Solitary Waves Research Articles

Effect of squared, cubic and quartic on-site potentials on heat conduction in a nonlinear silicon lattice

This paper explains the process of heat conduction in a nonlinear silicon lattice governed by squared, cubic and quartic on-site potentials along with squared and quartic interaction potentials. The tanh method is employed to construct solutions to the resulting nonlinear equations. The results show the transfer of heat in the form of solitary waves which are presented graphically.

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-wave is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are shown by the study of time variables. By virtue of the two breath-wave solutions, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

Diverse exact and solitary wave solutions to new extended KdV6 equation using IM extended tanh-function technique

Diverse exact and solitary wave solutions to new extended KdV6 equation using IM extended tanh-function technique

On solitary-wave solutions of Rosenau-type equations

The present paper is concerned with the existence of solitary wave solutions of Rosenau-type equations. By using two standard theories, Normal Form Theory and Concentration-Compactness Theory, some results of existence of solitary waves of three different forms are derived. The results depend on some conditions on the speed of the waves with respect to the parameters of the equations. They are discussed for several families of Rosenau equations present in the literature. The analysis is illustrated with a numerical study of generation of approximate solitary-wave profiles from a numerical procedure based on the Petviashvili iteration.

Nonplanar ion acoustic waves in e-p-i plasmas with non-Maxwellian distributed electrons and positrons

The process of propagation of cylindrical and spherical ion acoustic solitary waves in an electron, positron, and ion plasma system is studied. The electrons are considered to have the generalized (r, q) distribution, while the positrons are supposed to have the kappa distribution, also known as the Lorentzian distribution. The classic nonlinear equation, Korteweg de-Vries (KdV), has been numerically solved by employing appropriate mathematical transformations. The role of various plasma parameters and nonplanar geometry on the widths and amplitudes of nonlinear ion acoustic structures has been numerically investigated. The current study could be helpful to understand the significance of cylindrical and spherical solitary waves in laboratory, astrophysical as well as in space plasmas.

Nonlinear ion-acoustic waves with Landau damping in non-Maxwellian space plasmas

The dynamics of nonlinear ion-acoustic solitary waves in the presence of kinetic (Landau type) damping have been investigated in a collisionless, non-magnetized electron-ion plasma. A cold ion fluid model, coupled to a Vlasov-type kinetic equation for the electron dynamics, has been adopted as a starting point. The electron population was assumed to be in a kappa-distributed state, in account of the non-Maxwellian behavior of energetic (suprathermal) electrons often observed in Space. A multiscale perturbation technique has led to an evolution equation for the electrostatic potential, in the form of a modified Korteweg-de Vries (KdV) equation, incorporating a non-local term accounting for Landau damping (associated with the electron statistics). Exact analytical solutions have been obtained, representing solitary waves undergoing amplitude decay over time. The combined effect of Landau damping and non-Maxwellian electron statistics (via the kappa parameter) on the characteristics of IASWs has been examined. Numerical integration of the evolution equation has been undertaken, to elucidate the importance of kinetic Landau damping on a shock-shaped initial condition. The results of this investigation aim to improve our understanding of the dynamics of nonlinear electrostatic waves under the influence of Landau damping in various space plasma environments.

Applications of nonlinear waves; lump wave, rogue wave, periodic wave, breather wave for a nonlinear model

In this work, we study the Chafee–Infante differential equation (CIDE) for various types of nonlinear waves (NW) like breather waves (BW), periodic cross-kink (PCKW), lump waves (LW), rogue waves (RW), periodic wave (PW), lump soliton (LS) with one kink, LS with two kink and periodic-cross lump waves (PCLW). The CIDE is a nonlinear evolution equation (NLEE) introduced by Nathaniel Chafee and Ettore Infante. This equation has been widely investigated in the context of pattern generation and soliton solutions, and it plays an essential role in understanding the complex nonlinear dynamics across a wide range of scientific fields. BW is a localized, non-dispersive solution to NW equations, maintaining shape and amplitude during travel. An LS is a type of solitary wave solution in NLEEs. In mathematics, RW refers to rare and unusually large solutions of NW equations. These waves have garnered significant attention due to their unpredictable and extreme nature. A PW is a type of wave that repeats its shape and behavior at regular intervals in both time and space. These waves are characterized by their periodicity, which means that they have a consistent and predictable pattern of oscillation. We’ll offer a graphical explanation of our newly discovered answers toward the conclusion.

Tsunami Squares: Leapfrog scheme implementation and benchmark study on wave–shore interaction of solitary waves

Impulse waves are generated by rapid subaerial mass movements including landslides, avalanches and glacier break-offs, which pose a potential risk to public facilities and residents along the shore of natural lakes or engineered reservoirs. Therefore, the prediction and assessment of impulse waves are of considerable importance to practical engineering. Tsunami Squares, as a meshless numerical method based on a hybrid Eulerian–Lagrangian algorithm, have focused on the simulation of landslide-generated impulse waves. An updated numerical scheme referred to as Tsunami Squares Leapfrog, was developed which contains a new smooth function able to achieve space and time convergence tests as well as the Leapfrog time integration method enabling second-order accuracy. The updated scheme shows improved performance due to a lower wave decay rate per unit propagation distance compared to the original implementation of Tsunami Squares. A systematic benchmark testing of the updated scheme was conducted by simulating the run-up, reflection and overland flow of solitary waves along a slope for various initial wave amplitudes, water depths and slope angles. For run-up, the updated scheme shows good performance when the initial relative wave amplitude is smaller than 0.4. Otherwise, the model tends to underestimate the run-up height for mild slopes, while an overestimation is observed for steeper slopes. With respect to overland flow, the prediction error of the maximum flow height can be limited to ± 50% within a 90% confidence interval. However, the prediction of the front propagation velocity can only be controlled to ± 100% within a 90% confidence interval. Furthermore, a sensitivity analysis of the dynamic friction coefficient of water was performed and a suggested range from 0.01 to 0.1 was given for reference.

A Dynamical Analysis and New Traveling Wave Solution of the Fractional Coupled Konopelchenko–Dubrovsky Model

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software.

The interactions of dark, bright, parabolic optical solitons with solitary wave solutions for nonlinear Schrödinger–Poisson equation by Hirota method

The interactions of dark, bright, parabolic optical solitons with solitary wave solutions for nonlinear Schrödinger–Poisson equation by Hirota method

Some novel exact and non-standard finite difference schemes for a class of diffusion–advection–reaction equation

In this paper, a generalized diffusion–advection–reaction equation (GDARE) is considered and by using the solitary wave solution two new class of exact finite difference (EFD) schemes are proposed under certain functional relationship between the temporal and space step sizes. Suitable denominator functions are identified from these EFD schemes and a new non-standard finite difference (NSFD) scheme is developed. This NSFD scheme preserves positivity and boundedness properties of the solution of GDARE. The L 2 , L ∞ errors and CPU times are obtained for the proposed NSFD and EFD schemes to show the proficiency and accuracy of the schemes.

Diversity of solitonic wave structures to the M‐truncated dynamical system in ultrasound imaging

The utilization of ultrasound imaging has become extensively prevalent and well‐established in clinical practice. The fundamental technologies that serve as the foundation for various applications in the field include transducers, beam shaping, pulse compression, tissue harmonic imaging, contrast agents, methodologies for quantifying blood flow and tissue motion, and three‐dimensional imaging. This article focuses on the examination of ultrasonic propagation, which involves the transmission of mechanical vibrations within the molecules or particles of a material. It quantifies the velocity of sound propagation in the medium of air. The third‐order nonlinear M‐fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The recently developed computational methods have been applied in this study. The different wave structures are secured in various forms of solitary wave solutions including bright, dark, and combo solitons. In the domains of medical imaging and therapy, the investigation of sound wave propagation and high‐amplitude phenomena is facilitated by the utilization of wave structures. The effectiveness of these solutions extends to acoustic cavitation, acoustic levitation, underwater acoustics, and facilitating the process of ultrasonic propagation in tissue. Ultrasound imaging technologies currently find application in the medical field, enabling the visualization and examination of internal human tissue. This technology exhibits a wide array of applications in the fields of industry and medicine. A representation of the graphs is produced using the appropriate parametric values. The results suggest that the chosen approaches exhibit effectiveness, viability, and adaptability when implemented in complex systems in various fields, with particular emphasis on ultrasonic imaging.

Lie symmetry analysis and solitary wave solution of biofilm model Allen-Cahn

The investigation presented in this study delves into the analysis of Lie symmetries for the bistable Allen-Cahn (BAC) equation with a quartic potential, specifically applied to the biofilm model. By employing the Lie symmetry method, we have acquired the Lie infinitesimal generators for the considered model. Using a transformation method, the nonlinear partial differential equations (NPDEs) are converted into various nonlinear ordinary differential equations (NLODEs), providing the numerous closed-form solitary wave solutions. The obtained solutions manifest in various forms including dark, bright, kink, anti-kink, and periodic types using diverse strategies. To enhance the physical interpretation, the study presents 3D, 2D, and contour plots of the acquired solutions. Every graph’s wave-like structure contains information about the structural behaviour of the bacteria that build biofilms on surfaces where rectangles have different densities. This analysis enhances comprehension of the complex dynamics present in areas like fluid dynamics, fiber optics, biology, ocean physics, coastal engineering, and nonlinear complex physical systems.

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Experimental study of internal solitary wave evolution beneath an ice keel model

Internal solitary waves (ISWs) propagating in polar seas are affected by the sea ice at upper boundary of seas and thus exhibit complex evolution characteristics. Herein, spatiotemporal changes in the wave element, flow field, and energy of ISWs beneath an ice keel model were investigated to examine the evolution of ISWs. For this purpose, laboratory experiments were conducted using dye-tracing labeling, conductivity probes, Schlieren technology, and particle image velocimetry. The results show that ice keel causes an increase in the thickness of the pycnocline and even the occurrence of breaking and internal surging of ISW. Additionally, the waveform becomes narrower or wider at different positions, and wave amplitude and speed decrease, with a maximum reduction 30%–40%. Furthermore, the ice keel strengthens the shear of the ISW-induced flow field, generating vortices and mixing. The energy of ISWs undergoes internal conversion majorly at the front slope of the ice keel, while energy dissipation occurs largely at the back slope, with dissipation rates as high as 60%.

Some optical solitons and modulation instability analysis of (3 + 1)-dimensional nonlinear Schrödinger and coupled nonlinear Helmholtz equations

In this study, we acquired some optical solitons of (3 + 1) dimensional nonlinear Schrödinger equation (3DNLSE) and coupled nonlinear Helmholtz equations (CNLHE) by using generalized G′G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left(\\frac{{G}^{\\prime}}{G}\\right)$$\\end{document}-expansion method (GEM). We are able to derive exponential, trigonometric, and hyperbolic solutions. We notice that Mathematica 11.2 offer the equations for these answers. Aside from that, we display some of the solution’s graphic performance. Recently, accurate traveling-wave solutions for nonlinear partial differential equations (NLPDEs) have been acquired using this technique. Through linear stability analysis, we derive an analytical expression for the instability gain and analyze its main characteristics. Finally, we assess the robustness and stability of the solitary waves through numerical simulations and compare the accuracy of the analytical and numerical results. The obtained ultra-short optical solitons can be widely used as a guide for practical applications in telecommunications domain.

The modified cubic Benjamin–Ono equation describing internal solitary waves in the deep ocean and its related properties

In this article, we investigate the propagation of internal solitary waves in deep ocean. Based on the principles of nonlinear theory, perturbation expansion, and multi-scale analysis, a time-dependent modified cubic Benjamin–Ono (mCBO) equation is derived to describe internal solitary waves in the deep ocean with stronger nonlinearity. When the dispersive term ∂3f∂X3 vanishes, the mCBO equation transforms into the cubic BO equation. Similarly, when the dispersive term ∂3f∂X3 becomes zero and the nonlinear term ∂f3∂X degenerates into ∂f2∂X, the mCBO equation reduces to the BO equation. Furthermore, if the integral term ∂2∂X2ℵ(f) disappears, it simplifies to the mKdV equation. To gain deeper insight into the characteristics of solitary waves, conservation of mass and momentum associated with them are discussed. By employing Hirota's bilinear method, we obtain soliton solutions for the mCBO equation and subsequently investigate interactions between two solitary waves with different directions, leading to the occurrence of important events such as rogue waves and Mach reflections. Additionally, we explore how certain parameters influence Mach stem while drawing meaningful conclusions. Our discoveries reveal the complex dynamics of internal solitary waves within the deep ocean and contribute to a broader understanding of nonlinear wave phenomena.

Localized wave structures: Solitons and beyond

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves.

Exploring optical solitary wave solutions in the (2+1)-dimensional equation with in-depth of dynamical assessment

The current study explores the (2+1)-dimensional Chaffee-Infante equation, which holds significant importance in theoretical physics renowned reaction-diffusion equation with widespread applications across multiple disciplines, for example, ion-acoustic waves in optical fibres, fluid dynamics, electromagnetic wave fields, high-energy physics, coastal engineering, fluid mechanics, plasma physics, and various other fields. Furthermore, the Chaffee-Infante equation serves as a model that elucidates the physical processes of mass transport and particle diffusion. We employ an innovative new extended direct algebraic method to enhance the accuracy of the derived exact travelling wave solutions. The obtained soliton solutions span a wide range of travelling waves like bright-bell shape, combined bright-dark, multiple bright-dark, bright, flat-kink, periodic, and singular. These solutions offer valuable insights into wave behaviour in nonlinear media and find applications in diverse fields such as optical fibres, fluid dynamics, electromagnetic wave fields, high-energy physics, coastal engineering, fluid mechanics, and plasma physics. Soliton solutions are visually present by manipulating parameters using Wolfram Mathematica software, graphical representations allow us to study solitary waves as parameters change. Observing the dynamics of the model, this study presents sensitivity in a nonlinear dynamical system. The applied mathematical approaches demonstrate its ability to identify reliable and efficient travelling wave solitary solutions for various nonlinear evolution equations.

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.