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Investigated in this work is some high-order localized waves and their dynamical properties of a typical (3+1)-dimensional shallow water wave model. Firstly, N-solitons and M-order breathers were studied by applying Hirota's bilinear method, and we also investigated the parameter regulation of different structures of high-order breathers. Secondly, we studied high-order lump solutions and their trajectory equations, with a focus on the interaction behaviour between lump waves and n-solitons (where n is any positive integer), including cross-type and Y-type n-solitons, as well as the trajectory equations before and after collision, and provided computational proofs. Thirdly, we also investigated the interaction behaviour and collision trajectory between the lump solution and the breathers. Meanwhile, some structures and properties of the obtained solution are displayed through a large number of two-dimensional and three-dimensional plots. [ABSTRACT FROM AUTHOR]

We construct a family of singly periodic entire solutions to the Allen–Cahn equation in R 3 . Our approach relies on the relationship between this equation and the two-dimensional Toda lattice, an important integrable system. The level sets of these solutions closely approximate the graphs of suitable scaled lump solutions of the Toda lattice. [ABSTRACT FROM AUTHOR]

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions. [ABSTRACT FROM AUTHOR]

In this work, the Hirota bilinear symbolic computational method along with test functions and the generalized exponential rational function method are capitalized to secure soliton and lump solutions to the Sharma–Tasso–Olver–Burgers equation. Several novel soliton solutions are observed in unique patterns such as periodic, exponential, hyperbolic, dark, singular, and combo forms. Additionally, we also extract interaction, lump and breather solutions of the governing model. The novel characteristic of this work is the attained results, which were not before computed. Modulation instability of the governing equation is also examined via linear stability theory. To demonstrate the physical aspects and configuration of the attained solitons, some distinct graphs are plotted in different shapes. The validity of the solutions is verified by using Mathematica. The constructed outcomes are very encouraging and entail that the concerned methods can be utilized to acquire assorted improved, innovative, and advantageous outcomes for miscellaneous remarkable nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is systematically investigated based on the Hirota bilinear method. The explicit N-soliton solution and the bright and dark multi-soliton solutions of it are first derived. Next, various bright and dark higher-order breather solutions, including the periodic line wave solutions, as well as the hybrid solutions composed of solitons, breathers, and periodic line waves, are proposed by virtue of the complex conjugate constraints on the parameters. Then, applying the long wave limit to the N-soliton solution, the bright and dark lump solutions and line rogue wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation are constructed. The semi-rational solutions composed of breathers, lumps, solitons, and line rogue waves are further discussed. These new exact solutions all appear in pairs of bright and dark, which can be interpreted by the uplifts and collapses of energy. In addition, the dynamic behaviors of these exact nonlinear wave solutions are vividly demonstrated by their corresponding three-dimensional diagrams, sectional drawings, and density plots with contours. [ABSTRACT FROM AUTHOR]

In this work, a more accurate analytical solution of nonlinear partial differential equation is sought by setting the generalized activation function in the model of multiple bilinear neural network method. As an example, the 3-2-2-1, 3-2-3-1, 3-3-2-1 and 3-3-3-1 models are selected to study the new (2+1) dimensional nonlinear wave equation equations. Exact analytical solutions with arbitrary activation functions are obtained by selecting different activation functions and the dynamical properties are demonstrated through three-dimensional, two-dimensional and density plots. [ABSTRACT FROM AUTHOR]

The Lie symmetry analysis and generalized Riccati equation expansion methods were performed on the inviscid and viscous incompressible magnetohydrodynamic equations. Using the Lie symmetry analysis method, symmetries and similarity reductions of (2 + 1)- and (3 + 1)-dimensional magnetohydrodynamic equations were derived. Different forms of trigonometric function solutions and rational solutions were obtained, which yielded periodic solutions, single soliton solutions, and lump solutions. Furthermore, using the generalized Riccati equation expansion method, we obtained abundant new solutions of magnetohydrodynamic equations, including kink, kink-like, breather, and interaction solutions. Moreover, the stability of magnetohydrodynamic equations was investigated from both qualitative and quantitative perspectives. The exact solutions and stability analysis could provide accurate mathematical descriptions and theoretical basis for numerical analysis and regulation of magnetohydrodynamic systems. [ABSTRACT FROM AUTHOR]

In this article, we consider a (3 + 1) -dimensional negative-order KdV–CBS equation which represents interactions of long wave propagation dynamics with remarkable applications in the field of fluid mechanics and quantum mechanics. We investigate the integrability aspect of the considered model in the framework of Hirota bilinear differential calculus, construct infinitely many conservations laws and formulate a Lax pair. At first, we introduce the concept of Bell polynomial theory and utilize it to obtain the Hirota bilinear form. We introduce a two-field condition to determine the bilinear Bäcklund transformation. We use the Cole–Hopf transformation in bilinear Bäcklund transformation and linearize it to obtain the Lax pair formulation. The existence of infinitely many conservation laws has been checked through the Bell polynomial theory. Moreover, we derive one-kink, two-kink and three-kink soliton solution from the Hirota bilinear form. We have successfully investigated the existence of traveling wave solution for the (3 + 1) -dimensional negative-order KDV–CBS equation and the conditions for the existence of the solution are reported. The traveling wave solutions are extracted in the form of incomplete elliptic integral of second kind and Jacobi elliptic function. Particularly, the use of long wave limit yields kink soliton solutions. Furthermore, we exhibit necessary and sufficient condition for extracting lump solutions of (3 + 1) -dimensional nonlinear evolution equations, which have few particular types of Hirota bilinear form. The lump solutions are exploited by means of well-known test function in the Hirota bilinear form. This method reduces the number of algebraic equations to solve in deriving lump solutions of variety of NLLEs in comparison with the previously available methods in literature. Finally, two new forms of test functions are chosen and lump-multi-kink solutions have been determined. [ABSTRACT FROM AUTHOR]

In this paper, we investigate lump solutions of the (3+1)-dimensional gCH-KP equation employing the Hirota’s bilinear method and symbolic computation method. The bilinear form of the equation is first derived via the Hirota’s bilinear method under the appropriate transformations. Based on the bilinear form, some new 1-lump, 2-lump, 3-lump and 6-lump solutions of the (3+1)-dimensional gCH-KP equation are obtained with aid of the Maple via the symbolic computation method. The lump solutions are also illustrated by two-dimensional, three-dimensional and contour plots. The results obtained in this paper will be used to describe some interesting nature phenomena in the physics, oceanography, gas dynamics, biophysics, nonlinear optics and so on.

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.

The (3+1)-dimensional generalized shallow water wave equation is systematically investigated in this paper based on the Hirota bilinear method. The N-soliton solution and the higher-order kink-shaped breather solutions of the (3+1)-dimensional generalized shallow water wave equation are first proposed. Then, the line rogue wave solution and various hybrid solutions consisting of the breather, the kink-shaped soliton and the periodic solutions are discussed. Furthermore, the lump solutions of it are derived by using the long wave limit of the N-soliton solution. In addition, the diverse semi-rational solutions composed of lumps, kink solitons, line rogue wave and breathers enrich the research contents of the (3+1)-dimensional generalized shallow water wave equation. The dynamic behaviors of these exact solutions are vividly presented by their respective three-dimensional diagrams and density plots with contours. [ABSTRACT FROM AUTHOR]

Based on the bilinear form of the generalized (3+1)-dimensional KdV equation, the lump solution, the interaction solution and the breather solution of the equation were obtained. The obtained lump solutions were proved to be rationally localized in all directions of the space, then the "fusion" and "fission" phenomena were observed during the interaction between the lump soliton wave and the one-stripe soliton. Finally, the breather solution of the equation was obtained. [ABSTRACT FROM AUTHOR]

Our work is to discuss some new exact solutions of the (2+1)-dimensional generalized Benjamin–Ono (gBO) equation, which describes small amplitude long waves on the surface of shallow water. After careful consideration, lump, breather soliton and new solitons solutions of gBO equation are gained by using the Hirota bilinear method, the test function and the idea of the homogeneous balance, which have not been studied yet. By improving the expression of the test function, the quadratic function with two squares is increased to the quadratic function with three squares, and various lump solutions are obtained. Combining trigonometric function with hyperbolic function, breather soliton is derived. Obviously, sometimes the test function form of tanh–coth and tan–cot methods are so simple that it can’t get the desired result. Utilizing the improved tanh–coth and tan–cot methods, whose solutions depend on the parameter n, we can get more soliton solutions. Finally, by determining different parameters, one can draw the three-dimensional plots and density plots at different times. By observing these figures, we analyze the dynamic behavior of (2+1)-dimensional gBO equation in detail. These results can help us to understand nonlinear systems better. [ABSTRACT FROM AUTHOR]

In this article, a new dynamical system equation is constructed, named the (3+1)-dimensional Hirota-bilinear-like equation. The new ‘like’ equation has more nonlinear terms than the original equation while they have the same bilinear form. The generalized Hirota bilinear operator and its logarithmic transformation are applied to obtain the new ‘like’ equation and to find its solutions. Some different types of symbolic solutions are obtained, which are the lump solutions and the lump-kink-type rogue wave solutions. The propagation images of the lump waves and the interaction which is between the lump-type waves and two-kink soliton-waves are simulated to analysis the shapes and movements of these solutions.

In this paper, based on the combination of Hirota's bilinear method and long wave limit technique, we investigate rational and semi-rational solutions to the third-type Davey–Stewartson (DS III) equation and its nonlocal version. Rational solutions to the DS III equation demonstrate to be kinks, lumps and line rogue waves, while semi-rational solutions display hybrids of solitons, lumps and line rogue waves. As to the nonlocal DS III equation, we derive (semi-)rational solutions and breather solutions. Semi-rational solutions show lumps on the periodic line backgrounds, hybrids of breathers and lumps, line rogue waves and line breathers on the periodic line background. [ABSTRACT FROM AUTHOR]

We solve the Ostrovsky equation in the absence of the rotation effect using the Hirota bilinear method and symbolic calculation. Some unique interaction phenomena have been obtained between lump solution, breather wave, periodic wave, kink soliton, and two-wave solutions. All the obtained solutions are validated by putting them into the original problem using the Wolfram Mathematica 12. The physical characteristics of the solutions have been visually represented to shed additional light on the acquired results. Furthermore, using the novel conservation theory, the conserved vectors of the governing equation have been generated. The discovered results are helpful in understanding particular physical phenomena in fluid dynamics as well as the dynamics of nonlinear higher dimensional wave fields in computational physics and ocean engineering and related disciplines.

The anomalous interactions of dark lumps that are completely different from the general lump collisions of the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation were constructed. Different from the normal interaction mechanisms, lumps with same velocity and depth were obtained by extending the expression of rational solution based on Hirota’s Bilinear method. Moreover, interactions between dark lumps and soliton were also investigated. In particular, rogue wave excited from the lumps and stripe soliton in the interaction duration were observed. The dynamic behavior of dark lumps was analyzed in detail, and the results obtained were illustrated graphically.

A combined fourth-order (2 + 1)-dimensional nonlinear partial differential equation which contains four fourth-order nonlinear terms and all second-order linear terms is formulated. This equation covers three generalized KP, Hirota–Satsuma–Ito, and Calogero–Bogoyavlenskii–Schiff equations as examples, which have physical applications in the study of various nonlinear phenomena in nature. In terms of some settings of the coefficients, a class of lump solutions is constructed by the Hirota bilinear method and the solutions are calculated through the symbolic computation system of Maple. Meanwhile, the relation between the coefficients and the solution is explored. Two special lump solutions are generated by taking proper values for the involved coefficients and parameters, and their dynamic behaviors are studied, as illustrative examples. The primary advantage of the Hirota bilinear method is to transform a nonlinear equation into a bilinear one so that the targeted equation can be easily studied. [ABSTRACT FROM AUTHOR]

In this article, we investigate the generalized (2+1)-dimensional shallow water wave equation which enables an unidirectional propagation of shallow-water waves. By noticing that the system is integrable, we could get the diverse forms of the solitary wave solutions by using the rogue wave and semi-inverse variational principle (SIVP) schemes. In particular, we investigate four solutions including rogue wave, soliton, bright soliton, dark soliton, and lump solutions. To achieve this, an illustrative example of the Helmholtz equation is provided to demonstrate the feasibility and reliability of the used procedure in this study. The effect the free parameters on the behavior of acquired figures to a few obtained solutions containing two nonlinear rational exact cases was also analyzed due to the nature of nonlinearities. The dynamic properties the obtained results are shown and analyzed by some density, two and three-dimensional images and also are presented the physical nature of obtained solutions.

Under investigation is the (2+1)-dimensional version of fifth-order equation, which is an extension of the (1+1)-dimensional fifth-order equation. First, the N -soliton solutions of this (2+1)-dimensional equation are constructed by using its Hirota bilinear form. Second, through using long-wave limit technique and selecting appropriate parameters for the N -soliton solutions, diverse localized wave structures including kink soliton, breather, lump solution and their mixed interaction behaviors are obtained, and relevant dynamics features are discussed and analyzed graphically. Finally, some mathematical properties and special physical phenomena are summarized. The results obtained in this paper might be helpful to deepen the understanding of the interaction evolution of nonlinear localized waves. [ABSTRACT FROM AUTHOR]

In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail. [ABSTRACT FROM AUTHOR]

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation's dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas. [ABSTRACT FROM AUTHOR]

The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized (2 + 1)-dimensional Camassa-Holm-Kadomtsev-Petviashvili (CHKP) equation, which contains first-order, second-order, and third-order waves solutions. At the critical point, the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value. For the case, the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole. Also, the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms. In the meanwhile, the cross-kink wave and periodic wave solutions can be gained by the Hirota operator. The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values. We alternative offer that the determining method is general, impressive, outspoken, and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering.

The Hirota bilinear method is applied to construct the new dynamics of complex N-soliton solutions for a nonlocal breaking equation. By using the auxiliary traveling wave function, the different-order soliton solutions, bifurcation solutions and lump solutions of the model are obtained. The complex variable for Hirota bilinear form of this breaking equation is derived firstly via logarithmic variable transform. The physical phenomena and soliton propagation behavior of these solutions are explored, and the obtained results verify the proposed soliton solutions.

The Kadomtsev-Petviashvili (KP) equation is instrumental in characterizing nonlinear wave phenomena in fluid dynamics. With the incorporation of three-dimensional disturbances, the (3+1)-dimensional KP equation emerges. In this study, we employ the Hirota bilinear method to investigate a (3+1)-dimensional KP equation. We successfully derive both lump and lump-chain solutions for this equation. The characteristics of the resulting lump are thoroughly examined. It is observed that those lumps propagate along a specific ray direction, with the origin of these rays dependent on the imaginary part of σ ˆ j k = σ j − σ k with 1 ≤ j , k ≤ 3 and j ≠ k. We have simultaneously obtained the propagation direction of these rays. The periodicity of those lumps is denoted as T j k dependent on spectrum parameter. Furthermore, we explore two potential lines, including the characteristic line associated with the lumps, to gain a deeper understanding of the nonlinear waves controlled by this system. • We successfully derive both lump and lump-chain solutions for this equation. • It is observed that those lumps described by this equation propagate along a specific ray direction. • We have simultaneously obtained the propagation direction of these rays. The periodicity of those lumps is obtained. • We explore both of the two potential lines for this equation. [ABSTRACT FROM AUTHOR]

In this paper, a (4+1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients in fluid mechanics is investigated. Based on the improved positive quadratic function method, the lump, lump-soliton and rogue-soliton solutions are obtained. The interaction between lump wave and periodic wave is studied. Further, the breather wave solutions are presented. The nonlinear dynamics for different nonautonomous wave structures solutions are described in 3D and contour plots. [ABSTRACT FROM AUTHOR]

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.

We investigate rational and semi-rational solutions to the (2+1)-dimension Mel’nikov equation based on the Hirota bilinear method and the long wave limit approach. It is shown that the derived rational solutions display lump solitons in (x,y)-plane and line rogue waves in (x,t)-plane. As to the semi-rational solutions consisting of the hybrid of the soliton and rational solution, they exhibit the interaction between solitons and lumps in (x,y)-plane. While in (x,t)-plane the hybrid solutions demonstrate the interaction between line rogue waves and solitons. Dynamics of solutions are analysed with plots.

In this paper, we study lump solutions and rogue wave solutions for three (1+1)-dimensional nonlinear dynamic systems. By applying the Hirota direct method, lump and rogue wave solutions are presented with the aid of symbolic computations. The solutions are expressed in terms of a set of restricted parameters with necessary and sufficient conditions that guarantee their existence.

In this paper, we are concerned with the (2 + 1) dimensional Maccari's equation. With the aid of Truncated Painlevé Approach (TPA), the localized solutions have obtained in terms of arbitrary functions. By using the suitable arbitrary functions present in the solution, we have generated the multi-rogue waves, rogue wave doublet and lump solutions. Moreover, through selecting the values of control parameters, dynamical behaviors for the obtained multi-rogue wave, rogue wave doublet and lump solutions are graphically illustrated with the aid of Mathematica tool. The results obtained through this investigation will be beneficial for understanding the dynamics of nonlinear waves in higher dimensional Maccari systems. [ABSTRACT FROM AUTHOR]

In this article, the (3+1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, which describes the interactions of long wave propagations and has diverse applications in physics, mathematics and engineering, is investigated. Lump and multi-lump wave solutions, one-, and two-soliton solutions, exploding and periodic wave solutions, localized and breather wave solutions, and interaction of lump waves with solitary waves are constructed through the symbolic computational method. In addition, multi-soliton solutions in complex form via using Hirota's simple method and long-wave method are obtained. The dynamic behaviors of all obtained solutions are analyzed and illustrated in figures by choosing appropriate values of the parameters all of the obtained solutions are verified by a direct substitution in the original equation. [ABSTRACT FROM AUTHOR]

In this work, the (2 + 1) -dimensional Sawada–Kotera-like (SK-like) equation is derived through the generalized bilinear equation from SK equation. By using "3-2" and "3-2-2" neural network models, three trial functions are constructed. Classic lump, generalized lump solutions and new rogue wave solutions are acquired by giving a number of specific parameters. Through a variety of curve plots of three-dimensional contour plotting, these rogue waves are intuitively displayed. [ABSTRACT FROM AUTHOR]

In this study, we successfully apply Hirota's bilinear method (HBM) to retrieve the different wave structures of the general r th dispersionless Dym equation by considering the test function approaches. The studied model is used to describe the dynamics of deep water waves. We formally retrieve some novel lump periodic, some other new interaction, and breather wave solutions. Moreover, the physical behavior of the reported results is sketched through several three-dimensional, two-dimensional and contour profiles with the assistance of suitable parameters. The acquired results are valuable in grasping the elementary scenarios of nonlinear fluid dynamics as well as the dynamics of engineering sciences in the related nonlinear higher-dimensional wave fields. The gained results are checked and found correct by putting them into the governing equation with the aid of Mathematica. Thus, our strategies through the fortress of representative calculations give a functioning and intense mathematical execution for tackling complicated nonlinear wave issues. [ABSTRACT FROM AUTHOR]

The order- n periodic solutions for the (2+1)-D complex modified Korteweg–de Vries (cmKdV) equations are investigated with the aid of Darboux transformation (DT) method. By using Taylor expansion considering the limits λ 2 k − 1 → λ 0 = − a 2 + c i (k = 1 , ... , n) , order-n rational solutions are obtained, among which the order-1 and order-2 solutions are analyzed in detail. By varying different parameter l 1 , two kinds of rational solutions are deduced, namely, the line rogue wave solutions and the lump solutions. Dynamical properties of these solitons, including speed, amplitude, and extreme values, are investigated. It is shown that the line rogue wave solutions appear and disappear, while the lump solutions are localized traveling wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the interaction of the rational function, hyperbolic as well as exponential function for the (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation with the aid of Hirota's bilinear approach. We find several families of lump-type solutions. This method is a powerful and advantageous mathematical tool for establishing abundant lump solutions of nonlinear partial differential equations. In order to illustrate their dynamic properties, some figures are plotted with determined parameters. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new nonlinear evolution equation, which is (4 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation. First, according to the Hirota bilinear method, we get some exact solutions of the equation, including lump solution, lump-soliton solution, rogue-soliton solution and lump-kink solution. Then, we obtain some new exact solutions by generalizing the form of the lump solution on a further solution. Finally, based on the symbolic calculation method with Mathematica, the characteristics of the interaction solutions are shown in the graphs and we analyze the dynamic change of the solutions. Furthermore, we discuss the applications of these solutions in physics via the analysis. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (BLMP-like equation) is introduced by the generalized bilinear operators Dp associated with p = 3. The lump solutions, lump-soliton interaction phenomena and breather-soliton solutions are discussed to the (3+1)-dimensional BLMP-like equation based on the generalized bilinear method with symbolic computation system Mathematica. In order to observe the behavior of those solutions, we fix the value of z, then give the 3D-graphs of some solutions at different times. We find a lump solution moved in oblique direction; a lump-soliton interaction phenomenon is appeared and disappeared along with the time. We also see a kink-breather soliton moved in oblique direction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate lump solutions of the (3+1)-dimensional gCH-KP equation employing the Hirota's bilinear method and symbolic computation method. The bilinear form of the equation is first derived via the Hirota's bilinear method under the appropriate transformations. Based on the bilinear form, some new 1-lump, 2-lump, 3-lump and 6-lump solutions of the (3+1)-dimensional gCH-KP equation are obtained with aid of the Maple via the symbolic computation method. The lump solutions are also illustrated by two-dimensional, three-dimensional and contour plots. The results obtained in this paper will be used to describe some interesting nature phenomena in the physics, oceanography, gas dynamics, biophysics, nonlinear optics and so on. • The bilinear form of the (3+1)-dimensional gCH-KP equation is derived. • Some new lump solutions are obtained. • We illustrate the lump solutions by plotting their graphs for particular values of the parameters. [ABSTRACT FROM AUTHOR]

Localized interaction solutions of the (2+1)-dimensional Ito equation with free parameters are obtained by using a Hirota bilinear transformation. Various plots with particular choices of the involved parameters are made to show energy distributions and dynamical properties of the special exact solutions. This phenomenon may provide us with interesting information on dynamics in the higher-dimensional nonlinear world. [ABSTRACT FROM AUTHOR]

In this paper, we study some exact solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles. Based on a given transformation and the bilinear form, together with symbolic computations, we obtain the multi-soliton solutions and periodic solutions including X-periodic, Y-periodic and 2-periodic wave solutions. By using of the long wave limit method, we obtain lump solutions and lump-periodic solutions that lumps moving on a background with different periodic waves. The properties of the solutions are analyzed graphically.

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, N -soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model. [ABSTRACT FROM AUTHOR]

Based on Hirota bilinear formulation, the lump solutions to dimensionally reduced generalized variable coefficient B-type Kadomtsev-Petviashvili equation are obtained. The solution process is figured out and the solution properties are illustrated graphically. The present method can be extended to other non-linear equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Starting with Hirota bilinear form of (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada equation, a class of lump solutions, a strip soliton, a pair of resonance solitons as well as the rogue wave have been obtained through symbolic computation. Firstly, two lump solutions are derived, consisting of six free parameters, while the other three accord with non-zero determinant condition which guarantee locality and analyticity for the solutions. Secondly, the interaction phenomenon between a lump wave and a stripe soliton is reflected. Interaction figures show that the lump wave can be absorbed by the stripe wave or separate from it. Thirdly, by a novel ansatz of hyperbolic function and quadratic function, a fresh transformation for the rogue wave and a pair of resonance solitons have been discovered, that a peak profiled rogue wave emerges from one of the resonance wave and transfers to another, and finally fades away. At the same time, numerical simulations are demonstrated to visualize the mechanism of the above solutions.

Based on symbolic computation and Hirota bilinear form, a class of lump solutions for the (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation is given. As a result, the lump solution shows a new perspective to knowledge them. Meanwhile, variable separation solutions are also constructed by applying a simple ansatz method. These variable separation solutions contain the shock solution, the peakon-type breather solution and the periodic solution. They showed some special localized structures in 2D and 3D-plots.