Lump Soliton Solutions Research Articles

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

Lump, lump-periodic, lump-soliton and multi soliton solutions for the potential Kadomtsev-Petviashvili type coupled system with variable coefficients

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

Dynamics of Rational and Lump-Soliton Solutions to the Reverse Space-Time Nonlocal Hirota-Maccari System

We mainly construct lump-soliton solutions of the (2 + 1)-dimensional reverse space-time Hirota-Maccari (HM) equation by using the KP hierarchy reduction method. Meanwhile, with the help of a long wave limit, rational solutions to nonlocal HM equation are studied. According to the appropriate parameter selections, these solutions can be divided into two types: line soliton solutions and lump-soliton solutions. Moreover, we obtain one-lump, two-lump and W-type soliton to the nonlocal HM equation. These new lump-soliton solutions expand the structure of nonlocal nonlinear systems and aid in the comprehension of physical phenomena.

Lump and kink soliton phenomena of Vakhnenko equation

Understanding natural processes often involves intricate nonlinear dynamics. Nonlinear evolution equations are crucial for examining the behavior and possible solutions of specific nonlinear systems. The Vakhnenko equation is a typical example, considering that this equation demonstrates kink and lump soliton solutions. These solitons are possible waves with several intriguing features and have been characterized in other naturalistic nonlinear systems. The solution of nonlinear equations demands advanced analytical techniques. This work ultimately sought to find and study the kink and lump soliton solutions using the Riccati–Bernoulli sub-ode method for the Vakhnenko equation (VE). The results obtained in this work are lump and kink soliton solutions presented in hyperbolic trigonometric and rational functions. This work reveals the effectiveness and future of our method for solving complex solitary wave problems.

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

Variable-coefficient polynomial function method for finding the lump-type solutions of integrable system with variable coefficients

In this work, we study a ([Formula: see text])-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. The new lump and lump-soliton solutions are obtained by the variable-coefficient polynomial function method. We used 3D graphs, contour plots and density graphs to show that the amplitude and velocity of solitons are affected by some variable coefficients. It is proved that the polynomial function method with variable coefficients is very direct and effective for solving lump-type solutions in variable coefficient integrable systems, and more new conclusions can be obtained.

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable (3+1)-dimensional combined potential Kadomtsev--Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation (pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new (3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new (3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined pKP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

Winding real and order-parameter spaces via lump solitons of spinor BEC on sphere

The three condensate wavefunctions of a F = 1 spinor Bose–Einstein condensate on a spherical shell can map the real space to the order-parameter space that also has a spherical geometry, giving rise to topological excitations called lump solitons. The homotopy of the mapping endows the lump solitons with quantized winding numbers counting the wrapping between the two spaces. We present several lump-soliton solutions to the nonlinear coupled equations minimizing the energy functional. The energies of the lump solitons with different winding numbers indicate coexistence of lumps with different winding numbers and a lack of advantage to break a higher-winding lump soliton into multiple lower-winding ones. Possible implications are discussed since the predictions are testable in cold-atom experiments.

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Exploring lump soliton solutions and wave interactions using new Inverse $$(G'/G)$$-expansion approach: applications to the (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation

Analytical study on two new (3 + 1)-dimensional Painlevé integrable equations: Kink, lump, and multiple soliton solutions in fluid mediums

In this work, two new (3 + 1)-dimensional integrable wave equations are investigated. The complete Painlevé integrability of the two suggested equations will be investigated using Mathematica. We employ the method of Hirota to formally derive two sets of multiple soliton solutions for the two suggested models. Additionally, using symbolic computation with Maple, we provide a variety of lump solutions for the two suggested models. Other exact solutions of distinct structures, such as periodic, singular, and many other physical nonlinear structures, will be determined. We should mention here that the proposed two new models will assist many authors that are working in the field of fluids and plasma physics, in understanding the scenarios of the nonlinear waves that arise in different physical systems. Also, this study will contribute to understanding the nature of nonlinear waves that arise in the seas and oceans.

Study of compressive and rarefactive lump solitons structures in dusty plasma with double spectral (r,q) distributed electrons

Nonlinear features are revealed by one of the most competent and powerful approaches, i.e. soliton theory. The present article starts with the dust collisionless magnetized plasma where electrons are following double spectral ( r , q ) distribution. From the hydrodynamical governing equations set, Kadomtsev–Petviashvili (KP) equation is derived using the reductive perturbation technique. The derived KP equations are converted to standard KP equations by a suitable transformation to use the Hirota bilinear method (HBM) more conveniently. By using HBM, an auxiliary function is constructed, and through symbolic computation, three arbitrary constant sets are formed. These sets associate with three lump soliton solution sets. Finally, all these lump soliton solutions are returned back by the inverse transformation that was used to make the derived KP equation to the standard KP equation and get the lump soliton solution of the associated system. It is investigated that associated plasma parameters have a significant impact on the lump soliton structures while tracing figures using these plasma parameters within the justified range of the system. While lump soliton features' are analysed for various parameters' effect on it, one most important aspect is revealed. Double spectral index r and q play a significant role in the lump solitons structures and depict compressive and rarefactive lump soliton structures with the associated system. It is found that these double spectral indices r and q are the decision-making parameters for the lump soliton structures to be compressive or rarefactive.

Interaction solutions for the second extended (3+1)-dimensional Jimbo–Miwa equation

Based on the Hirota bilinear method, the second extended (3+1)-dimensional Jimbo–Miwa equation is established. By Maple symbolic calculation, lump and lump-kink soliton solutions are obtained. The interaction solutions between the lump and multi-kink soliton, and the interaction between the lump and triangular periodic soliton are derived by combining a multi-exponential function or trigonometric sine and cosine functions with quadratic functions. Furthermore, periodic-lump wave solution is derived via the ansatz including hyperbolic and trigonometric functions. Finally, 3D plots, 2D curves, density plots, and contour plots with particular choices of the suitable parameters are depicted to illustrate the dynamical features of these solutions.

The soliton solutions and combined solutions of a high-dimensional wave soliton equation

In this paper, a high-dimensional wave soliton equation is considered and the simple Hirota method and bilinear backlund transformation are applied to construct the new soliton solutions, lump soliton solutions, breather solutions and their combined solutions. Then, through long-wave limit method and appropriate parameter constraints, one-soliton, one-breather, combined soliton-lump and lump-breather and soliton-breather solutions to nonlinear wave equation are obtained. Moreover, the physical interaction and frontal collision phenomena to the solutions of this equation are explored. In addition, a kind of hyperbolic and trigonometric ansatz is employed to derive other solutions such as kink soliton solution, periodic solutions for the high-dimensional wave equation.The obtained results verify the proposed solutions.

Superthermal electron’s effects on lump solitons structures in magnetized auroral plasma

Based on the considerations of the acknowledgements of the numerous space observations by satellites in the auroral plasma, astrophysics plasmas, spacecraft observations, various plasma models have been framed and revealed different interesting features, like electrostatic structures, solitary waves, double layers, supersolitons, etc. Soliton theory is a very efficient and competent way to describe nonlinear features. Using Viking satellite data in the auroral plasma, we have derived lump soliton solutions of the Kadomstev-Petviashvili (KP) equation by employing Hirota bilinear method. Due to its wide range of applications, the study of lump soliton is very attractive and important too. It has been shown that the lump solitons structures as well as in the one-dimensional form of lump soliton are varied with associated parameters in the auroral magnetized plasma. During the analysis of the features of the lump solitons, it is found that the system parameters play a pivotal role on the lump solitons structures.

Interaction phenomenon and breather wave to the extend (3 + 1)-dimensional Kadomtsev-Petviashvili equation

In this paper, we investigated the extend (3+1)-dimensional Kadomtsev-Petviashvili (eKP) equation. Based on the Hirota direct method, we derived lump soliton solutions and breather solutions for eKP equation in the cases of z=x and z=y after dimension reduction. In the process of solving the equation, we all make use of Maple symbolic computation. At the same time, in order to further understand the dynamic behavior of these solutions, by choosing appropriate parameters, we give the plots of the solutions.

Multiple lump, generalized breathers, Akhmediev breather, manifold periodic and rogue wave solutions for generalized Fitzhugh-Nagumo equation: Applications in nuclear reactor theory

Based on ansatz functions technique, we construct many types of novel wave structures and multiple lump-soliton solutions to the generalized Fitzhugh-Nagumo (gFN)-equation. In particular, we obtain entirely exciting lump, lump 1-kink, lump 2-kink, lump-periodic, manifold periodic-soliton, periodic cross-rational, kink cross-rational (KCR) solutions, interaction of the multi-lump and periodic solutions as well as breather style of solitary wave solutions. Using a transformation of dependent variable, which contain a controlling parameter (can control the direction, wave height and angle of the traveling wave), we build generalized breathers, rogue wave, Akhmediev breather, homoclinic breather, Ma breather, Kuznetsov-Ma breather and their relating rogue waves, multiwave, M-shaped rational and some various interactions. We show the results of our solutions graphically in distinct dimensions by assigning different values to the parameters.

Mixed lump–kink solutions and lump–soliton solutions to the generalized BKP equation with some second order terms

A generalized ( 2 + 1 ) -dimensional BKP equation with more second order nonlinear terms is studied. The different types of lump–kink solutions and lump–soliton solutions are constructed by using a combination of Hirota’s bilinear method and Maple symbolic computations. By assuming the solution of Hirota’s bilinear equation has a linear combination of quadratic function and exponential function, we have eight classes of lump–kink solutions and four classes of lump–soliton solutions. Two illustrative examples of these mixed solutions are studied and their dynamics are analyzed.

Dynamics of mixed lump-soliton solutions to the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli like equation

In this paper, a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) like equation is proposed based on the generalized bilinear operators in a prime number p = 3 . By combining an exponential function with a quadratic function, an interaction between a lump wave and one kink soliton is generated. Furthermore, the lump and the interaction solutions are illustrated by three-dimensional plots, density plots, contour plot and propagation pattern along x -axis by selecting the appropriate parameters.

Lump and interaction solutions for a (2 + 1)-dimensional combined pKP-BKP equation in fluids

In this paper, a number of exact solutions of (2 + 1)-dimensional combined pKP-BKP equation are given. By taking into account the Hirota bilinear method and making use of quadratic function, a number of lump solutions are obtained. Combining quadratic function with exponential function, the interaction phenomenon between lump and soliton, and the lump-soliton solutions are obtained. When the trigonometric function is combined with the exponential function, the breather wave solutions are obtained. Finally, by picking appropriate parameter values, the dynamic properties of these exact solutions are presented by three-dimensional, density and contour plots.

Abundant wave solutions for generalized Hietarinta equation with Hirota’s bilinear operator

In this work, we study the generalized ([Formula: see text])-dimensional Hietarinta equation by utilizing Hirota’s bilinear method. In addition, the lump solution and breather solution are presented. Using suitable mathematical assumptions, the new types of lump, singular, and breather soliton solutions are derived and established in view of the hyperbolic, trigonometric, and rational functions of the governing equation. Moreover, we give a lot of graphs in some subsections to determine the analysis of behavior solutions for the generalized ([Formula: see text])-dimensional Hietarinta equation. The results are useful for obtaining and explaining some new soliton phenomena.