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In this paper, we investigate the exact solutions of the (3+1)-dimensional pKP-BKP equation for describing certain nonlinear fluctuation phenomena in nonlinear optics, plasma physics and hydrodynamics. By imposing novel constraints on the N-soliton solution of this equation, we derive Y-type soliton solutions. Subsequently, through the application of complex conjugation techniques, these N-soliton solutions are converted into P-order breathing wave solutions. Furthermore, the interaction solutions are obtained by combining the Y-type soliton solutions with the breathing wave solutions during the partial degeneracy process of the N-soliton solutions. Finally, we select specific parameters to demonstrate the dynamic behaviors of the obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, we show the interaction phenomena about the first-order lump wave and multiple solitons by four theorems. The interaction phenomena are divided into two types, one type is absorbing collision phenomenon and the other type is elastic collision phenomenon. Six sets of dynamic plots over time are showed the phenomenon of absorbing interactions with energy transfer process, the phenomenon of elastic interactions with restoration of the initial state after collision. We apply the four theorems to an equation which can characterize the Rossby waves' amplitude, this equation can be used to describe the resonance phenomenon of Rossby waves. Different test functions with the same parameter values can present different wave-wave interaction properties. This study contributes to in-depth investigation of the mechanism of lump wave generation, evolution and interaction. [ABSTRACT FROM AUTHOR]

This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear β -fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of β -fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we show the interaction phenomena about the first-order lump wave and multiple solitons by four theorems. The interaction phenomena are divided into two types, one type is absorbing collision phenomenon and the other type is elastic collision phenomenon. Six sets of dynamic plots over time are showed the phenomenon of absorbing interactions with energy transfer process, the phenomenon of elastic interactions with restoration of the initial state after collision. We apply the four theorems to an equation which can characterize the Rossby waves’ amplitude, this equation can be used to describe the resonance phenomenon of Rossby waves. Different test functions with the same parameter values can present different wave-wave interaction properties. This study contributes to in-depth investigation of the mechanism of lump wave generation, evolution and interaction.

Abstract This research work focuses on investigating the propagation of ultrasonic waves, which propagate mechanical vibrations of molecules or particles inside materials. Ultrasound imaging is extensively used and deeply rooted in the medical field. The key technologies that form the basis for many different uses in the area include transducers, contrast agents, pulse compression, beam shaping, tissue harmonic imaging, techniques for measuring blood flow and tissue motion, and three-dimensional imaging. The third-order non-linear $$\beta$$ β -fractional Westervelt model has been used as a governing model in the imaging process for securing the different wave structures. The exact solutions of different types, including mixed, dark, singular, bright-dark, bright, complex and combined solitons are extracted. These solutions are obtained by using two newly introduced techniques, namely modified generalized Riccati equation mapping method and modified generalized exponential rational function method. Moreover, breather, lump and other waves are extracted by the assistance of logarithmic transformation and different test functions. The used methodologies are extremely effective and possess substantial computing capacity to effectively address the different solutions with a high level of accuracy in these systems. The techniques used are well-known for being effective, simple, and flexible enough to integrate multiple soliton systems into a unified framework. In addition, we provide 2D and 3D graphs that explain the behavior of the solution at various parameter values, under the influence of $$\beta$$ β -fractional derivatives. The results offered in this study may improve the comprehension of the nonlinear dynamic behavior of the specific system and confirm the efficacy of the approaches used. We expect that our approaches will be beneficial for a wide range of nonlinear models and other problems in the related fields.

We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order γ on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. [ABSTRACT FROM AUTHOR]

The main orientation of the current research is to look into the generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation (BKPE) for the water waves. By exerting the Cole-Hopf transform, we extract its Hirota bilinear equation. First, the weight algorithm (WA) together with the linear superposition principle(LSP) is carried out to look for the resonant multiple soliton solutions (RMSSs). Two different types of the RMSSs are obtained by introducing the parameterization of the wave numbers and frequencies. Second, the N-soliton solutions (NSSs) are also explored by using Hirota bilinear equation. On this basis, the resonance conditions of the soliton molecules on the (x, y)-, (x, z)- and (y, z)-planes are extracted and the soliton molecules are found. Finally, the ansatz function scheme, together with the symbolic computation is manipulated to look into the interaction solutions (ISs). Two different interaction solutions of the sin-cosh type and cos-cosh type are developed. A comparison between the RMSSs and the N-soliton solutions are elaborated in detail. Additionally, the dynamics of the solutions are displayed graphically to expound the physical interpretation. The proposed methods in this work can be also employed to inquire into the similar exact solutions of the other PDEs. [ABSTRACT FROM AUTHOR]

This paper initiates an exploration into the exact solutions of the variable coefficient Date-Jimbo-Kashiwara-Miwa equation, first utilizing the Painlevé analysis method to discuss the integrability of the equation. Subsequently, By employing the Hirota bilinear method, N-soliton solutions for the equation are constructed. The application of the Long wave limit method to these N-soliton solutions yields rational and semirational solutions. Various types of localized waves, encompassing solitons, lumps, breather waves, and others, emerge through the careful selection of specific parameters. By analyzing the image of the solutions, the evolution process and its dynamical behavior are studied. [ABSTRACT FROM AUTHOR]

In this article, we focuss primarily on a ( 3 + 1 )-dimensional nonlinear equation in fluids and the dynamic behaviour of the lump solution and new interaction solutions of the equation. First, the bilinear form of the equation is investigated using the Bell polynomials and the Hirota bilinear method. The Bell-polynomial-type Bäcklund transformation of the equation and the corresponding Lax pair are obtained by using the transformation technique. Based on the obtained bilinear form, we use the positive quadratic function method to obtain the lump solutions of the equation and study its dynamic behaviour in detail. On the other hand, we constructed two types of interaction solutions, lump-soliton and lump-periodic, and combined the images to vividly show the interaction phenomenon. Furthermore, the homoclinic breather solutions of the equation are constructed using the homoclinic test method. Finally, we hope that the solutions we obtained can explain some nonlinear phenomena in the fluid mechanics and shallow water wave fields. [ABSTRACT FROM AUTHOR]

Three types of solutions (namely, localized wave solutions and their two distinct types of interaction solutions) to a (3+1)-dimensional nonlinear evolution equation were acquired based on the Hirota bilinear method. The basic steps commenced with obtaining the Hirota bilinear form of the original equation, then solving it by using mathematical software, and ultimately the solutions to the original equation were derived by the bilinear transformation. The solutions of the Hirota bilinear equation were assumed as the superposition of arbitrary number of functions. For the first type of localized wave solutions, the solutions of the corresponding bilinear equation were the superposition of n quadratic positive functions. The second type of mixed solutions which reflected the interaction between localized waves and multi-kink waves, consisted of n positive quadratic function, n 1 exponential function and n 2 hyperbolic cosine functions. The last type solutions were the interaction solutions between the localized wave solutions and the k-kink ( k = 1 , 2 ) wave solutions. All three types of solutions contained localized wave solutions which are not homogeneous. Finally, an analysis of the dynamic properties of each type of solutions was conducted by introducing specific parameter values and generating plots. [ABSTRACT FROM AUTHOR]

The (3+1)-dimensional BKP-Boussinesq equation is widely used to describe and understand nonlinear wave phenomena. In this article, a single hidden layer neural network model is constructed using the bilinear neural network method to obtain lump solutions, interaction solutions, and breather solutions of the equation. Based on the single-layer model, a '4-2-2-1' neural network model was constructed. By assigning different weight coefficients and thresholds, interference wave solutions and periodic solutions of the equations are obtained. During the solving process, some weight coefficients being zero may lead to the degeneration of the bilinear neural network model, and this phenomenon can be mitigated by appropriately enhancing the model's performance. Furthermore, the study shows that due to the universal approximation property of neural networks, the bilinear neural network method offers a more flexible and simpler way to solve nonlinear problems. It can yield a greater number of novel analytical solutions and promote the development of the generality of solving nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order γ on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations.

In this paper, we mainly concentrate on the exact solutions of the (3 + 1) -dimensional Ito equation for describing certain nonlinear phenomena in fluid dynamics and plasma, including the breathing solutions, the lump solutions, the Y-type soliton solutions, and the interaction solutions between the breathing wave and the Y-type soliton. Particularly, the Y-type soliton solutions and the interaction solutions have been the focus of scholars' attention recently. Firstly, the single and double breathing wave solutions are given via the three-wave approach. Applying the parametric limit method, the breathing wave solutions are degenerated into lump solutions. Then, the Y-type soliton solutions are constructed based on the N-soliton solutions, which are novel soliton solutions. Next, employing the parametric complex conjugation technique, the N-soliton solutions are transformed into P-breathing wave solutions. Finally, the interaction solutions between breathing waves and Y-type solitons are investigated by the partial degeneration of Y-type soliton solutions. The corresponding visualization graphs exhibit the dynamic behavior of the solutions. [ABSTRACT FROM AUTHOR]

The Kadomtsev–Petviashvili (KP) equation is widely used to model nonlinear wave phenomena occurring in plasmas and fluids, such as water waves with weak nonlinear restoring forces, and shock waves propagation phenomena in plasma. In this paper, we investigate a new extended ( 3 + 1 )-dimensional KP equation. Based on the Hirota bilinear method, the N-soliton solutions for the equation are derived. By introducing a new constraint to the N-soliton solutions, namely considering different dispersion coefficients to zero, the fission and fusion phenomena in multi solitons have been discovered. Subsequently, the M-lump wave and rogue wave solutions were derived using the long wave limit, and systematically analyzed the dynamics of the lump waves through numerical simulations. Additionally, based on the method mentioned earlier, combined with the complex conjugate construction method, we investigated some novel interaction phenomena such as lump waves with fusion (fission) waves, fusion (fission) waves with breather waves, line rogue waves with lump waves, and so on. These results may provide some explanation for the fission or fusion phenomena occurring in nonlinear models, as well as the complex nonlinear superposition between waves in fluid systems. [ABSTRACT FROM AUTHOR]

This paper focuses on some novel exact solutions of the (3 + 1)-dimensional Kudryashov-Sinelshchikov equation (KSE). Based on the linear superposition principle (LSP) and weight algorithm (WA), we construct the complexiton solutions by introducing the pairs of the conjugate parameters. Besides, the Y-type soliton solutions are extracted by imposing the resonant conditions on the N-soliton solutions. Finally, two different types of the interaction solutions (ISs) are also explored by exerting the test function method and symbolic computation. The graphical representations of the corresponding solutions are presented to reveal the dynamical characteristics. To the best of our knowledge, the attained outcomes of this article are brand new and have never been reported in other literature, and can enable us make sense of the physical behaviors of the (3 + 1)-dimensional KSE better.

Under investigation in this paper is the dynamics of localized waves for the higher-order nonlinear Schrödinger equation with self-steepening and cubic–quintic nonlinear terms, which describes the propagation of ultrashort pulses in optical fibers. Firstly, based on Lax pair, the Nth-fold Darboux transformation is constructed. Secondly, the N-soliton solutions are obtained and the interactions of solitons are analyzed graphically. Moreover, the Akhmediev breather, space-time period breather and line breather are derived. The interactions of two breathers are shown and discussed. In addition, the first-order usual rogue wave and line rogue wave are given and investigated. And the different structures of second- and third-order rogue wave are observed. Finally, the interaction solutions between rogue wave and one-breather are constructed. These results in the present work could be used to understand related physical phenomena in nonlinear optics and relevant fields. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to construct the breather and interaction solutions for a (3 + 1) -dimensional generalized shallow water wave equation. Such equation has been utilized to describe the long water waves in an ocean, impoundment or estuary, and used in the tsunami predictions, river/tidal-wave/irrigation flow studies, as well as weather simulations. Via the Kadomtsev–Petviashvili hierarchy reduction method and taking the long-wave limit technique, we derive the breather and interaction solutions in terms of the Gramian, which have never been investigated before. Each interval of the breather possesses one peak and one valley, and the interaction between two breathers is elastic. When we take the long-wave limit in the breather solutions, interaction solutions are demonstrated. Furthermore, we investigate graphically the dynamical behaviors of interaction solutions and find that the interaction between lump and breather is also elastic. [ABSTRACT FROM AUTHOR]

In this work, a generalized ( 2 + 1 )-dimensional variable-coefficient shallow water wave equation is investigated, which describes the interaction between Riemann waves propagating along the y-axis and long waves propagating along the x-axis in a fluid. Based on the Hirota bilinear form and three-wave method, the breather wave solution is obtained. The interaction solutions between lump wave and periodic wave are presented. The interaction solutions between lump wave and solitary waves are also studied. None of these obtained solutions have been found in other literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with the new extended (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE), which is regarded as an extension of the well-known (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation and acts a key role in the incompressible fluid. By means of the Cole-Hopf transform, the Hirota bilinear equation of the studied equation is developed. Then based on the Hirota bilinear equation, we give a detailed investigation into the new exact solutions of the BLMPE. Firstly, the linear superposition principle combined with the weight algorithm is utilized to construct the resonant multiple wave solutions by introducing the parameterization for the wave numbers and frequencies. Secondly, the new homoclinic approach is applied to seek for the periodic wave solutions. Finally, the symbolic computation with the ansatz function scheme is adopted to construct the interaction solutions. The 3-D plots and 2-D contours are figured out to reveal the nonlinear dynamic behaviors of the different solutions. To the best of our knowledge, the results obtained in this work are all new and have not been reported elsewhere yet. The attained solutions can help us understand the nonlinear dynamic behaviors of the new extended (3 + 1)-dimensional BLMPE better. What's more, the proposed methods in this work can be also adopted to study the resonant multiple wave, periodic wave and interaction solutions of the other PDEs arising in physics. [ABSTRACT FROM AUTHOR]

After introducing an auxiliary variable, the (4 + 1) -dimensional variable-coefficient Fokas (4D vc-Fokas) equation is rewritten and its bilinear form is constructed by using the Bell polynomial. Based on the bilinear form, the bilinear B a ¨ cklund transformation is obtained and some solitary wave solutions are derived. In addition, some new lump solutions and interaction solutions are given, and the special solutions influenced by the parameter choice are analyzed. [ABSTRACT FROM AUTHOR]

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Constructing solutions to the extended (2+1)-dimensional Kadomtsev–Petviashvili equation is the major goal of this article, which includes resonance solutions, rational solutions, breather solutions, and various hybrid solutions. The main approaches are conjugate complex method, long-wave limit method, and introducing new restrictive conditions. We are able to better explain the complex dynamical behavior by plotting the images of the solutions (fusion, fission, and collision phenomena, trajectory changes, etc.) and facilitate to help in the study of fluid mechanics. [ABSTRACT FROM AUTHOR]

Variable coefficients (3+1)-generalized shallow water wave equation (GSWE) is investigated via modified Hirota bilinear method. This method is presented for the first time. Compared with other methods, it solves solution without setting solution and calculates transformations without making logarithmic transformations. The rational transformation is first utilized to transform GSWE. According to homogeneous balance principle, the relation between F and G in rational transformation can be calculated by utilizing. Solutions that included rogue wave solutions, interaction solutions, breather solutions and so on, are obtained and depicted graphically. Figures are given out to the dynamic characteristics of the solution. Furthermore, the results obtained demonstrate that this approach is more direct, generalized, effective and holds for many nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this research article, we study a new (3+1)-dimensional Hirota bilinear equation which can describe the dynamics of ion-acoustic wave and Alvin wave of small but finite amplitude in plasma physics and describe the propagation process of nonlinear waves in shallow water. First, we apply two methods to study the equation, namely the Hirota bilinear method and long-wave limit method M-lump solution, and line rogue waves are reported. Furthermore, we investigate the velocity, propagation trajectory, and interaction phenomenon of M-lump solution(M=2,3). Then, based on the multi-solitons, two cases of high-order breather solution are constructed by selecting some special parameters. Finally, four types interaction solutions are successfully obtained by employing long-wave limit method and selecting some special parameters. More importantly, we explore physical collision phenomenon of the interaction between nonlinear waves. In order to better illustrate the characteristics of the interaction solutions, the results are shown in three-dimensional plots and numerical simulation. To our knowledge, all of the obtained solutions in this article are novel. The results of this article may be provide an important theoretical basis for explaining some nonlinear phenomena in the field of fluid mechanics and shallow water. [ABSTRACT FROM AUTHOR]

The purpose of this work is the examine the three-waves, double exponential, homoclinic breather, and periodic cross kink-soliton solutions for a third-order fractional nonlinear Schrödinger equation (3-FNLS). This model which is vital in fiber optics, explains the propagation of light in optical fibers when ultra-short pulses are produced. We define the model in terms of the conformable time fractional derivative operator so that the model could give a better description of the physical aspect. By picking the activation function in Hirota bilinear form, as the hyperbolic, trigonometric and exponential function along with suitable dispose of parameters, we acquire the three-waves, double exponential, homoclinic breather, and periodic cross kink-soliton auspiciously. Additionally, we have elucidated some three-dimensional and contour portraits to foresee the wave dynamics. These acquired new solutions that do not exist in the literature include some free constants and thereby can be significant to describe variety in qualitative aspects of wave phenomena. At last, we also check the stability of the governing model. By executing the modulation instability analysis, we scrutinize the stability analysis of the yielded exact solutions and the movement role of the waves. [ABSTRACT FROM AUTHOR]

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

With the support of bilinear method, specific N-soliton solutions of a ( 3 + 1 )-dimensional nonlinear model are proposed, and various solutions are constructed through the different degeneration processes of N-soliton solutions. On the basis of defining a novel constraint condition on N-soliton solutions, resonance Y-type solitons are found out. Under complex conjugating method (complexification method), N-soliton solutions could convert to high-order breathers (T-order breathers). Resorting to long wave limit technique, high-order lumps (M-order lumps) are obtained. During the partial degeneration of N-soliton solutions, combining complexification method and long wave limit technique, abundant interaction solutions composed of soliton, breather and lump are yielded. The corresponding numerical simulations are presented to demonstrate the dynamic behaviors of the obtained solutions. [ABSTRACT FROM AUTHOR]

The lump solutions and interaction solutions are mainly investigated for the (2 + 1)-dimensional KPI equation. According to relations of the undetermined parameters of the test functions, the N-soliton solutions are showed by computations of the Maple using the Hirota bilinear form for (2 + 1)-dimensional KPI equation. One type of the lump solutions for (2 + 1)-dimensional KPI equation has been deduced by the limit method of the N-soliton solutions. In addition, the interaction solutions between the lump and N-soliton solutions of it are studied by the undetermined interaction functions. The sufficient conditions for the existence of the interaction solutions are obtained. Furthermore, the new breather solutions for the (2 + 1)-dimensional KPI equation are considered by the homoclinic test method via new test functions including more parameters than common test functions. [ABSTRACT FROM AUTHOR]

Eq. (2) is equivalent to HT ht Graph (4) The organization of this paper is as follows. [Extracted from the article]

In this paper, the solution in the form of Grammian of the Kadomtsev–Petviashvili I equation is employed to investigate a new type of multiple-lump solution. The bound state called lump molecules appears in a period of time. A generalized form of the N-lump solutions in N-order determinant possessing the structure of lump molecules is explicitly given. Utilizing the nonzero constant matrices and a non-homogeneous polynomial, the interaction solutions between the multiple-lump waves and the line solitons are constructed. The interactions among the N-lumps and between lumps and solitons are investigated with the aid of numerical simulation. The results extend the understanding of the multiple lumps and interaction dynamical behaviors of the Kadomtsev–Petviashvili I equation. [ABSTRACT FROM AUTHOR]

Abstract In this study, we investigate a new fourth-order integrable nonlinear equation. Firstly, by means of the efficient Hirota bilinear approach, we establish novel types of solutions which include breather, rogue, and three-wave solutions. Secondly, with the aid of Lie symmetry method, we report the invariance properties of the studied equation such as the group of transformations, commutator and adjoint representation tables. A differential substitution is found by nonlinear self-adjointness (NSA) and thereafter the associated conservation laws are established. We show some dynamical characteristics of the obtained solutions through via the 3-dimensional and contour graphs.

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

Fission and fusion are important phenomena, which have been observed experimentally in many physical areas. In this paper, we study the above two phenomena in the ( 2 + 1 )-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. By introducing some new constraint conditions to its N-solitons, the fission and fusion are obtained. Numerical figures show that the two types of solutions look like the capital letter Y in spatial structures. Then, by taking a long wave limit approach and complex conjugation restrictions, some hybrid resonance solutions are generated, such as the interaction solutions between the L-order lumps and Q-fission (fusion) solitons, as well as hybrid solutions mixed by the T-order breathers and Q-fission (fusion) solitons. Dynamical behaviors of these solutions are analyzed theoretically and numerically. The results obtained can be helpful for understanding the fusion and fission phenomena in many physical models, such as the organic membrane, macromolecule material and even-clump DNA, plasmas physics, and so on. [ABSTRACT FROM AUTHOR]

Based on the Hirota bilinear method, we study some kinds of novel interaction solutions of the new integrable (2 + 1) -dimensional extended shallow water wave equation. Using the symbolic computations, we obtain a rich variety of exact solutions such as the lump solutions, lump-kink solutions, breather solutions and two-kink solutions and these exact solutions are shown analytically and graphically by selecting the proper parameters. These new exact solutions contribute us to our understanding of the physical phenomena of the integrable nonlinear shallow water wave model. [ABSTRACT FROM AUTHOR]

In this work, the consistent tanh expansion (CTE) method is developed for a high-order classical Boussinesq- Burgers (HCBB) equation. Via the CTE method, we obtain many exact significant solutions including soliton-resonant solutions, soliton-periodic wave interactions and soliton-rational wave interactions. The CTE related nonlocal symmetries are also proposed. The nonlocal symmetries can be localized to find finite Backlund transformations by prolonging the model to an enlarged one. [ABSTRACT FROM AUTHOR]

The aim of this work is to analyze and explore the dynamics of two extensions of the Bogoyavlenskii–Schieff equation. The Hirota bilinear method is applied to the equations that arise in plasma physics. The N-soliton, fusion, rational solutions and breather solutions, as well as the interaction between M-lump and soliton solutions, are retrieved for both extended equations. The M-lump wave solutions are created from the soliton wave solutions, which are established via the Hirota bilinear method, by considering a longwave limit to the soliton wave solutions and presenting suitable conjugation conditions. The solutions for a given choice of constants are represented graphically to better understand the associated physical phenomena such as the propagation behaviors and the types of collisions. [ABSTRACT FROM AUTHOR]

In this paper, new trial functions are constructed via extended "3-3-2-3-1" and "3-3-2-3-2-1" network models based on the bilinear neural networks method. The new lump-type solution, interaction solution, plentiful arbitrary function solutions and periodic lump solutions of the dimensionally reduced p -generalized Burgers–Kadomtsev–Petviashvili equation are solved. To analyze the dynamic properties of the solutions, appropriate parameters and different activated functions are defined in arbitrary function solutions. Through the three-dimensional and density plots, the dynamical characteristics of the solutions are shown well. [ABSTRACT FROM AUTHOR]

This paper focuses on the modulation instability, conservation laws, and interaction solutions of the generalized coupled Fokas–Lenells equation. Based on the theory of linear stability analysis, distribution pattern of modulation instability gain G in the (K, k) frequency plane is depicted, and the constraints for the existence of rogue waves are derived. Subsequently, we construct the infinitely many conservation laws for the generalized coupled Fokas–Lenells equation from the Riccati-type formulas of the Lax pair. In addition, the compact determinant expressions of the N-order localized wave solutions are given via generalized Darboux transformation, including higher-order rogue waves and interaction solutions among rogue waves with bright-dark solitons or breathers. These solutions are parameter controllable: (m i , n i) and (α , β) control the structure and ridge deflection of solution, respectively, while the value of |d| controls the strength of interaction to realize energy exchange. Especially, when d = 0 , the interaction solutions degenerate into the corresponding order of rogue waves. [ABSTRACT FROM AUTHOR]

The Boussinesq equation has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave overtopping, inundation, and near-shore wave process. Consequently, the study deals with the dynamics of localized waves and interaction solutions to a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions by Hirota's bilinear method. The integrability of the equation of interest was confirmed via the Painlevé test. Taking the long-wave limit approach in coordination with some constraint parameters available in the N-soliton solutions, the localized waves and interaction solutions are constructed. The interaction solutions can be obtained among the localized waves, such as (1) one breather or one lump from the two solitons, (2) one stripe and one breather, and one stripe and one lump from the three solitons, and (3) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. The interactions among the solitons are found to be elastic. The energy, phase shift, shape, period, amplitude, and propagation direction of each of the localized waves and interaction solutions are found to be influenced and controlled by the parameters involved. The dynamical characteristics of these localized waves and interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water. [ABSTRACT FROM AUTHOR]

In this paper, a generalized Darboux transformation of multicomponent nonlinear Schrödinger (NLS) equations is constructed. Nth-order rogue wave solutions of the discussed multicomponent NLS equations are obtained by the resulting generalized Darboux transformation. As two illustrative examples, different kinds of solutions of three-component and six-component NLS equations are obtained, which include rogue wave solutions and interaction solutions. [ABSTRACT FROM AUTHOR]

Based on the N-soliton solutions, the resonant line wave soliton and interaction solutions are derived through some constraints in the (2+1)-dimensional nonlinear wave equation. General resonant line wave soliton solutions are firstly presented and their changing routes are illustrated. Then, several interaction solutions including a nonlinear superposition of resonant line wave soliton with breather wave, a hybrid between resonant line wave soliton and lump wave are constructed via long-wave limit method and module resonant mechanism. The characteristics and properties of these interaction solutions are discussed analytically and graphically. Localized wave and interaction solutions of the nonlinear wave models have a great impact on oceanography and physics. The results may be useful in investigating the physical phenomena in shallow water waves and nonlinear optics.

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]

In this article, we take into account the (1 + 1) -dimensional dispersive long-wave equation which is reduced from the Wu–Zhang equation. Firstly, the B a ¨ cklund transformation and the non-local symmetry are successfully acquired from the truncated Painlev e ´ expansion. At the same time, the non-local symmetry is transformed to Lie point symmetry by a suitable prolonged system. Then some solitary solutions are derived without a hitch via new B a ¨ cklund transformation which originates from Lie point symmetry. Secondly, by utilising the consistent Riccati expansion method and the consistent tanh-expansion method, we find the interaction solutions between soliton and cnoidal wave by using the Jacobi elliptic function. Lastly, the conservation laws which are related to symmetries of the equation are successfully obtained by Ibragimov’s method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple. [ABSTRACT FROM AUTHOR]

In this study, we investigate a new fourth-order integrable nonlinear equation. Firstly, by means of the efficient Hirota bilinear approach, we establish novel types of solutions which include breather, rogue, and three-wave solutions. Secondly, with the aid of Lie symmetry method, we report the invariance properties of the studied equation such as the group of transformations, commutator and adjoint representation tables. A differential substitution is found by nonlinear self-adjointness (NSA) and thereafter the associated conservation laws are established. We show some dynamical characteristics of the obtained solutions through via the 3-dimensional and contour graphs. [ABSTRACT FROM AUTHOR]