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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.