Lump-kink Solutions Research Articles

The non-autonomous perturbed potential Kadomtsev–Petviashvili equation: its integrability, kinky-quasiperiodic, kink-like breather, lump-kink solutions with mixed backgrounds

The integrability of (2+1)-dimensional non-autonomous perturbed potential Kadomtsev–Petviashvili (NpPKP) problem has been demonstrated in this manuscript using Lax pairs and the Bäcklund transformation. The bilinear form of the NpPKP problem has been generated with the help of Hirota polynomial technique. Using test function and N-dimensional Riemann theta function, the kink-like breathers, lump-kink, interaction of lump and shock and the kinky quasiperiodic are explored analytically in light of the vector selections in the solution space. Outcomes of the solutions are plotted as illustration using particular values of the relevant parameters. These findings may aid in our understanding of intriguing physical events and mechanisms in potential systems.

Rogue wave, lump, kink, periodic and breather-like solutions of the (2+1)-dimensional KdV equation

In this paper, the ([Formula: see text])-dimensional KdV equation is investigated by using the bilinear neural network method (BNNM). We construct six neural network models, extending beyond single hidden layer models to create deeper and broader network structures (e.g. [3-3-1], [3-4-1], [3-1-3-1], [3-4-1-1], [3-2-2-1] and [3-2-3-1-1] models). Introducing specific activation functions into the neural network model enables the generation of various test functions, resulting in novel solutions for equations that include rogue wave solutions, lump-kink solutions, periodic soliton solution, breather-like solutions and lump solutions. The physical properties of these novel solutions are vividly depicted through three-dimensional plots, density plots, and curve plots. The findings contribute to a better understanding of the propagation behavior of shallow water waves.

Bilinear auto-Bäcklund transformations, breather and mixed lump–kink solutions for a (3+1)-dimensional integrable fourth-order nonlinear equation in a fluid

In this paper, we investigate a (3+1)-dimensional integrable fourth-order nonlinear equation that can model both the right- and left-going waves in a fluid. Based on the Hirota method, we derive three sets of the bilinear auto-Bäcklund transformations along with some analytic solutions. Through the extended homoclinic test approach, we obtain some breather solutions. We find that the breather propagates steadily along a straight line, with one hole and one peak in each period. Graphical investigation indicates that the coefficients in that equation affect the location and shape of the breather. Moreover, we construct some mixed lump–kink solutions. Fusion and fission between a lump wave and a kink soliton are analyzed graphically. The solutions addressed in this paper may be applied to mimic some complex waves in fluids.

Dynamics of breather waves, lump-kink solutions and interaction solutions for a (3+1)-dimensional generalized shallow water waves equation

PurposeThis paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.Design/methodology/approachHirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.FindingsThe fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.Originality/valueThis paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.

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.

Novel interaction phenomena of the new (2 + 1)-dimensional extended shallow water wave equation

Based on the Hirota bilinear method, we study some kinds of novel interaction solutions of the new integrable [Formula: see text]-dimensional extended shallow water wave equation. Using the symbolic computations, we obtain a rich variety of exact solutions such as the lump solutions, lump-kink solutions, breather solutions and two-kink solutions and these exact solutions are shown analytically and graphically by selecting the proper parameters. These new exact solutions contribute us to our understanding of the physical phenomena of the integrable nonlinear shallow water wave model.

Abundant soliton wave solutions and the linear superposition principle for generalized (3+1)-D nonlinear wave equation in liquid with gas bubbles by bilinear analysis

In this article, we study the generalized (3+1)-dimensional nonlinear wave equation where is investigated in soliton theory and by employing the Hirota’s bilinear method the bilinear form is obtained, and the N-soliton solutions are constructed. In addition, some Lump solutions, Lump-kink solutions, Lump-two kink solutions, Lump-periodic solutions, its Interaction solutions, Cross-kink wave, Breather-type, Multi wave, Periodic wave solutions, and Solitary wave solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic. Moreover, we employ the linear superposition principle to determine N-soliton wave solutions for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles.

Lump, lump-stripe, and breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation in fluid mechanics

The present study investigates the lump, one-stripe, lump-stripe, and breather wave solutions to the (2+1)-dimensional Sawada-Kotera equation using the Hirota bilinear method. For lump and lump-stripe solutions, a quadratic polynomial function, and a quadratic polynomial function in conjunction with an exponential term are assumed for the unknown function f giving the solution to the mentioned equation, respectively. On the other hand, only an exponential function is considered for one-stripe solutions. Besides, the homoclinic test approach is adopted for constructing breather wave solutions. The propagations of the attained lump, lump-stripe, and breather wave solutions are shown through some graphical illustrations. The graphical outputs demonstrate that the lump wave moves along the straight line y=−x and exponentially decreases away from the origin of the spatial domain. On the other hand, lump-kink solutions illustrate the fission and fusion interaction behaviors upon the selection of the free parameters. Fission and fusion processes show that the stripe soliton splits into a stripe soliton and a lump soliton, and then the lump soliton merges into one stripe soliton. In addition, the achieved breather waves illustrate the periodic behaviors in the xy-plane. The outcomes of the study can be useful for a better understanding of the physical nature of long waves in shallow water under gravity.

Characteristics of lump-kink and their fission-fusion interactions, rogue, and breather wave solutions for a (3+1)-dimensional generalized shallow water equation

In this study, lump and two classes of interaction, multi-stripe, and breather wave solutions for the (3+1)-dimensional generalized shallow water equation are presented via the Hirota bilinear method. Interaction solutions are found between one-lump and one-stripe, and one-lump and two-stripes solutions by combining a quadratic function and an exponential function, and a quadratic function and a hyperbolic cosine or double exponential functions, respectively. Dynamical behaviours of some obtained valid solutions are presented through some graphs. The physical interpretation of fission-fusion dynamics is also explained graphically through lump-kink interaction solutions. During the fission-fusion interaction process, it is seen that stripe solitons split into a stripe and a lump soliton, and then the lump and stripe solitons fuse together. During this process, a rogue wave is found between one lump and twin stripes soliton at Furthermore, multi-stripe and breather wave solutions are investigated by choosing the appropriate functions and the values for the free parameters. The multi-stripe waves are found to be nonsingular and rectangular hyperbolic shaped. On the other hand, breather waves are found to be periodic, which can evolve periodically along a straight line in the xy-plane. The produced wave solutions might be helpful to understand the propagation behaviour of waves in shallow water.

Abundant lump-type solutions for the extended (3+1)-dimensional Jimbo–Miwa equation

In this work, we research the mixed lump–kink solutions and their dynamic properties of the extended (3+1)-dimensional Jimbo–Miwa equation. Mixed lump–kink solutions are triggered by the interaction between lump soliton and exponential function, the interaction between lump soliton and hyperbolic function, and the interaction between lump soliton and periodic wave. Their dynamics properties are shown by means of many graphics and corresponding density plots.

Localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations* *Project supported by the Fundamental Research Funds for the Central Universities of China (Grant No. 2018RC031), the National Natural Science Foundation of China (Grant No. 71971015), the Program of the Co-Construction with Beijing Municipal Commission of Education of China (Grant Nos. B19H100010 and B18H100040), and the Open Fund

We focus on the localized characteristics of lump and interaction solutions to two extended Jimbo–Miwa equations. Based on the Hirota bilinear method and the test function method, we construct the exact solutions to the extended equations including lump solutions, lump–kink solutions, and two other types of interaction solutions, by solving the under-determined nonlinear system of algebraic equations for associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed.

M-lump and lump–kink solutions of $$(2+1)$$-dimensional Caudrey–Dodd-Gibbon–Kotera–Sawada equation

In this paper, M-lump and interaction between lumps and kink solitons of the $$(2+1)$$ -dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation are studied based on Hirota bilinear form. M-lump solutions are derived by taking a ‘long wave’ limit of the N-soliton solutions, and the lump–kink solutions are presented consequently. In addition, evolutions of solutions are shown by choosing certain parameters.

New types of interaction solutions to the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili equation

In this paper, with the aid of symbolic computation system Maple, and based on the simplified Hirota method and ansatz technique, we discussed the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation with [Formula: see text] to obtain lump solutions, lump–kink solutions and three classes of interaction solutions. Comparing our new results with other researchers’ results shows that using this method gives the more opportunity to solve the nonlinear partial differential equations that appear in mathematics, physics, biological engineering and other fields. We also presented profiles of new lump solution, lump–kink solutions and interaction solutions as illustrative examples.

Lump and interaction solutions for a generalized (3 + 1)-dimensional propagation model of nonlinear waves in fluid dynamics

In this paper, we focus on the lump and interaction solutions for a generalized (3 + 1)-dimensional propagation model of nonlinear waves in fluid dynamics. With symbolic computation, some new exact solutions which include lump, lump-periodic, lump-kink and lump-soliton solutions are presented by using the Hirota's bilinear method and test function method. The lump and interaction solutions are illustrated by three-dimensional plots and contour plots.

Multiple-soliton and lump-kink solutions for a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation

In recent years, searching for analytic solutions to nonlinear evolution equations has become a popular topic. In this paper, a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation is proposed. With the help of symbolic calculation, the multiple-soliton and lump-kink solutions of the equation are obtained in two different ways. Those analytic solutions are presented, and their dynamic properties are discussed through graphical presentation. The final result is helpful for studying the interaction between solitons in nonlinear mathematical physics.

Study of lump and lump-kink solitons of a coupled reduced Hirota bilinear equation

By symbolic computation and searching for the solutions of the positive quadratic functions of the related bilinear equations, two kinds of lump solutions of the (3[Formula: see text]+[Formula: see text]1)-dimensional weakly coupled Hirota bilinear equation are derived, and the practicability of this method is verified. Then we add an exponential function to the original positive quadratic function, and obtain a new solution of the Hirota bilinear equation. The interaction between the lump solutions and lump-kink solutions is included in the new solution. On this basis, we give the possibility of adding multiple exponential functions. Finally, we give the coupled reduced Hirota bilinear equation lump-kink solitons by combining the above two methods. In order to ensure the analyticity and reasonable localization of the block, two sets of necessary and sufficient conditions are given for the parameters involved in the solution. The local characteristics and energy distribution of bulk solution are analyzed and explained.

Lump and Lump–Kink Soliton Solutions of an Extended Boiti–Leon–Manna–Pempinelli Equation

Abstract In this paper, the extended Boiti–Leon–Manna–Pempinelli equation (eBLMP) is first proposed, and by Ma’s [1] method, a class of lump and lump–kink soliton solutions is explicitly generated by symbolic computations. The propagation orbit, velocity and extremum of the lump solutions on (x,y) plane are studied in detail. Interaction solutions composed of lump and kink soliton are derived by means of choosing appropriate real values on obtained parameter solutions. Furthermore, 3-dimensional plots, 2-dimensional curves, density plots and contour plots with particular choices of the involved parameters are depicted to demonstrate the dynamic characteristics of the presented lump and lump–kink solutions for the potential function v = 2ln( f(x)) x .

Lump solitons and interaction phenomenon to a (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation

This paper studies lump solutions and interaction solutions for a (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like equation. With the help of symbolic computation and Hirota’s bilinear form, we obtain bright–dark lump solutions, lump-soliton solutions, and lump-kink solutions. Meanwhile, the dynamics of the obtained three classes of solutions are analyzed and exhibited mathematically and graphically. These results provide us with useful information to grasp the propagation processes of nonlinear waves.

Non-traveling lump solutions and mixed lump–kink solutions to (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation

Through Hirota bilinear form and symbolic computation with Maple, we investigate some non-traveling lump and mixed lump–kink solutions of the (2[Formula: see text]+[Formula: see text]1)-dimensional variable-coefficient Caudrey–Doddy–Gibbon–Kotera–Sawada equation by an extended method. Firstly, the non-traveling lump solutions are directly obtained by taking the function [Formula: see text] as a quadratic function. Secondly, we can get the interaction solutions for a lump solution and one kink solution by taking [Formula: see text] as a combination of quadratic function and exponential function. Finally, the interaction solutions between a lump solution and a pair of kinks solution can be derived by taking [Formula: see text] as a combination of quadratic function and hyperbolic cosine function. The dynamic phenomena of the above three types of exact solutions are demonstrated by some figures.

Painlevé analysis, lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painlevé–Bäcklund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.