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We discuss two models in 1 + 1 dimensional space-time for real scalar field multiplets on the root space of g $$ \mathfrak{g} $$ 2 and su $$ \mathfrak{su} $$ (4) Lie algebras. The construction of these models is presented and the corresponding BPS solutions are found.

In this paper, we investigate a ( $$2+1$$ )-dimensional Sawada–Kotera (SK) equation for the atmosphere, rivers, lakes, oceans, as well as the conformal field and two-dimensional quantum gravity gauge field. Bilinear form and N-soliton solutions, which are different from those in the existing literatures, are derived, where N is a positive integer. The higher-order breather, lump and hybrid solutions for the ( $$2+1$$ )-dimensional SK equation are also constructed based on the N-soliton solutions. Three kinds of the first-order breathers are obtained, and the higher-order breathers are constructed. The higher-order lump solutions are also derived via the long-wave limit method. Hybrid solutions composed of the solitons, breathers and lumps are worked out, and interaction between the waves is discussed graphically. Finally, similar solutions for a generalized form of the ( $$2+1$$ )-dimensional SK equation are given.

In this paper, a generalized $$(2+1)$$-dimensional nonlinear wave equation is obtained by extending the generalized $$(2+1)$$-dimensional Hirota bilinear equation into a more generalized form. The obtained new equation is useful in describing nonlinear wave phenomena in nonlinear optics, shallow water and oceanography. Based on the bilinear method, the N-soliton solutions of the generalized $$(2+1)$$-dimensional nonlinear wave equation are obtained. M-lump solutions are investigated by applying the long wave limit to the N-soliton solutions. The propagation orbits and velocities of the M-lump wave are analyzed. The high-order breather waves are obtained by establishing the complex conjugate relations in the parameters of the N-solitons. Furthermore, the interaction hybrid solutions are constructed, which contain hybrid solutions composed of breathers, solitons and lumps. The dynamical behaviors of the hybrid solutions are systematically analyzed via numerical simulations. The obtained results will enrich the study of theory of the nonlinear localized waves.

In this paper, the multivariate trilinear operators in the ($$3+1$$)-dimensional space are applied to a ($$3+1$$)-dimensional GBK equation. The resulting trilinear form is used to study its wave dynamics. Particularly, we generate a type of new interaction solutions between breather lump-type solitons and other multi-kink solitons, thereby formulating a kind of breather lump–kink solitons. By setting time constants, we change the coordinates of kink solitons to make them collide with the breather lump-type soliton, during which breather lump-type soliton is swallowed eventually by those kink solitons. The evolution behaviours of the breather lump–kink solitons are depicted by plotting 3-D and density graphs from the perspective of wave characteristics.

We investigate a reduced generalized (3 + 1)-dimensional shallow water wave equation, which can be used to describe the nonlinear dynamic behavior in physics. By employing Bell’s polynomials, the bilinear form of the equation is derived in a very natural way. Based on Hirota’s bilinear method, the expression of N-soliton wave solutions is derived. By using the resulting N-soliton expression and reasonable constraining parameters, we concisely construct the high-order breather solutions, which have periodicity in x,y-plane. By taking a long-wave limit of the breather solutions, we have obtained the high-order lump solutions and derived the moving path of lumps. Moreover, we provide the hybrid solutions which mean different types of combinations in lump(s) and line wave. In order to better understand these solutions, the dynamic phenomena of the above breather solutions, lump solutions, and hybrid solutions are demonstrated by some figures.

We present recent results on dynamical chiral symmetry breaking in (2 + 1)-dimensional QED with N four-component fermions. The results of the 1 / N expansion in the leading and next-to-leading orders were found exactly in an arbitrary nonlocal gauge.

Graphical abstract Periodic parabolic solitons with different energies have been presented. The purpose of changing the period and span of the parabolic solitons has been achieved by adjusting the corresponding parameters., In this paper, a system with controllable parameters for describing the evolution of polarization modes in nonlinear fibers is studied. Using the Horita’s method, the coupled nonlinear Schrödinger equations are transformed into the bilinear equations, and the one- and two- bright soliton solutions of system (3) are obtained. Then, the influencing factors on velocity and intensity in the process of soliton transmission are analyzed. The fusion, splitting and deformation of the solitons caused by their interactions are discussed. Finally, a method for adjusting the inconsistencies of sine-wave soliton transmission is given. The conclusions of this paper may be helpful for the related research of wavelength division multiplexing systems.

The (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation with weak nonlinearity, dispersion and perturbation can denote the development of the long waves and the surface waves in fluid dynamics. In this paper, the KP equation is illustrated with the symbolic computation. The mixed interaction solutions of local wave, solitary wave, breather wave, exploding wave and periodic wave for the equation are derived by the Hirota method. The effects of dispersion, nonlinearity and other parameters on the interactions are investigated. The solitary wave can be amplified via introducing the local wave. Adjusting the parameters can make the transmission of localized and breather wave more stable. Moreover, a new exploding and periodic wave is observed. It is useful for enriching the dynamic patterns of the wave solutions.

Korteweg-de Vries equation governs the weakly nonlinear long wave whose phase speed reaches a simple maximum of wave with the infinite length in shallow water wave. The exponential-form variable separation solution of (2+1)-dimensional Kortweg-de Vries equation is found via the two-function method, and this solution covers many special combined solutions including sinh-cosh,sin-cos,sech-tanh,csch-coth,sec-tan and csc-cot solutions. From the exponential-form solution with choosing suitable functions, inelastic interactions between special multi-valued solitons with two loops such as anti-bell-shaped, anti-peak-shaped semifoldons and anti-foldon are graphically and analytically studied. By the asymptotic analysis, phase shift and its difference during interactions between multi-valued solitons are analytically given.

This paper deals with localized waves in the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation in the incompressible fluid. Based on Hirota’s bilinear method, N-soliton solutions related to Boiti–Leon–Manna–Pempinelli equation are constructed. Novel nonlinear wave phenomena are obtained by selecting appropriate parameters to N-soliton solutions, and time evolutions of different kinds of solitary waves are investigated in detail. Rich elastic interactions are illustrated analytically and graphically. More specifically, the inelastic interactions, i.e., fusion and fission of solitary waves, are constructed by choosing special parameters on kink solitons and breathers. The analysis of the influence of parameters on propagation is revealed in three tables. The results have potential applications in fluid mechanics.

An effective and simple method to solve nonlinear evolution partial differential equations is the self-similarity transformation, in which one utilizes solutions of the known equation to find solutions of the unknown. In this paper, we employ an improved similarity transformation to transform the $$(2+1)$$-dimensional (D) sine-Gordon (SG) equation into the $$(1+1)$$-D SG equation and obtain non-rational solutions of the $$(2+1)$$-D SG equation by utilizing the known solutions of the $$(1+1)$$-D SG equation. Based on the solutions obtained, and with the help of special choices of the involved solution parameters, several localized structures of the $$(2+1)$$-D SG model are analyzed on a finite background, such as the embedded hourglass, split silo, dumbbell, and pie solitons. Their spatiotemporal profiles are displayed, and their properties are discussed.

This paper considers a generalized (2+1) dimensional nonlinear evolution equation depending on two nonzero arbitrary constants. We derive the Lie point symmetry generators and Lie symmetry groups. This symmetry analysis leads us the reductions equations, through one of which we obtain solutions. We also get the low-order conservation laws of the equation that have been obtained using the corresponding symmetries of the family. We will present a classification of conservation laws for this equation and we will apply Lie symmetry analysis to the equation in order to obtain exact solutions.

Under investigation in this letter is an extended (3 + 1)-dimensional Jimbo–Miwa (eJM) equation, which can be used to describe many nonlinear phenomena in mathematical physics. With the aid of Hirota bilinear method and long-wave limit method, M-order lumps which describe multiple collisions of lumps are derived. The propagation orbit, velocity and extremum of the 1-order lump solutions on (x, y) plane are investigated in detail. Resorting to the extended homoclinic test technique, we obtain the breather–kink solutions, rational breather solutions and rogue wave solutions for the eJM equation. Meanwhile, through analysis and calculation, the amplitude and period of breather–kink solutions increase with p increasing and the extremum of rational breather solution and rogue waves are also derived. T-order breathers are obtained by means of choosing appropriate complex conjugate parameters on N-soliton solutions. Periods of the 1-order breather solutions on the (x, y) plane are determined by $$k_{12}$$ and $$k_{12}p_{11}+k_{11}p_{12}$$, while locations are determined by $$k_{11}$$ and $$k_{11}p_{11}-k_{12}p_{12}$$. Furthermore, hybrid solutions composed of the kink solitons, breathers and lumps for the eJM equation are worked out. Some figures are given to display the dynamical characteristics of these solutions.

Liquids with gas bubbles are very common in science, engineering, physics, nature and life. Under investigation in this paper is a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles. With respect to the velocity of the liquid-gas bubble mixture, we obtain the lump, rogue wave, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions via the symbolic computation. Based on the mixed lump-stripe soliton solutions, we construct the rogue wave and investigate the fusion and fission phenomena between a lump and the one-stripe soliton. We graphically study the mixed lump-stripe soliton under the influence of the parameters β , γ , δ and ξ , which represent the dispersion, perturbed effect, disturbed wave velocities along the y and z (i.e., the two transverse) directions, respectively. With the decreasing value of β to -1, the graph from a lump and one-stripe soliton shows a soliton; with the increasing value of γ to -3, location of the lump moves along the negative x axis; with the value of δ increasing to 0.5, location of the lump moves along the positive x axis; with the increasing value of ξ to -3, location and range of the lump soliton keep unchanged. With respect to the velocity of the mixture, we obtain the interaction between a rogue wave and a pair of stripe solitons according to mixed rogue wave-stripe soliton solutions. A lump is provided.

In this work, we extend the (2+1)-dimensional Date- Date–Jimbo–Kashiwara–Miwa equation to a new version with time-dependent coefficients. The integrability property of this developed equation is demonstrated by exhibiting the Painleve test. We show that this time-dependent equation admits multiple real and multiple complex soliton solutions. Moreover, we derive other solitonic and periodic solutions.

This paper investigates the (2 + 1)-dimensional coupled Burgers equations (CBEs) which is an important nonlinear physical model. In this respect, by making use of the generalized unified method (GUM), a series of double-wave solutions of the (2 + 1)-dimensional coupled Burgers equations are derived. The Lie symmetry technique (LST) is also utilized for the symmetry reductions of the (2 + 1)-dimensional coupled Burgers equations and extracting a non-traveling wave solution. Through some figures, we discussed the wave structures of the double-wave solutions of the CBEs for different values of parameters in these solutions.

By using a direct method, we construct the Hirota bilinear form for an extended (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Based on this bilinearization, the lump solitons and rogue wave solutions are investigated. Constraint conditions for the wave propagation and velocity for lump solitons are found and verified by figures. Also the lump-stripe interaction was investigated to show that the lump solitons will be swallowed by the stripe soliton. Finally, the dynamic behaviour for the obtained lump solution, rogue wave and lump-stripe soliton interaction by suitable special parameters is shown graphically.

An integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations is investigated in this paper. We will refer to this integrable extension as the $$(4+1)$$-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the $$(4+1)$$-dimensional Fokas equation are constructed based on the Hirota’s bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the $$(1+1)$$-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of $$(3+1)$$-dimensional KP equation by reducing the solutions of the $$(4+1)$$-dimensional Fokas equation. All these results show that the $$(4+1)$$-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, nonlinear optics, and photonics, ion-acoustic waves in plasmas, matter waves in Bose–Einstein condensates, and sound waves in ferromagnetic media.

In this work, the (4+1)-dimensional Fokas equation, which is an important physics model, is under investigation. Based on the obtained soliton solutions, the new rational solutions are successfully constructed. Moreover, based on its bilinear formalism, a concise method is employed to explicitly construct its rogue-wave solution and interaction solution with an ansätz function. Finally, the main characteristics of these solutions are graphically discussed. Our results can be helpful for explaining some related nonlinear phenomena.

This paper presents a robust rack position control method based on a disturbance observer to enhance maneuverability and steering robustness during steering situation. It is shown that the rack and pinion system of Steer-by-Wire system with irregular parameters and external disturbance from the tires and mechanical elements can be modelled as a simple second-order system. In control design, robust control method, e.g., two-degree-of-freedom control (2-DOF) with a disturbance observer, is used for improving the steering robustness. Moreover, A friction compensator is designed to eliminate the friction generated by mechanical gear contact that hinders precise position control. The simulation model of the Steer-by-Wire system is designed with a permanent magnet synchronous motor (PMSM) and mechanical parts using AMESim program. The proposed controller was verified through simulation using AMESim and Matlab/Simulink cosimulation with different references.

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics and plasma physics. With the help of symbolic computation, we obtain and disc...

Influenced by the modern experiments on spin chain materials to CoPt3 nanowire system, we theoretically examine the dynamics of ferromagnetic nano-wire with Dzyaloshinskii-Moriya interaction. This classification is described with a background inthe Landau - Lifshitz - Gilbert equation. We analytically develop the electric field illustrations of both lump soliton and soliton collision interactions. Similarly, we designate this system aiming at the related parametrically (2 + 1) dimensional equation employing generalized unified method and give an estimated systematic solution that defines the soliton collision interaction. Furthermore, we establish that the soliton can be confined to nanowire system with a suitable condition.

In this section the Successive approximate method (S.A.M) introduced for solving the Wu-Zhang systems, a (1 + 1)-dimensional nonlinear dispersive wave equation, this method shows us that the technique provided without disorder, in this model of convergence power series with a simple calculated ingredients and gives effective results.

With the aid of symbolic computation, some new types of breathing wave solutions to a (3+1)-D Jimbo-Miwa equation are obtained by the extended homoclinic test method. Its homoclinic breather-wave solution, periodic oscillating soliton and doubly-soliton solution are investigated.

This work aims to study a new $({2 + 1} )\ KdV $ equation that is recently introduced in (Phys. Lett. A. 383: 728–731, 2019). By using the method of dynamical systems, we examine the bifurcation and construct exact travelling wave solutions for a (2+1) KdV equation. Exact parametric representations of all wave solutions are introduced and they are clarified graphically.

A (2+1)-dimensional generalized potential Kadomtsev-Petviashvili (gpKP) equation which possesses a Hirota bilinear form is constructed. The lump waves are derived by using a positive quadratic function solution. By combining an exponential function with a quadratic function, an interaction solution between a lump and a one-kink soliton is obtained. Furthermore, an interaction solution between a lump and a two-kink soliton is presented by mixing two exponential functions with a quadratic function. This type of lump wave just appears to a line $k_2x+k_3y+k_4t+k_5 \sim 0$. We call this kind of lump wave is a special rogue wave. Some visual figures are depicted to explain the propagation phenomena of these interaction solutions.

Based on the generalized bilinear method, diversity of exact solutions of the (3 + 1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation is successfully derived by using symbolic computation with Maple. These new solutions, named three-wave solutions and periodic wave have greatly enriched the existing literature. Via the three-dimensional images, density images and contour plots, the physical characteristics of these waves are well described. The new three-wave solutions and periodic solitary wave solutions obtained in this paper, will have a wide range of applications in the fields of physics and mechanics.

Under investigation is a generalized (3+1)-dimensional Kadomtsev-Petviashvili equation which can be used to describe nonlinear wave propagation in dissipative media. Via the bilinear transformation method, the mixed lump and soliton solutions are obtained for the equation. The asymptotic behavior of the mixed solutions are analyzed. Furthermore, the fusion and fission behaviors of the lump and soliton are observed for the first time. The lump and soliton can merge into a whole soliton over time, or, on the contrary, the soliton may differentiate into a lump and a new soliton. During the processes, the amplitude of the lump will greatly vary, while the amplitude of the soliton will change slightly.

In this paper, the new exact solutions of nonlinear conformable fractional partial differential equations(CFPDEs) are achieved by using auxiliary equation method for the nonlinear space-time fractional Klein-Gordon equation and the (2+1)-dimensional time-fractional Zoomeron equation. The technique is easily applicable which can be applied successfully to get the solutions for different types of nonlinear CFPDEs. The conformable fractional derivative(CFD) definitions are used to cope with the fractional derivatives.

We aim to generalize the (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation, passing the three-soliton test, to a new one which still has diverse solution structures. We add all second-order derivative terms to the HSI equation but demand the existence of lump solutions. Such lump solutions are formulated in terms of the coefficients, except two, in the resulting generalized HSI equation. As an illustrative example, a special completely generalized HSI equation is given, together with a lump solution, and three 3d-plots and contour plots of the lump solution are made to elucidate the characteristics of the presented lump solutions.

Under investigation in this work is a (2+1)-dimensional generalized Korteweg-de Vries equation, which can be used to describe many nonlinear phenomena in plasma physics. By using the properties of Bell's polynomial, we obtain the bilinear formalism of this equation. The expression of $ N $-soliton solution is established in terms of the Hirota's bilinear method. Based on the resulting $ N $-soliton solutions, we succinctly show its breather wave solutions. Furthermore, with the aid of the corresponding soliton solutions, the $ M $-lump solutions are well presented by taking a long wave limit. Two types of hybrid solutions are also represented in detail. Finally, some graphic analysis are provided in order to better understand the propagation characteristics of the obtained solutions.

In this paper, through symbolic computations, we obtain two exact solitary wave solitons of the (2 + l)-dimensional variable-coefficient Caudrey-Dodd- Gibbon-Kotera-Sawada equation. We study basic properties of l-periodic solitary wave solution and interactional properties of 2-periodic solitary wave solution by using asymptotic analysis.

Through symbolic computation with Maple, the $ (2+1)- $dimensional B-type Kadomtsev-Petviashvili(BKP) equation is considered. The generalized bilinear form not the Hirota bilinear method is the starting point in the computation process in this paper. The resulting lump solutions contain six free parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are arbitrary. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.

In this paper, the bifurcation theory of dynamical system is applied to study the traveling waves of the (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq (KP-Boussinesq) equation. By transforming the traveling wave system of the KP-Boussinesq equation into a dynamical system in R3, we derive various parameter conditions which guarantee the existence of its bounded and unbounded orbits. Furthermore, by calculating complicated elliptic integrals along these orbits, we obtain exact expressions of all possible traveling wave solutions of the (3+1)-dimensional KP-Boussines equation.

Fractional derivatives can be utilized as a promising tool for characterizing systems with embedded memory or describing viscoelasticity of advanced materials. Motivated by the significance of fractional derivatives, we provide assorted of analytical representations for the solution of higher-dimensional fractional differential equations that involve multi-memory indices. Then, an iterative parallel scheme of the power series method with underlying these representations is applied to extract fractal closed-form and supportive approximate solutions for several multi-memory models. Some of the obtained closed-form solutions are given in terms of the generalized exponential and hyperbolic functions which might be more suitable for representing nonlinear physical behaviors. MSC (2010): 26A33, 41A58, 35R11, 35C10, Keywords: Fractional derivatives, Fractional differential equations, Series solutions

This paper aims to study a generalized first extended (3+1)- dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \begin{document}$ ({G'}/{G})- $\end{document} expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.

In this work, we present the interaction solutions and abundant exact solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation based on the Hirota's bilinear form and a direct function. The obtained interaction solutions contain the interaction between the rational function and the $\tanh$ function and the interaction between the rational function and the $\cos$ function. The dynamical properties of these resulting solutions are analyzed and shown in three-dimensional plots, corresponding contour graphs and plane figures.

In this paper, we study on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions. To this purpose an effective method which is the sine-Gordon expansion method is used. The 2D and 3D surfaces under some suitable values of parameters are also plotted.

In this paper, the 2+1 dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation which obtained from the potential Boiti-Leon-Manna-Pempi nelli (pBLMP) equation, is introduced. Through the bilinear method and ansatz technique, the rational solutions consisting of rogue wave and lump soliton solutions are constructed, where we discuss the condition of guaranteeing the positiveness and analyticity of the lump solutions. The collection of a quadratic function with an exponential function describing rationalexponential solutions is proved, the interaction consisting of one lump and one soliton with fission and fusion phenomena. The second kind of interaction comprises the line rogue wave and soliton solution, which is inelastic. With the usage of the extended homoclinic test approach, the homoclinic breather-wave solution is derived. The characteristics of these various solutions are exhibited and illustrated graphically.

In this article, some new travelling wave solutions of the (3+1) dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation are obtained using the modified exponential function method. When the solution functions obtained are examined, it is seen that functions with periodic functions are obtained. Two and three dimensional graphs of the travelling wave solutions of the BLMP equation are drawn by selecting the appropriate parameters

The (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is studied, which describes the incompressible fluid. By virtue of an ansatz functions and the bilinear form, many entirely new complexiton solutions and periodic-soliton solutions are derived. With the aid of symbolic computation, their dynamical behaviors are demonstrated in some three-dimensional plots by choosing different values of the parameters.

In this article, meromorphic exact solutions for the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation are obtained by using the complex method. With the applications of our results, traveling wave exact solutions of the breaking soliton equation are achieved. The dynamic behaviors of exact solutions of the (2 + 1)-dimensional gCBS equation are shown by some graphs. In particular, the graphs of elliptic function solutions are comparatively rare in other literature. The idea of this study can be applied to the complex nonlinear systems of some areas of engineering.

This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding density plots.

In this paper we perform Lie group classification of a generalized coupled (2+1)-dimensional hyperbolic system, viz., \begin{document}$ u_{tt}-u_{xx}-u_{yy}+f(v) = 0,\,v_{tt}-v_{xx}-v_{yy}+g(u) = 0, $\end{document} which models many physical phenomena in nonlinear sciences. We show that the Lie group classification of the system provides us with an eleven-dimensional equivalence Lie algebra, whereas the principal Lie algebra is six-dimensional and has several possible extensions. It is further shown that several cases arise in classifying the arbitrary functions \begin{document}$ f $\end{document} and \begin{document}$ g $\end{document} , the forms of which include, amongst others, the power and exponential functions. Finally, for three cases we carry out symmetry reductions for the coupled system.

In many physical situations, variable-coefficient nonlinear partial differential equations provide us with more realistic aspects of media inhomogeneities and boundary nonuniformities than constant-coefficient. In this research, we extract some new structures of lump solutions to some important nonlinear equations with variable coefficients. The equations under consideration are the (2+1)-dimensional Burger's equation (BE) and the (2+1)-dimensional Chaffee-infante equation (CIE). The Hirota bilinear form (HBF) is the basic idea behind the novel lump solutions. By utilizing the HBF, various kind of lump solutions are derived. Additionally, numerical results are presented through the three-dimensional and contour outlooks to observe the advantages of the variable coefficients.