Lump Wave Solutions Research Articles

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.

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

On lump and solitonic wave structures for the (3+1)-dimensional nonlinear evolution model

In order to understand many complex situations in wave propagation, such as heat transfer, fluid dynamics, optical fibers, electrodynamics, physics, chemistry, biology, condensed matter physics, ocean engineering, and many other branches of nonlinear science, the majority of natural processes are routinely modelled and analysed using nonlinear evolution equations. In this study, the (3+1)-dimensional nonlinear evolution equation is investigated analytically. Initially, the Hirota bilinear approach is used to develop the bilinear version of the higher dimensional nonlinear model. Consequently, we are able to design periodic wave soliton solutions, lump wave and single-kink soliton solutions, and collisions between lumps and periodic waves. Later on, the unified method is applied to develop several new travelling wave solutions for the governing model substantially. Furthermore, numerous exact solutions are analyzed graphically to explore many fascinating nonlinear dynamical structures with the aid of 3D, contour, and 2D visualizations. A variety of higher dimensional nonlinear evolution models can also be investigated by employing present approaches arising in many fields of contemporary science and technology.

Phase transitions of lump wave solutions for a (2+1)-dimensional coupled Maccari system

Phase transitions of lump wave solutions for a (2+1)-dimensional coupled Maccari system

A study of (2+1)-dimensional variable coefficients equation: Its oceanic solitons and localized wave solutions

In this work, shallow ocean-wave soliton, breather, and lump wave solutions, as well as the characteristics of interaction between the soliton and lump wave in a multi-dimensional nonlinear integrable equation with time-variable coefficients, are investigated. The Painlevé analysis is used to verify the integrability of this model. Based on the bilinear form of this model, we use the simplified Hirota's method obtained from the perturbation approach and various auxiliary functions to construct the aforementioned solutions. Besides, the interaction between the soliton and lump wave solutions is also examined. In addition, by imposing specific constraint conditions on the N-soliton solutions, we further derive higher-order breather solutions. To show the physical characteristics of this model, several graphical representations of the discovered solutions are established. These graphs show that the time-variable coefficients result in a variety of novel dynamic behaviors that differ significantly from those for integrable equations with constant coefficients. The acquired results are useful for the study of shallow water waves in fluid dynamics, marine engineering, nonlinear sciences, and ocean physics.

Some new nonlinear wave solutions and dynamical behavior of the (3+1)-dimensional Ito equation

Some new nonlinear wave solutions of the (3+1)-dimensional Ito equation, such as three-wave type solutions, lump wave solutions, rogue wave solutions and hybrid solutions of different forms, are studied by the bilinear neural network method. Besides, three-wave solutions are investigated via the single-layer neural network ‘4-3-1’ model, and hybrid soliton solutions between lump-type and strip-like soliton in ‘4-5-1’ model are discussed. Furthermore, rogue wave solutions are studied by the double-layer neural network the ‘4-2-3-1’ and the ‘4-3-3-1’ models, and the interference phenomena between them are analyzed using visual images. We also employed the extended parameter limit method to investigate the collapse degradation behavior of both solutions. Lump solutions and Y-type semi-rational solutions are derived separately.

Variety of solutions and dynamical behavior for YTSF equations

We construct non-homogeneous polynomial lump wave solutions of the Yu-Toda-Sasa-Fukuyama (YTSF) equation, based on a bilinear approach, enriching the formal diversity of lump waves. By studying the interaction between the lump solutions of the YTSF equation and the solitary wave solutions, we find a new aggregation effect and elastic collision effect. We obtain exact solutions, such as the solution of separated variables and periodic nonlinear wave solutions, by applying the Lie symmetry group method and the bilinear method.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/82/abstr.html

Lump wave solutions, lump-stripe soliton inelastic collision phenomena and rogue-type wave solutions for a generalized breaking soliton system in (3+1)-dimensions

Based on five types of structures of the functional multi-layer neural network model, some explicit exact solutions for a (3+1)-dimensional generalized breaking soliton system including lump waves, lump-stripe solitons and rogue-type waves are explored by using bilinear neural network method without the datasets training. The 3-dimensional plots, 2-dimensional curves and density plots of these results obtained are provided to display their dynamic characteristics. In particular, the complete processes of lump wave and stripe soliton fusion phenomena are presented. We hope the formulations established in this paper could be applied to study the exact traveling wave solutions of other integrable nonlinear evolution equations.

Resonant collisions among X-type, multi-lump, generalized breathers, N-solitons and rogue waves in plasma

This paper studies the (1+1)-dimensional Kuramoto–Sivashinsky equation (1D-KSE) also known as canonical evolution equation within the context of flame front propagation, plasma instabilities, and phase turbulence in reaction–diffusion systems. We use Hirota’s derivatives to formulate bilinear equations and subsequently compute various types of solitons. We obtain lump solution, lump one strip, lump two strip, X-type, lump periodic solution, generalized breather and rogue wave solutions for our governing model with the help of Hirota bilinear method (HBM) and the ansatz approach. Breathers solutions may be used to improve the efficiency of fiber optic communication systems and solitons in plasma waves. Rogue wave solutions may be used for the safety of oil platforms and ships while lump wave solutions can be utilized in manipulating and controlling laser beams for materials processing or laser surgery. We also discuss one, two and other soliton interactions for 1D-KSE under some constraint conditions. These interactions may be discussed in plasma stability and confinement, which can control fusion as a future energy source.

Solitons, Lumps, breathers and rouge wave solutions to the (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt model

The mixed localized wave solutions and their interaction have fascinated the vast age of plasma physics, fluid mechanics, and ocean dynamics, thanks to advances in nonlinear shallow water waves theory. In this paper, we conduct an analytical investigation of the (3+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt model. For a deeper understanding of the integrability characteristics of the governing model, the Hirota’s bilinear form has been used to exhibit a variety of wave patterns, such as the Rogue wave solution and interactions between lump wave and periodic wave solutions. Additionally, by using the exp(−ϕ(ξ)) expansion method, certain novel travelling wave shapes are created. In order to understand the dynamics of the governing model, many exciting graphical representations are demonstrated at the conclusion in the form of 3D, 2D, and contour plots. As a result, we created unique bell-shaped solutions, bright soliton solutions, and periodic wave solutions. The discovered results are highly positive and can also be used to assess the characteristics of different complex dynamical models that have emerged in more modern science and engineering.

Applications of the generalized nonlinear evolution equation with symbolic computation approach

In this work, we will try to find lump solutions, interaction between lump wave and solitary wave solutions, kink-solitary wave solutions and shock wave-type solutions to [Formula: see text]-dimensional generalized nonlinear evolution equation arising in the shallow water waves. The lump solutions, the interaction between lump wave and solitary wave solutions and kink-solitary wave solutions are derived with symbolic computation based on a logarithmic derivative transform which is derived by the help of Hirota’s simple method. The shallow water waves in this equation are associated with some natural problems such as tides, storms, atmospheric currents and tsunamis. For the physical presentation of the solutions, we draw 3D and counter graphics by giving the suitable values to include the free parameters. We believe that disciplines such as mathematical physics, nonlinear dynamics, fluid mechanics and engineering sciences can benefit from this study.

Lump solutions to an integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations in shallow water

This study uses the Hirota bilinear method and Maple, a symbolic computation program, to derive lump solutions for a new integrable (3 + 1)-dimensional Boussinesq equation and its dimensionally reduced equations. Furthermore, lump solutions with free parameters have been constructed using the dimensionally reduced new form of the (3 + 1)-dimensional Boussinesq equation. The derived lump solutions show it has two trough positions and one crest position. The amplitudes and shapes of lump waves don't vary during propagation but they change their positions. By making three-dimensional, two-dimensional, and density plots for specific values of the relevant free parameters, the propagations of the obtained lump wave solutions are displayed. They also demonstrate how the trough and crest positions of a lump wave change over time with constant velocity. The phase shifts, propagation directions, and energy distributions can be seen from the graphical outputs that the free parameters of the model play a significant role in changing the shapes and amplitudes of the waves. The resulting solutions and their physical characteristics may help to understand how the waves propagate in shallow water in oceanography.

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of active research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of water waves with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. Particularly, we construct some explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear Bäcklund transformation. We explore their propagation and transformation/modulation dynamics due to different spatial backgrounds through categorical analysis and clear graphical demonstrations for a complete understanding of the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into manipulated periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type arbitrary spatial backgrounds. The present results will be an important addition to the context of engineering nonlinear waves with non-vanishing controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.

Multiple solitons with bifurcations, lump waves, M-shaped and interaction solitons of three component generalized (3+1)-dimensional Breaking soliton system

<abstract><p>The generalized (3+1)-dimensional Breaking soliton system (gBSS) has numerous applications across various scientific fields. This manuscript presents a study on important exact solutions of the gBSS, with a focus on novel solutions. Using the Hirota bilinear technique, we derive the general solution of the proposed system and obtain the novel solutions by considering different types of auxiliary functions. Our analysis includes the study of multi-solitons, multiple bifurcation solitons, lump wave solutions, M-shaped solitons, and their interactions. We also observe several hybrid solitons, including tuning fork-shaped, X-Y shaped, and double Y shaped. Our results are presented through graphical representations.</p></abstract>

Structures of interaction between lump, breather, rogue and periodic wave solutions for new (3 + 1)-dimensional negative order KdV-CBS model

The characteristics of lump solutions are investigated using a new (3+1)-dimensional negative order KdV-Calogero-Bogoyavlenskii-Schiff model with a bilinear representation. Various ansatzes have been used to discover a lump, breather wave, Lump with periodic waves, Lump with rogue waves, and Periodic cross lump wave solutions for the model's function patterns. Under consideration of the parameter values, numerical simulations in the form of line plots, contour plots, and 3-D profiles have been carried out to provide more insight into the dynamics of the acquired solutions.

Degenerate lump wave solutions of the Mel’nikov equation

Degenerate lump wave solutions of the Mel’nikov equation

Nonlinear dynamics for different nonautonomous wave structure solutions

Abstract Based on the positive quadratic function method, the rich nonautonomous solutions of a generalized (2+1)-dimensional variable-coefficient breaking soliton equation with different wave structures are given. In this case, due to the influence of nonlinearity and dispersion, the characteristics, amplitude and velocity of nonautonomous wave will change with time. The breather wave and the interaction among lump wave, solitary wave and periodic wave solutions are studied. For different choices of arbitrary functions in these solutions, the corresponding dynamic properties are demonstrated.

Charged space debris induced nonlinear magnetosonic waves using inertial magnetohydrodynamics

The excitations of nonlinear magnetosonic waves in the presence of charged space debris objects in the low Earth orbital (LEO) plasma region are investigated by taking into account the effects of electron inertia in the framework of classical magnetohydrodynamics (MHD); which is also referred to as inertial magnetohydrodynamics. The magnetosonic waves are found to be governed by a forced Kadomtsev–Petviashvili (KP) equation with the forcing term representing the effects of space debris objects. The dynamical behaviours of both slow and fast magnetosonic solitary waves are explored in detail. Exact pinned accelerated magnetosonic lump wave solutions are shown to be stable for the entire region in the parameter space of slow waves and a large region in the parameter space of fast waves. In a similar way, exact pinned curved magnetosonic solitary wave solutions become stable for a small region in the parameter space of fast waves. These exact solutions with special properties are derived for some specific choices of the forcing debris functions. These novel results can have potential applications in the scientific and technological aspects of the space debris detection and mitigation in the near-Earth space.

SOLITARY AND LUMP WAVES INTERACTION IN VARIABLE-COEFFICIENT NONLINEAR EVOLUTION EQUATION BY A MODIFIED ANSÄTZ WITH VARIABLE COEFFICIENTS

<abstract> In this work, we examine variable-coefficient nonlinear evolution equations that often describe complex physical models more than constant coefficient models. A modified ansätz with variable coefficients is used for studying the solitary and lump waves interaction in these variable-coefficient nonlinear evolution equations. We discuss the variable-coefficient Kadomtsev-Petviashvili equation to achieve this goal. We present lump wave and interaction solutions between solitary and lump waves for this model. By choosing appropriate values of the variable coefficients, 3d plots and corresponding contour plots are drawn to illustrate the dynamical behaviors of the obtained solutions. </abstract>