stable solitons in the absence of dissipation is analyzed, and the nature of arbitrary amplitude solitary waves (obtained via the Sagdeev potential approach) is discussed. It is found through theoretical and numerical investigation that different plasma compositional parameters (such as the trapping effect of electrons (β e ) and positrons (β p ), the obliquity effect (θ), electron-to-ion number density ratio (µ e ), the magnetic field effect (via Ω) and the viscous effect (via η)) have a significant influence on the dynamics of ion-acoustic solitary and shock waves. The theoretical and numerical investigations in this study may be helpful in describing the nature of localized structures in different plasma contexts, e.g. space and astrophysical plasmas and experimental plasmas where electron–positron-ion plasmas exist.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

Analyzing solitary wave solutions of the nonlinear Murray equation for blood flow in vessels with non-uniform wall properties

Solitary wave solutions are of great interest to bio-mathematicians and other scientists because they provide a basic description of nonlinear phenomena with many practical applications. They provide a strong foundation for the development of novel biological and medical models and therapies because of their remarkable behavior and persistence. They have the potential to improve our comprehension of intricate biological systems and help us create novel therapeutic approaches, which is something that researchers are actively investigating. In this study, solitary wave solutions of the nonlinear Murray equation will be discovered using a modified extended direct algebraic method. These solutions represent a uniform variation in blood vessel shape and diameter that can be used to stimulate blood flow in patients with cardiovascular disease. These solutions are newly in the literature, and give researchers an important tool for grasping complex biological systems. To see how the solitary wave solutions behave, graphs are displayed using Matlab.

New solitary waves in a convecting fluid

It is critical to examine the effects of small perturbations on solitary wave solutions to nonlinear wave equations because of the existence of the unavoidable perturbations in real applications. In this work, we aim to study the solitary wave solutions for the dissipative perturbed mKdV equation which is proposed to model the evolution of shallow wave in a convecting fluid. By using the geometric singular perturbation theorem, the three-dimensional traveling wave system is reduced to a planar dynamical system with regular perturbation. By examining the homoclinic bifurcations via Melnikov method, we show that not only some solitary waves with particularly chosen wave speed persist but also some new type of solitary waves appear under small perturbation. The theoretical analysis results are illustrated by numerical simulations.

Ion acoustic solitary waves in an adiabatic dusty plasma: roles of superthermal electrons, ion loss and ionization

Abstract The propagation of dust ion acoustic solitary wave (DIASW) is investigated in the multicomponent dusty plasma with adiabatic ions, superthermal electrons and stationary dust. The reductive perturbation method is employed to derive the damped Korteweg-de Vries (DKdV) equation which describes DIASW. The result reveals that, the adiabaticity of ions significantly modifies the basic features of the DIASW. The ionization effect makes the solitary wave growing, while collisions reduce the growth rate and even lead to the damping. With the increase in ionization cross section $\frac{{\Delta \sigma }}{{{\sigma _0}}}$, ion-to-electron density ratio ${{\delta}_{\rm{ie}}}$ and superthermal electrons parameter $\kappa $, the effect of ionization on DIASW enhances.

Abundant soliton solutions on fractional kraenkel manna merle model (FKMM) via new extended of generalized exponential rational function approach (GERFA)

Abstract In this paper, we use the generalized exponential rational function approach (GERFA) for constructing new solitary wave solutions for the fractional Kraenkel Manna Merle (FKMM) model, which is saturated ferromagnetic materials (MF) with a field outside that has very little conductivity, reflects the nonlinear of an ultrashort pulse. The fractional performance of the proposed model is assessed using the beta-derivative. As a result, this investigation demonstrates the efficacy and simplicity of the offered strategy. Some of the revealed solutions' 3D graphs are also employed in the numerical simulations. The findings show that the model theoretically has extraordinarily rich soliton structures.

A novel perspective for simulations of the Modified Equal-Width Wave equation by cubic Hermite B-spline collocation method

In the current study, the Modified Equal-Width (MEW) equation will be handled numerically by a novel technique using collocation finite element method where cubic Hermite B-splines are used as trial functions. To test the accuracy and efficiency of the method, four different experimental problems; namely, the motion of a single solitary wave, interaction of two solitary waves, interaction of three solitary waves and the birth of solitons with the Maxwellian initial condition will be investigated. In order to verify, the validity and reliability of the proposed method, the newly obtained error norms L2 and L∞ as well as three conservation constants have been compared with some of the other numerical results given in the literature at the same parameters. Furthermore, some wave profiles of the newly obtained numerical results have been given to demonstrate that each test problem exhibits accurate physical simulations. The advantage of the proposed method over other methods is the usage of inner points at Legendre and Chebyshev polynomial roots. This advantage results in better accuracy with less number of elements in spatial direction. The results of the numerical experiments clearly reveal that the presented scheme produces more accurate results even with comparatively coarser grids.

Interaction between internal solitary waves and the seafloor in the deep sea

Interaction between internal solitary waves and the seafloor in the deep sea

Exploring travelling wave solutions, bifurcation, chaos, and sensitivity analysis in the (3+1)-dimensional gKdV-ZK model: A comprehensive study using Lie symmetry methodology

This article presents a study on the generalized Korteweg-de Vries-Zakharov-Kuznetsov (gKdV-ZK) model, which is a nonlinear system that demonstrates the effect of magnetic fields on weak ion-acoustic waves in plasma consisting of cold and hot electrons. The research entails investigating the reduction of symmetry through Lie group analysis, scrutinizing the characteristics of the dynamic structure using bifurcation phase diagrams, and examining the dynamic behaviour of the perturbed dynamical system employing chaos theory. Methods such as 3D and 2D phase portraits, time series analysis, Poincaré maps, exploration of multistability in the autonomous structure across various initial conditions, Lyapunov exponents, and bifurcation diagrams are exercised to demonstrate chaotic behaviour. Additionally, the research establishes general forms of solitary wave solutions, encompassing hyperbolic, trigonometric, and rational soliton solutions, through the utilization of a modified auxiliary equation approach to analytically address the examined problem. These findings are visually depicted as 2D and 3D graphs with carefully selected parameters, accompanied by their corresponding constraint conditions. Furthermore, the sensitivity analysis of the studied equation is deliberated upon and visually illustrated. The uncovered findings are captivating, innovative, and potentially beneficial for comprehending various physical phenomena in engineering and science.

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

Transcending classical diffusion models: nonlinear dynamics and solitary waves in the fractional Chaffee–Infante equation

When discrete fronts and pulses form a single family: FPU chain with hardening-softening springs

We consider a version of the classical Hamiltonian FPU (Fermi–Pasta–Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink–antikink bundle. Direct numerical simulations and Floquet analysis of linear stability suggest that all of the obtained solutions are effectively stable. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem both qualitatively and quantitatively.

Solitary waves in slightly dispersive quasi-incompressible hyperelastic materials

Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys.

New solitary wave solutions of space-time fractional dynamical models

New solitary wave solutions of space-time fractional dynamical models

On Multiple-Type Wave Solutions for the Nonlinear Coupled Time-Fractional Schrödinger Model

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics.

Magnetoacoustic waves in spin-1/2 dense quantum degenerate plasma: nonlinear dynamics and dissipative effects

Abstract Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments.

Unveiling three-dimensional sea surface signatures caused by internal solitary waves: insights from the surface water ocean topography mission

Unveiling three-dimensional sea surface signatures caused by internal solitary waves: insights from the surface water ocean topography mission

Learning Traveling Solitary Waves Using Separable Gaussian Neural Networks

In this paper, we apply a machine learning approach to learning traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture called the Separable Gaussian Neural Network (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs, which treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into what is called the co-traveling wave frame. This adaptation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as “leftons”) within the (1 + 1)-dimensional b-family of PDEs. In addition, we expand our investigation and explore not only peakon solutions in the ab-family but also compacton solutions in the (2 + 1)-dimensional Rosenau–Hyman family of PDEs. A comparative analysis with a multi-layer perceptron (MLP) reveals that the SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader applications in solving complex nonlinear PDEs.