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This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili (KdV-KP) equation. This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation. The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric, hyperbolic, and rational solutions. The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations. Furthermore, the obtained solutions have not been reported in the previous literature and might have significant impact on future research.

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

In this paper, a generalized (3+1)-dimensional Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation which describes the fluid flow in the case of an offshore structure, is investigated. Here, making use of the bilinear form and symbolic computation, we construct four kinds of rogue wave solutions consisting of independent breathers. Among these solutions, the fourth order rogue wave solution is rarely considered in nonlinear system. Exact locations of the highest and lowest peaks as well as the extreme values of the wave heights are systematically analyzed. The obtained rogue waves observe certain “circularity structure”, the highest or lowest peaks both sit at the same circular. Moreover, we show that the rogue waves are stable during the propagation.

The potential Korteweg-de Vries equation arises in the study of water waves and is reported in the dynamics of tsunami waves. The fractional order potential Korteweg-de Vries equation is more flexible and generalized than its classical form. In this work, the modified auxiliary equation technique and residual power series method are utilized to build new exact and analytical approximate solutions of the time-fractional potential Korteweg-de Vries equation. The dynamics of the solutions obtained are explored by drawing them in two and three dimensions. Comparisons between the new results and the solutions available in literature show that the presented approaches of nonlinear problem resolution are highly effective and reliable. The obtained solutions will be helpful to understand the dynamical framework of many nonlinear physical phenomena.

Different resonance constraints enrich the behavior of soliton solutions. The soliton molecules, which are the bound states of solitons, can be set off by the velocity resonance. The lump waves, which are localized in all directions in space, are theoretically regarded as a limit form of soliton in some ways. In this paper, a (2+1)-dimensional Sharma–Tasso–Olver–Burgers (STOB) equation is investigated. Soliton (kink) molecule, half periodic kink(HPK) and HPK molecule are studied. Then the lump solution is obtained and the interactions between lump and kink molecule are discussed. The kink molecule-lump solutions exhibit a fusion phenomenon and a rogue (instanton) phenomenon, respectively.

Within the Hirota bilinear formulation, we construct N -soliton solutions and analyze the Hirota N -soliton conditions in (2+1)-dimensions. A generalized algorithm to prove the Hirota conditions is presented by comparing degrees of the multivariate polynomials derived from the Hirota function in N wave vectors, and two weight numbers are introduced for transforming the Hirota function to achieve homogeneity of the related polynomials. An application is developed for a general combined nonlinear equation, which provides a proof of existence of its N -soliton solutions. The considered model equation includes three integrable equations in (2+1)-dimensions: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, and the (2+1)-dimensional Hirota–Satsuma–Ito equation, as specific examples.

In this paper, the new travelling wave solutions of the (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff (ECBS) equation are investigated. The main aim of this work is to find the new exact solutions with the aid of relatively new ( G ′ G ′ + G + A ) -expansion method. Moreover, the physical interpretation of the nonlinear phenomena is reported through the exact solutions, which indicate the efficacy of the proposed method. Furthermore, the recovered solutions are periodic and solitary wave solutions which are presented graphically.

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.

The deformed soliton solutions of the (2+1)-dimensional nonlinear Schrodinger Maxwell–Bloch (NLS-MB) equations are investigated by the n-fold Darboux transformation method based on the seed function $$q=p=v=0,\,\eta =1$$ . New soliton solutions, including order-1 and order-2 types, are obtained and analyzed in detail. For the order-1 solutions, polynomial, trigonometric, and hyperbolic solutions are discussed. Especially, the analytical formulas of $$|q^{[1]}|$$ and soliton trajectories are obtained, which are involved by an arbitrary smooth function $$f(y+2\lambda t)$$ . For the order-2 solutions, besides the polynomial, trigonometric, and hyperbolic types, the mixed-type solutions are also derived and exhibited through several typical examples.

In this paper, we study the $$(2 + 1)$$ -dimensional variable-coefficient Kadomtsev–Petviashvili equation, which has certain applications in fluids and plasmas. Via the Kadomtsev–Petviashvili hierarchy reduction, we derive two types of the breather solutions in terms of Gramian. Based on the first type breather solutions, we observe the breathers and periodic waves, while we observe the breathers and solitons according to the second type breather solutions. Taking the long-wave limits technique for the first type breather solutions, we derive semi-rational and rational solutions. The semi-rational solutions describe the interactions between the rogue waves/lumps and breathers, while the rational solutions give birth to the rogue waves and lumps.

In this paper, we consider a generalized $$(2+1)$$ -dimensional nonlinear wave equation. Based on the bilinear method, the N-soliton solutions are obtained. The resonance Y-type soliton, which is similar to the capital letter Y in the spatial structure, and the interaction solutions between different types of resonance solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions is presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the resonance Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.

In this paper, we investigate a (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation in fluid dynamics. Based on the Hirota method, we give a bilinear auto-Backlund transformation. Via the truncated Painleve expansion, we get a Painleve-type auto-Backlund transformation. With the aid of the symbolic computation, we derive some one- and two-kink soliton solutions. We present the oblique and parallel elastic interactions between the two-kink solitons. Via the extended homoclinic test technique, we construct some breather-wave solutions. Besides, we derive some lump solutions with the periods of the breather-wave solutions to the infinity. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. Based on the polynomial-expansion method, travelling-wave solutions are constructed.

In this paper, nonlocal residual symmetry of a generalized (2+1)-dimensional Korteweg–de Vries equation is derived with the aid of truncated Painleve expansion. Three kinds of non-auto and auto Backlund transformations are established. The nonlocal symmetry is localized to a Lie point symmetry of a prolonged system by introducing auxiliary dependent variables. The linear superposed multiple residual symmetries are presented, which give rise to the n th Backlund transformation. The consistent Riccati expansion method is employed to derive a Backlund transformation. Furthermore, the soliton solutions, fusion-type N -solitary wave solutions and soliton–cnoidal wave solutions are gained through Backlund transformations.

Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of nonlinear waves. Based on a special Cole–Hopf transformation and Hirota bilinear method, the bilinear form of the equation is obtained and this form has never been given. High-order breather solutions, lump solutions and mixed solutions are obtained by using complex conjugate parameters and long-wave limit method. Then, the influence of the coefficient $$g_{t}(t)$$ of the bilinear equation on the interaction of these solutions is analyzed by means of images. It can be found that $$g_{t}(t)$$ changes the interaction of the solutions by influencing the positions and trajectories of higher-order breather solutions, lump solutions and mixed solutions. We find that different values of g(t) make the interaction of solutions different. Finally, the mixed solution of the equation including a breather wave and a line rogue wave is obtained by using the test function, and its dynamic properties are illustrated by means of images.

In this paper, a new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation is developed. Its integrability is verified by the Painleve analysis. The bilinear form, multiple-soliton, breather and lump solutions are obtained via using the Hirota bilinear method, a symbolic computation scheme. Furthermore, the abundant dynamical behaviors for these solutions are discovered. It is interesting that there are splitting and fusing phenomena when the lump waves interact. The results can well simulate complex waves and their interaction dynamics in fluids.

In this paper, we investigate a (3+1)-dimensional Hirota–Satsuma–Ito-like system for the shallow water waves. We obtain a Painlev $$\acute{\mathrm{e}}$$ integrable condition of the system. By virtue of the truncated Painlev $$\acute{\mathrm{e}}$$ expansion, we get an auto-Backlund transformation under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. Based on the bilinear form, we give a bilinear auto-Backlund transformation and a Lax pair under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. We obtain that a breather and kink waves propagate under certain Painlev $$\acute{\mathrm{e}}$$ integrable condition. The breather has a peak and a trough and the height of the kink wave periodically increases or decreases during the propagation. Furthermore, we get the lump-periodic-wave and solitary-wave solutions and observe that the lump-periodic and solitary waves propagate under certain Painlev $$\acute{\mathrm{e}}$$ integrable conditions. During the propagation, the heights of the lump-periodic waves keep unchanged and height of the solitary wave periodically increases or decreases.

In this paper, based on the bilinear form, we give multi-solitary wave solutions of the $$(4+1)$$ -dimensional Fokas equation. From the obtained multi-solitary wave solutions, with special parameters, we derive resonant solutions of N-solitary waves. For complex conjugate parameters, we analyze interactions of two periodic waves, interactions of a solitary wave and a periodic wave by making use of their phase shifts. Particularly, the intermediate processes of elastic interactions are analyzed in detail, and interesting fusional and fissionable phenomena are found. The asymptotic interactional behaviors for these solutions are analyzed theoretically and illustrated graphically.

This paper investigates some novel exact analytical solutions, including lump-type wave, breather wave as well as BK-type wave solutions for a (3+1)-dimensional p ¯ -gKP equation. Firstly, the trilinear p ¯ -gKP equation is well established by the multivariate trilinear operators. Secondly, three types of wave solutions for the p ¯ -gKP equation are obtained by utilizing the trilinear method. A kind of new interaction solutions among the breather wave and the kink-type waves are also generated, i.e. the so called BK-type waves, which enriches the types of solutions of PDEs. Through the way of resetting different space constants, we adjust the coordinates of kink-type waves in order for them colliding with the breather wave, after that, the transformed kink-type waves gradually swallow the breather wave. Finally, the solitary wave solutions for the (3+1)-dimensional gKP equation are successfully constructed through using a perturbation method.

In this work, we study the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation. We employ the extended tanh function method and the simple equation method to achieve analytical soliton solutions. Moreover, numerical treatment for this equation is introduced by the finite difference method. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs.

New wave excitations are revealed for a (3 + 1)-dimensional nonlinear evolution equation to enrich nonlinear wave patterns in nonlinear systems. Based on a new variable separation solution with two arbitrary variable separated functions obtained by means of the multilinear variable separation approach, localized excitations of N dromions, $$N\times M$$ lump lattice and ring soliton lattice are constructed. In addition, it is observed that soliton molecules can be composed of diverse “atoms” such as the dromions, lumps and ring solitons, respectively. Elastic interactions between solitons and soliton molecules are graphically demonstrated.

In this paper, an efficient $$(G^{\prime}/G,\,1/G)$$ -expansion method is adopted to resolve a famous (2 + 1)-dimensional fractional Ablowitz–Kaup–Newell–Segur (AKNS) water wave equation for the non-conservative system that plays a significant role in understanding the wave propagation. This work addresses the physical and dynamic behavior of some new exact trigonometric, hyperbolic, and rational solitary wave solutions in the form of 3D-plots and contour plots using different measures of parameters. The obtained results show the efficiency of the proposed method for the analytical treatment of nonlinear problems in mathematics, science and engineering and may be helpful in better understanding the propagating wave dynamics in diverse situations.

We study the transformed nonlinear waves of the (2+1)-dimensional generalized breaking soliton (gBS) equation by analyzing characteristic lines. The N-soliton solution of the gBS equation is obtained by virtue of the Hirota bilinear method, from which the 1-order and 2-order breather wave solutions of the gBS equation are derived by the complexification method. Then, we obtain the condition of the breather wave transformation analytically. Under the condition that the two characteristic lines of the 1-order breather wave are parallel to each other, we show that the 1-order breather wave can be converted into many other types of nonlinear waves, such as M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, quasi-periodic soliton, etc. Moreover, we give four deformation modes of the 2-order breather wave, including intersection mode of a transformed wave and a breather wave; parallel mode of a transformed wave and a breather wave; intersection mode of two transformed waves; parallel mode of two transformed waves. Finally, we present the graphical analysis of the resulting solutions in order to better understand their dynamical behaviors.

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 meth...

To construct a class of new multiwave interaction solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation, we calculate different types of interaction solutions among solitons, periodic waves and rational waves using the direct algebraic method together with the inheritance solving skill. Moreover, a new algorithm is proposed with the aid of the simplified Hirota method, the conjugated parameters assignment and long wave limit strategies, from which multiwave interaction solutions among solitons, breathers and lump waves are generated.

The 2-D generalized variable-coefficient Kadomtsev-Petviashvili-Burgers equation representing many types of acoustic waves in cosmic and/or laboratory dusty plasmas is reduced by the modified classical direct similarity reduction method to nonlinear ordinary differential equation of fourth-order. Using the extended Riccati equation mapping method for solving the reduced equation, many new shock wave, solitary wave and periodic wave solutions are obtained with some constraints between the variable coefficients. Finally, some physical interpretations for the obtained solutions as, bright and dark solitons, periodic solitary wave, and shock wave in dust plasma and quantum plasma are achieved.

In this work, the soliton molecules, three soliton molecules and four soliton molecules are obtained for the new (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation by the velocity resonance mechanism which is a new resonance condition and the N-soliton solution. The interactions between these kinds of soliton molecules are studied analytically and numerically by some figures with selecting suitable parameters. The other kinds of waves such as dark-singular, periodic-singular and plane-wave solutions are investigated by e x p ( − Φ ( ζ ) ) expansion method for this model to understand the physical properties for this model. Also, the dynamics for these waves are presented by some graphics.

This article studies a (2+1)–dimensional first extended Calogero-Bogoyavlenskii-Schiff equation, which was recently introduced in the literature. We derive Lie symmetries of this equation and then use them to perform symmetry reductions. Using translation symmetries, a fourth-order ordinary differential equation is obtained which is then solved with the aid of Kudryashov and $ (G'/G)- $expansion techniques to construct closed-form solutions. Besides, we depict the solutions with the appropriate graphical representations. Moreover, conserved vectors of this equation are computed by engaging the multiplier approach as well as Noether's theorem.

In this paper, we study the (3+1)-dimensional variable-coefficient Kudryashov–Sinelshchikov (vc-KS) equation, which characterizes the evolution of nonautonomous nonlinear waves in bubbly liquids. The nonautonomous lump solutions of the vc-KS equation are produced via the Hirota bilinear technique. The characteristics of trajectory and velocity of this wave are analyzed with variable dispersion coefficients. Based on the positive quadratic function assumption, we further discuss two types of interactions between the soliton and lump under the periodic and exponential modulations. Then, we give the breathing lump waves showing the periodic oscillation behavior. Finally, we obtain the second-order nonautonomous lump solution, which also shows periodic interactions if we select trigonometric functions as the dispersion coefficients.

In the present paper, taking advantage of the bilinear method, we attained N-soliton solutions of the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (gCH-KP) equation. By combining the long wave limit method and the complex conjugate method to N-soliton solutions, M-lump wave solutions including one-lump wave, two-lump wave and three-lump wave and two types of hybrid solutions including hybrid solution between lump wave and solitons and between M-lump wave and soliton were obtained. In addition, several groups of images exhibited their dynamic structure and physical properties with physically interpreted via symbolic computation. Finally, resonant multi-soliton wave solutions were obtained by carrying out the linear superposition principle and the progresses of wave motions with physical interpretation represented graphically. Furthermore, the received results have immensely augmented the exact solutions of (3+1)-dimensional gCH-KP equation on the available literature and enabled us to understand the nonlinear dynamic system deeply. In addition, the proposed methods will be a strong boost to the calculation method of nonlinear equations.

The (2+1)-dimensional generalized coupled nonlinear Schrodinger equation with the four-wave mixing (FWM) term is studied in this paper, which describes the optical solitons in a birefringent fiber. By virtue of the Hirota method, the one- and two-soliton solutions are derived. On the basis of solutions obtained, we discuss how the values of the FWM and some free parameters affect the solitons’ peformance. The FWM parameter can help to control the amplitude of the solitons. Meanwhile, by setting the values of certain free parameters, we can control the solitons’ propagation direction and speed, and reduce the interactions between them as well. In addition, the energy transfer of solitons during elastic collision and separation is also discussed. The conclusions here may be useful in improving the communication quality in multi-mode fibers.

In oceanography, acoustics and hydrodynamics, people pay attention to the Burgers-type equations for different wave processes, one of which is an extended coupled (2+1)-dimensional Burgers system hereby under investigation. Based on the scaling transformation, Bell polynomials, Hirota operators and symbolic computation, we structure out two hetero-Backlund transformations, each of which to a solvable linear partial differential equation, and construct two sets of the bilinear forms, with the relevant one- and two-soliton solutions. Results rely on the coefficients in the original system.

In this work, we study a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium. Bilinear form and N-soliton solutions are derived, where N is a positive integer. The higher-order breather and lump solutions are constructed based on the N-soliton solutions. Hybrid solutions comprising the solitons and breathers, breathers and lumps, as well as solitons and lumps are worked out. Amplitudes and velocities of the one solitons as well as periods of the first-order breathers are investigated. Amplitudes of the first-order lumps reach the maximum and minimum values at certain points given in the paper. Interactions between any two of those waves are discussed graphically.

Based on the bilinear method, rational lump and mixed lump-solitary wave solutions to an extended (2+1)-dimensional KdV equation are constructed through the different assumptions of the auxiliary function in the trilinear form. It is found that the rational lump decays algebraically in all directions in the space plane and its amplitude possesses one maximum and two minima. One kind of the mixed solution describes the interaction between one lump and one line solitary wave, which exhibits fission and fusion phenomena under the different parameters. The other kind of the mixed solution shows one lump interacting with two paralleled line solitary waves, in which the evolution of the lump gives rise to a two-dimensional rogue wave. This shows that these three interesting phenomena exist in the corresponding physical model.

In this work, the Date–Jimbo–Kashiwara–Miwa (DJKM) equation include time-dependent in (2+1)-dimensions that characterize the propagation of nonlinear dispersive waves in inhomogeneous media is stud...

The physics of the (4 + 1)-dimensional Fokas equation follows necessarily from the physical nature of the Kadomtsev–Petviashvili and Davey-Stewartson equations in wave theory. In this article, we consider the nonlinear ( 4 + 1 ) -dimensional Fokas partial differential equation. Two recognizable methods namely improved F-expansion and generalized exp ( - ϕ ( ξ ) ) -expansion methods are proposed to investigate the new wave structures of (4 + 1)-dimensional Fokas dynamical model which have never been constructed before. As a result, new solutions are in different solitons such as dark soliton, bright soliton, combined dark-bright soliton solutions, periodic, and solitary wave solutions. The new generalized solitary wave solutions can be constructed by assigning the specific values to those parameters involved in these methods. The achieved results are compared with existing results in the literature and we found that some of results are not available in literature. The derived results are explained graphically, to understand the phenomena of the proposed model. The constructed results rendering that the consider methods in this article are simple, effective, and easy to find the solution of many other nonlinear higher-dimensional models which arise in several areas of science and engineering.

By applying the Lie symmetry method, abundant group-invariant solutions are constructed for a (3+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation, which provides a water wave model for long waves of small amplitude with weakly non-linear restoring forces and frequency dispersion. Infinitesimal generators of symmetries, the commutation table of the symmetry Lie algebra, and an optimal system of one-dimensional Lie symmetry sub-algebras are presented. The governing gKP equation is reduced into several nonlinear ordinary differential equations utilizing thrice symmetry reductions. Consequently, abundant group invariant solutions are obtained in the shapes of different dynamical wave structures of solitons, multi-solitons, W-shaped solitons, doubly solitons, kink-type solitons, lump-type solitons interaction between parabolic waves and lump solitons, and annihilation multi-solitons profiles. The physical interpretation of the resulting soliton solutions is illustrated by three dimensional graphical through numerical simulation. The obtained group-invariant solutions involve many arbitrary functions, thereby exhibiting rich physical structures and including the existing solutions in the literature.

The effect of three wave mixing in the (2+1)-dimensional Ito equation is investigated by the Hirota bilinear method and the long wave limit approach. Among the mixed solutions, there are the solito...

In this paper, a generalized (2 + 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation is considered. Based on the Hirota bilinear method, three kinds of exact solutions, soliton solution, breather solutions, and lump solutions, are obtained. Breathers can be obtained by choosing suitable parameters on the 2-soliton solution, and lump solutions are constructed via the long wave limit method. Figures are given out to reveal the dynamic characteristics on the presented solutions. Results obtained in this work may be conducive to understanding the propagation of localized waves.

In this article, a new solution for the dimensions Boiti–Leon–Manna–Pempinelli (BLMP) equation using the sine Gordon expansion method and the extended tanh function method is given. The methods were chosen very carefully due to the precision of their solutions. Also, some of the two- and three-dimensional figures and some of the contours plots of the obtained solutions were presented. Finally, a discussion of the results was given.

In this study, based on the Hirota bilinear form, the exact analytic solutions of the (3 + 1) dimensional Vakhnenko–Parkes equation with various physical properties were constructed with the help of the Maple package program and symbolic computation. These solutions are the type of multi-waves, breather wave, lump–kink, lump–periodic solutions and interaction solutions (between lump and hyperbolic wave solutions). The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. By means of Maple package program, 3D and 2D graphs were drawn for the special choices of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in the propagation of high-frequency waves in a relaxing medium.

Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

On the basis of the binary Bell polynomial scheme, the bilinear form of the extended (3+1)-dimensional Jimbo-Miwa (JM) equation, is constructed. Then, a class of new type rogue waves solutions to the extended (3+1)-dimensional JM equation, is found. It mainly includes the lump solutions, lumpoff solutions and instanton solutions. Their nonlinear evolutionary processes by 3D- and 2D-graphs, are shown. Finally, a direct method which is called the tanh-function method was used to get solitary waves of this considered model. These results can help us better understand interesting physical phenomena and mechanism.

This paper deals with localized waves in the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation in the incompressible fluid. Based on Hirota's bilinear method, N-soliton solutions related to CDGKS equation are constructed. Taking the special reduction, the exact expression of multiple localized wave solutions comprising lump soliton(s) are obtained by using the long wave limit method. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear localized waves.

In this paper, we investigate some interesting solution structures of the (2+1)-dimensional bidirectional Sawada-Kotera (bSK) equation. We obtain general high-order breather solutions by utilizing the Hirota's bilinear method united with the perturbation expansion technique. Taking a long-wave limit of the obtained breather solutions and then making particular parameter constraints, smooth rational solutions are generated, which include high-order lumps and mixed solutions consisting of lumps and stripe. In order to easily explore the dynamical behaviors, some plots are presented to analyse these solutions.

In this paper, the nonlinear conformable time-fractional (3 + 1)- dimensional modified KdV-Zakharov-Kuznetsov equation is being explored using three well-established integration schemes named as: the expζ function method, the hyperbolic function and modified Kudryashov schemes. In returns, many new exact solitary wave solutions, including rational, dark, singular and combined dark-singular solitons, are obtained and have been compared with those given in the literature. Moreover, the obtained solutions are demonstrated by 2D and 3D graphs for suitable values of the parameters to observe the dynamical behavior of the secured solutions.

In this paper, outcomes of the study on the Backlund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Backlund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

In this paper, the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation is investigated to acquire the complexiton solutions by the Hirota direct method. It is essential to transform the equation into Hirota bi-linear form and to build N-compilexiton solutions by pairs of conjugate wave variables.

In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.

It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.