Solitary Waves Research Articles

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

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

Solitary waves and double layers of dust-acoustic waves propagating in a non-ideal dusty plasma

Solitary waves and double layers of dust-acoustic waves propagating in a non-ideal dusty plasma

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

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.

Solitary waves in slightly dispersive quasi-incompressible hyperelastic materials

Solitary waves in slightly dispersive quasi-incompressible hyperelastic materials

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.

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

Exact Solutions for the Modified Burgers Equation with Additional Time-dependent Variable Coefficient

In this article, we investigated new travelling wave solutions for the modified Burgers equation with additional time-dependent variable coefficient via the functional variable method. The performance of this method is reliable and effective and gives the exact solitary wave solutions. All solutions of this equation have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. The exact solutions have its great importance to reveal the internal mechanism of the physical phenomena. This method presents a wider applicability for handling nonlinear wave equations.

A meshfree method for the nonlinear KdV equation using stabilized collocation method and gradient reproducing kernel approximations

A gradient reproducing kernel based stabilized collocation method (GRKSCM) to numerically solve complicated nonlinear Korteweg-de Vries (KdV) equation is proposed in this paper. The acquisition of GRK through high-order consistency conditions reduces the complexity of RK derivative operations and remarkably improves the effectiveness of the proposed method by directly forming GRK approximations. Owing to the fulfillment of high-order integration constraints, the SCM achieves exact integration in the domain and on boundaries. Von Neumann analysis is utilized to establish the stability criteria for GRKSCM when combined with forward difference temporal discretization. The effectiveness of the GRKSCM method in solving the KdV equation is investigated through six numerical examples. The examples include the motion of the single solitary wave propagation, interaction between two solitary waves and interaction among three solitary waves. Furthermore, the behavior of a two-dimensional solitary wave and the propagation of a three-dimensional solitary wave are also numerically investigated. The numerical outcomes confirm that GRKSCM provides high accuracy in comparison with analytical solutions. In addition, the invariants and error analysis show the conservation of our proposed GRKSCM method.

A study on dust inertial Alfvén solitary waves in a magnetized dust-ion-electron plasma using sagdeev potential method

Abstract - The propagation characteristics of arbitrary amplitude Dust Inertial Alfvén Solitary Wave (DIASW) has been studied in a three-component plasma consisting of magnetized cold dust, magnetized cold ions and electrons with Boltzmann distribution. The range of Mach number where solitary waves can be formed has been established analytically. To discuss the behaviors of the DIASWs, the Sagdeev potential method is used. Both rarefactive and compressive DIASWs travelling at sub and super Alfvénic speed are generated is this plasma model.

Dust magnetoacoustic waves in an inhomogeneous cylindrical four-component dusty plasma in the presence of polarization force

Abstract Dust magnetoacoustic waves have been examined in an inhomogeneous, bounded, cylindrical dusty plasma containing oppositely polarized dust particles. Considering polarization force, dust dynamics in r − θ plane is studied in the presence of inhomogeneous external magnetic field along z axis. At equilibrium, the dusty plasma components are supposed to follow Gaussian density distribution. Using reductive perturbation method (RPM), a variable coefficient cylindrical Kadomtsev–Petviashvili (VCCKP) equation has been derived. For weak azimuthal perturbation, an analytical solution, obtained by Zhang (“Exact solutions of a kdv equation with variable coefficients via exp-function method,” Nonlinear Dynam., vol. 52, nos. 1–2, pp. 11–17, 2008) using Exp-function method, is chosen. Phase velocity of dust magnetoacoustic wave is found to be modified by the density inhomogeneities, polarization force, dust charge state ratio and ion-to-electron temperature ratio. Spatio-temporal evolution of the dust number densities has been noticed. Existence of the compressive electromagnetic solitary waves is observed numerically for the chosen dusty plasma parameter range. The impacts of the inhomogeneity, polarization force, dust charge state ratio and ion-to-electron temperature ratio on the relative amplitude of the dust magnetoacoustic wave are also discussed.

Envelope solitary waves in two-negative ions with stationary dust grains

Abstract Using a multi-fluid model, we look at how modulated electrostatic dust-ion-acoustic wave
packets move nonlinearly through a plasma made up of a three-ion fluid with Maxwellian
electrons and stationary dust grains. A nonlinear Schr¨odinger (NLS) equation describes
the electric potential envelope wave packet. The analysis reveals the existence of different
types of localized modes, namely bright, dark, and grey solitons. We numerically analyse
the coefficients of the NLS equation to identify stable or unstable regions for wave packet
propagation. It is found that higher relative density ratios increase the group velocity of the
wave packets. Stable pulses can become unstable when plasma parameters exceed certain
relative density ratio values. Stable pulses can exist within a crucial window of the relative
dust density ratio. Controlling the dust grain density ratio outside the zone can cause unstable
wave packets or bright envelope solitons to propagate.

Solitary Wave Solutions to the General Class of Nonlocal Nonlinear Coupled Wave Equations

In this paper, we study a general class of nonlocal nonlinear coupled wave equations that includes the convolution operation with kernel functions. For appropriate selections of the kernel functions, the system becomes well-known nonlinear coupled wave equations, for instance Toda lattice system, coupled improved Boussinesq equations. A numerical scheme is proposed for the solitary wave solutions of the system using the Pethiashvili method. Using the different kernels, the validity of the numerical method has been tested.

Partial dam-break wave characteristics due to partial gate opening

Understanding how to predict the characteristic height of a dam-break wave is crucial for effective risk management in dam-break scenarios. This study introduces an experimental framework to simulate partial dam-break waves triggered by partial gate openings. We conducted numerous experiments to explore how different factors, including water-level differences (hi), reservoir widths (L), dam-break openings (e), and submerged depths (hw) influence the characteristic heights of the main-wave peak (h1c), subwave peak (h2c), and main-wave trough (h1t). Our analysis led to the development of an equation for predicting h1c. Furthermore, we compared the partial dam-break wave profile with the theoretical profile of a solitary wave, identifying optimal conditions for solitary wave generation and examining the velocity characteristics of partial dam-break waves. The results reveal that: (1) h1c, h2c, and h1t have positive correlations with hi and L, but negative correlations with e and hw; (2) notably, the main-wave pre-peak profile aligns more closely with the solitary wave theory; and (3) the solitary wave theory accurately predicts the main-wave peak velocity. This study enhances our understanding of wave characteristic evolution during partial dam-breaks and can be used for validation in numerical simulations.

Dynamical analysis of exact optical soliton structures of the complex nonlinear Kuralay-II equation through computational simulation

In this paper, we successfully extracted the various types of soliton solutions for the complex nonlinear Kuralay-II equation through the improved F-expansion method with symbolic computational software Mathematica. The extracted soliton solutions for the Kuralay-II equation are interesting, novel and more general such as anti-kink wave solitons, dark solitons, kink wave solitons, bright solitons, periodic wave solitons, mixed solitons in bright-dark soliton shape, peakon solitons, and solitary wave structures. The graphical structure of some extracted solutions is visualized in 2D, 3D and contour plottings with imaginary, real, and absolute values of the functions by using the numerical simulation. The proposed research will contribute to advancing our knowledge about the complex nonlinear Kuralay-II equation and demonstrating the applicability to the proposed approach to investigate other higher-order complex nonlinear equations. The successful investigation demonstrated that the proposed method is effective, simple, more powerful, efficient and can be utilized on a variety of other nonlinear equations. The explored solitary waves and optical solitons will play an important role in the investigation of nonlinear phenomena in various domains of science and engineering.

Existence and stability of solitary waves to the rotation–Camassa–Holm equation

In this paper, we investigate the existence and stability of solitary waves to the rotation–Camassa–Holm equation which can be considered as a model in the shallow water for the long‐crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. We prove the existence of solitary waves by performing a phase plane analysis. Moreover, utilizing the approach proposed by Grillakis–Shatah–Strauss, we prove the stability of solitary waves.

Solitary Wave Solutions in (2+1) Dimensions: The KdV Equation Derived from Ideal Fluid Models

Solitary Wave Solutions in (2+1) Dimensions: The KdV Equation Derived from Ideal Fluid Models

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

Abstract 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-$\kappa$ (kappa) distribution function describing the plasma nonthermality as well as 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 analysed and the nature of arbitrary amplitude solitary waves (obtained via the Sagdeev potential approach) is discussed. It is investigated theoretically and numerically that different plasma compositional parameters [such as the trapping effect of electrons ($\beta_e$) and positrons ($\beta_p$), the obliquity effect ($\theta$), electron-to-ion number density ratio ($\mu_e$), the magnetic field effect (via $\Omega$) and the viscous effect ( via $\eta$)] have significant influence on the dynamics ion-acoustic solitary and shock waves. The theoretical and numerical investigations in this study may be helpful in describing the nature of localised structures in different plasma contexts, e.g., space and astrophysical plasmas, experimental plasmas where electron-positron-ion plasmas exist.