Lump Wave Research Articles

Two types of interaction phenomena of the lump wave for nonlinear model of Rossby waves

In this paper, we show the interaction phenomena about the first-order lump wave and multiple solitons by four theorems. The interaction phenomena are divided into two types, one type is absorbing collision phenomenon and the other type is elastic collision phenomenon. Six sets of dynamic plots over time are showed the phenomenon of absorbing interactions with energy transfer process, the phenomenon of elastic interactions with restoration of the initial state after collision. We apply the four theorems to an equation which can characterize the Rossby waves’ amplitude, this equation can be used to describe the resonance phenomenon of Rossby waves. Different test functions with the same parameter values can present different wave-wave interaction properties. This study contributes to in-depth investigation of the mechanism of lump wave generation, evolution and interaction.

Exact and Data-Driven Lump Wave Solutions for the (3+1)-Dimensional Hirota–Satsuma–Ito-like Equation

In this paper, the lump wave solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation are constructed by employing the Hirota bilinear method and quadratic function approach, and the corresponding propagation behaviors and nonlinear dynamical properties are also investigated. At the same time, the physics informed neural network (PINN) deep learning technique is employed to study the data-driven solutions for the HSIl equation from the derived lump wave solutions. The machine learning results show high effectiveness and accuracy, providing new techniques for discussing more nonlinear dynamics of lump waves and discovering new lump wave solutions.

Modelling of KdV-Soliton Through Fractional Action and Emergence of Lump Waves

Modelling of KdV-Soliton Through Fractional Action and Emergence of Lump Waves

Multiple solitons, multiple lump solutions, and lump wave with solitons for a novel (2+1)-dimensional nonlinear partial differential equation

The primary aim of this paper is to explore exact solutions to a novel (2+1)-dimensional water wave equation that models oceanic wave phenomena. We begin by applying the Hirota bilinear transformation method to derive multi-soliton solutions, including 3-soliton and 4-soliton solutions. Then, utilizing the bilinear form of the equation and the long-wave limit method, we identify multiple lump solutions and interaction solutions between lumps and solitons. These include 1-lump, 2-lump, and 3-lump solutions, as well as interactions between a 1-lump and a 1-soliton, and between a 1-lump and 2-solitons. The physical dynamics of these solutions are visually represented, offering insight into the corresponding oceanic wave phenomena.

Derivation of weakly interacting lumps for the (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation via degeneracy of lump chains

This paper introduces two distinct pathways for degenerating normally interacting lump chains into weakly interacting lump waves for Yu–Toda–Sasa–Fukuyama equation, which will enrich the correlation between lump chains and weakly interacting lump waves. The first pathway involves letting the periods of M similarly-velocity lump chains approach infinity directly. The second pathway first transforms M normally interacting lump chains into weakly interacting lump chains with similar dynamic behaviors. Then, by allowing their periods to approach infinity, MM+12 weakly interacting lump waves are produced. Distance between weakly interacting lump chains in this equation is proportional to ln|t|, while between weakly interacting lump waves is proportional to |t|3. These findings will contribute valuable theoretical insights to the study of wave theory, ocean science and related disciplines.

Solitons, Lumps, Breathers, and Interaction Phenomena for a (2+1)-Dimensional Variable-Coefficient Extended Shallow-Water Wave Equation

In this paper, soliton solutions, lump solutions, breather solutions, and lump-solitary wave solutions of a (2+1)-dimensional variable-coefficient extended shallow-water wave (vc-eSWW) equation are obtained based on its bilinear form. By calculating the vector field of the potential function, the interaction between lump waves and solitary waves is studied in detail. Lumps can emerge from the solitary wave and are semi-localized in time. The analytical solutions may enrich our understanding of the nature of shallow-water waves.

Soliton-based modeling of nano-ionic currents in transmission line

Many nonlinear evolution equations, such as the nano-ionic currents (NIC) equation, are used extensively in many scientific and technological domains particularly in nanoelectronics and bioelectronics. The mathematical modeling of NIC phenomena is vital for understanding their behavior and optimizing device performance. Our research leverages an array of mathematical methods, including multi-wave analysis, periodic wave solutions, lump soliton dynamics, breather wave phenomena, homoclinic breathers, M-shaped waveforms, and rogue wave analysis. Additionally, our investigation encompasses the exploration of single kink and double kink configurations, interactions between periodic and kink waves, interaction between M shaped with kink and rogue, interaction between M shaped with one kink, interaction between M shaped with kink and periodic, interaction between M shaped with two kinks as well as periodic wave interactions with lump waves. To further emphasize the structure of solutions derived from particular parameter choices, we include three-dimensional, two-dimensional, streamplot, and contour graphs.

Rossby waves with barotropic–baroclinic coherent structures and dynamics for the (2 + 1)-dimensional coupled cylindrical KP equations with variable coefficients

Starting from the classical quasi-geostrophic potential vorticity equation with equal depth two-layer fluid, the coupled cylindrical Kadomtsev–Petviashvili (KP) equations with variable coefficients for Rossby waves are studied. To be more general, the phase velocity is considered an indefinite integral about time and improves the analysis procedure. So the variable coefficients are obtained and some previous studies are reasonably explained. The cylindrical wave theory is therewith utilized to reduce the coupled cylindrical KP equations with variable coefficients, and based on the modified Hirota bilinear method, the lump solutions and interaction solutions are found. Through numerical simulations, the Rossby lump waves on both sides of the y axis move closer to the center, and their amplitude gradually decreases and tends to flatten with the generalized Rossby parameter growth. In the Rossby waves flow field, the dipole structures propagate to the east and lead to the appearance of the compress phenomenon during barotropic–baroclinic interaction. It is possibly useful for further theoretical research on atmospheric phenomena.

Analyzing chaos and superposition of lump waves with other waves in the time-fractional coupled nonlinear schördinger equation.

This article aims to study the time fractional coupled nonlinear Schrödinger equation, which explains the interaction between modes in nonlinear optics and Bose-Einstein condensation. The proposed generalized projective Riccati equation method and modified auxiliary equation method extract a more efficient and broad range of soliton solutions. These include novel solutions like a combined dark-lump wave soliton, multiple dark-lump wave soliton, two dark-kink solitons, flat kink-lump wave, multiple U-shaped with lump wave, combined bright-dark with high amplitude lump wave, bright-dark with lump wave and kink dark-periodic solitons are derived. The travelling wave patterns of the model are graphically presented with suitable parameters in 3D, density, contour and 2D surfaces, enhancing understanding of parameter impact. The proposed model's dynamics were observed and presented as quasi-periodic chaotic, periodic systems and quasi-periodic. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations.

Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.

Nonlinear waves and transitions mechanisms for (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation

In this paper, state transition waves are investigated in a (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation by analyzing characteristic lines. Firstly, the N-soliton solutions are given by using the Hirota bilinear method. The breather and lump waves are constructed by applying complex conjugation limits and the long-wave limit method to the parameters. In addition, the transition condition of breather and lump wave are obtained by using characteristic line analysis. The state transition waves consist of quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, W-shaped soliton, and quasi-periodic wave soliton. Through analysis, when solitary wave and periodic wave components undergo nonlinear superposition, it leads to the formation of breather waves and transformed wave structures. It can be used to explain the deformable collisions of transformation waves after collision. Furthermore, the time-varying property of transformed waves are studied using characteristic line analysis. Based on the high-order breather solutions, the interactions involving breathers, state transition waves, and solitons are exhibited. Finally, the dynamics of these hybrid solutions are analyzed through symbolic computations and graphical representations.

WITHDRAWN: Lump, lump with damped oscillation, double cross kinky lump with oscillation, collision of kinky and breather wave solutions to a broadened Hietarinta-like equation

One of the most significant solutions to linear and nonlinear partial differential equations is a lump soliton as it has seismic wave nature. Realizing seismic natures of such lump and oscillating lump wave, one can take initiative to mitigate its harmful energies by opposite external periodic forces. The main purpose of this study is to demonstrate various lump solutions from kinky breather-wave. In this regard, the Hirota bilinear scheme and the enhanced homoclinic test procedure are utilized to get kinky breather-wave solutions with second-order linear dispersive components that can be produced by a fourth-order nonlinear term of the Hietarinta-like equation. By making particular values of certain parameters, we derived bright- dark lump solitons from kinky-breather propagation wave solutions in a specific scenario. The cross-kinky wave in breather form, lump with damped oscillation, kink with damped oscillation, as well as lump solutions are also obtained from the Hirota bi-linear scheme constructing breather waves. A set of 3D, 2D and density graphs are used to investigate such dynamical behaviors of model numerically. These patterns are also illustrated numerous clarifications for others prototypes in the purviews of nonlinear science and engineering.

Multi-Soliton, Soliton–Cnoidal, and Lump Wave Solutions for the Supersymmetric Boussinesq Equation

Multi-Soliton, Soliton–Cnoidal, and Lump Wave Solutions for the Supersymmetric Boussinesq Equation

Resonant multiple soliton and rogue type multiple lump wave solutions of the modified KdV–KP equation

Resonant multiple soliton and rogue type multiple lump wave solutions of the modified KdV–KP equation

Modulation instability analysis, and characterize time-dependent variable coefficient solutions in electromagnetic transmission and biological field

This study integrates the telegraph model with the traveling wave coefficient. This model characterizes planar random motion of particles in fluid flow, pressure waves from pulsatile blood flow in arteries, electrical impulses in nerve and muscle cell axons, and electromagnetic waves in superconducting media. Using the efficient and well-established Enhanced Modified Simple Equation (EMSE) method, we investigate solitary wave solution with the variable coefficient of the telegraph model. To investigate the variable coefficient solitary wave solution, we apply a transformation variable η = h(t)x + g(t), which is dependent on the time variable function. The diverse functions of h(t) and g(t) yield many novel and exclusive solutions. Numerical solutions are plotted in 3D, density, and 2D. instanton soliton, kinky periodic wave, lump wave, kink wave, anti-kink wave, double periodic wave, diverse types periodic wave, lump with bell wave periodic wave, periodic lump wave, etc. Solitary waves explain complex wave phenomena through nonlinear effects like self-interaction and dispersion. Due to the classical nonlinear telegraph model's wave dynamics, they are used in brain function and communication system design. We conclude by testing the governing model's modulation instability. Dispersive and nonlinear processes modulating the stable state cause instability in high-order nonlinear equations. Examining the wave movement role and modulation instability analysis to assess solution stability shows that all solutions are accurate and stable. The computational difficulties and results demonstrate the approaches' clarity, effectiveness, and simplicity, suggesting they can be applied to evolutionary dynamic and static nonlinear equations in computational physics, other real-world situations, and numerous academic disciplines.

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.

Shallow-water waves through two new generalized multi-dimensional variable coefficient equations

The objective of this study is to propose and investigate two new forms of generalized variable coefficients within multi-dimensional equations describing shallow-water waves. We employ the Mathematica program to rigorously establish Painlevé's integrability for these two nonlinear equations. Subsequently, we constructed their bilinear forms and utilized Hirota's bilinear method to examine the dispersion relations and phase shifts of these two models that enable the derivative of multi-soliton solutions. Furthermore, diverse forms of lump-wave solutions are also considered. To illustrate the physical characteristics of these two models, we establish several graphical representations of the discovered solutions. These visualizations offer insights into the behavior, shape, and dynamics of both the multi-soliton, Peregrine soliton, lump wave, and rogue wave, enhancing our understanding of their physical significance. The two soliton solutions effectively replicate the shallow water waves, encompassing the T-, X-, and Y-types, along with other intricate interactions. Additionally, the lump and rogue wave structures are displayed to visually represent their spatial structures. These graphical representations offer a comprehensive view of the diverse wave phenomena observed in shallow water systems, aiding in the understanding of their spatial characteristics and interactions. Therefore, our findings indicate that the introduction of the two newly proposed integrable nonlinear evolution equations enhances the repertoire of integrable system models and aids in comprehending the distinctive characteristics of nonlinear dynamics in real-world applications.

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

Formation of solitary waves solutions and dynamic visualization of the nonlinear schrödinger equation with efficient techniques

This article investigates the non-linear generalized geophysical KdV equation, which describes shallow water waves in an ocean. The proposed generalized projective Riccati equation method and modified auxiliary equation method extract a more efficient and broad range of soliton solutions. These include U-shaped, W-shaped, singular, periodic, bright, dark, kink-type, breather soliton, multi-singular soliton, singular soliton with high amplitude, multiple periodic, multiple lump wave soliton, and flat kink-type soliton solutions. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Maple and Wolfram Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. Sensitivity and modulation instability analyses were performed to offer insights into the dynamics of the examined model. The observed dynamics of the proposed model were presented, revealing quasi-periodic chaotic, periodic systems, and quasi-periodic behaviour. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations.

Soliton structures for the (3 + 1)-dimensional Painlevé integrable equation in fluid mediums

The (3 + 1)-dimensional Painlevé integrable equation are a class of nonlinear differential equations with special properties, which play an important role in nonlinear science and are of great significance in solving various practical problems, such as many important models in fields such as quantum mechanics, statistical physics, nonlinear optics, and celestial mechanics. In this work, we utilize the Hirota bilinear form and Mathematica software to formally obtain the interaction solution among lump wave, solitary wave and periodic wave, which has not yet appeared in other literature. Additionally, using the (G′/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G)$$\\end{document}-expansion method, we provide a rich set of exact solutions for the (3 + 1)-dimensional Painlevé integrable equation, which includes two functions with arbitrary values. This method is the first to be applied to the (3 + 1)-dimensional Painlevé integrable equation. By giving some 3D graphics and density maps, the dynamic properties are analyzed and demonstrated, which is beneficial for promoting understanding and application of the (3 + 1)-dimensional Painlevé integrable equation.