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N-soliton solutions are investigated by the Hirota method for a modified Boussinesq equation, which can be related to the bad Boussinesq equation. The bilinear form of the modified Boussinesq equation is given through the bi-logarithmic transformation and two identical relations. N-soliton solutions and those in the Wronskian determinants are presented for the modified Boussinesq equation. Especially, the explicit one- and two-soliton solutions are discussed in detail. The complex solutions of the bad Boussinesq equation are correspondingly obtained through the Miura transformation simultaneously. From the structure of solutions and the difference of parameters, the solitons show the soliton or antisoliton type, which brings the interactions between the same and different types of waves. There exist three cases of the interactions between two solitons: soliton-soliton, soliton-antisoliton and antisoliton-antisoliton. Four types of interactions among three solitons/antisolitons are presented through graphics. The rational solutions, which admit the algebraic soliton, are derived by taking the limits of wave numbers and proper phase parameters.

In the present course of study, we examine a family of Boussinesq equations of distinct structures and dimensions. We investigate the complete integrability of these equations via Painlevé test. Real and complex multiple soliton solutions, for each considered model, are derived by mode of simplified Hirota’s method. Moreover, exponential expansion method has been employed to each equation, resulting into soliton solutions possessing rich spatial structure due to the presence of abundant arbitrary constants.

We study two (3 + 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type Kadomtsev–Petviashvili–Boussinesq equation. We use the simplified Hirota’s method to conduct this study and to find the general phase shift of these equations. We obtain one- and two-soliton solutions, for each equation, with the coefficients of the three spatial variables are left as free parameters. However, we also develop special conditions on the coefficients of the spatial variables guarantee the existence of three-soliton solutions for each of these two equation.

Associated with the prime number p = 3 , a combined model of generalized bilinear Kadomtsev–Petviashvili and Boussinesq equation (gbKPB for short) in terms of the function f is proposed, which involves four arbitrary coefficients. To guarantee the existence of lump solutions, a constraint among these four coefficients is presented firstly, and then, the lump solutions are constructed and classified via searching for positive quadratic function solutions to the gbKPB equation. Different conditions posed on lump parameters are investigated to keep the analyticity and rational localization of the resulting solutions. Finally, 3-dimensional plots, density plots and 2-dimensional curves with particular choices of the involved parameters are given to show the profile characteristics of the presented lump solutions for the potential function u = 2 ( ln f ) x .

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes.

In this manuscript, we utilize the algorithm of (G′/G) expansion method to construct new solutions of three important models, the Ablowitz–Kaup–Newell–Segur water wave equation, the (2+1)-dimensional Boussinesq equation, and the (3+1)-dimensional Yu–Toda–Sasa–Fukuyama equation, having numerous application in plasma physics, fluid dynamics, and optical fibers. Some new types of traveling wave solutions are acquired, which have not been obtained previously by using this our new technique. The achieved solutions appear with all necessary constraint conditions, which are compulsory for them to exist. The constructed new solutions have vital applications in applied sciences. To understand the physical phenomena of these models, we have also presented graphically movements of the obtained results. It is shown that the our technique provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear waves models in mathematics and physics.

Under investigation in this paper is the ( 3 + 1 ) -dimensional B-type Kadomtsev–Petviashvili–Boussinesq (BKP–Boussinesq) equation, which can display the nonlinear dynamics in fluid. By using Bell’s polynomials, we explicitly derive a bilinear equation for the equation via a very natural and effective way. Then, three types of exchange identities of Hirota’s bilinear operators are presented to derive its Bäcklund transformation. Based on that, we construct the traveling wave solutions, kink solitary wave solutions, rational breathers and rogue waves of the equation. Finally, some properties of interaction phenomena are also provided, which can be used to study the domain of lump solutions. It is hoped that our results can be used to enrich the dynamical behavior of the ( 3 + 1 ) -dimensional nonlinear evolution equations.

This study examines the Boussinesq equation, which is a nonlinear partial differential equation used to describe long wave propagation in shallow water and has broader applications, including nonlinear lattice waves, vibrations in nonlinear strings, and ion sound waves in plasma. The Boussinesq equation provides an insight into the nonlinear long wave propagation behavior in shallow water by taking wave phase into account. Its versatility extends its utility beyond fluid dynamics to various physical phenomena. By providing specific activation functions in the ` ` 2 - 3 - 1 ′ ′ and ` ` 2 - 5 - 1 ′ ′ neural network models, respectively, the generalized lump solution and the precise analytical solutions are produced using the bilinear neural network approach. These analytical solutions, together with the related rogue waves, dark soliton, and bright soliton, are derived using symbolic computation. These findings fill in the gaps in the current research about the Boussinesq equation. The dynamical properties of these waves are displayed on three-dimensional, contour, density, and two-dimensional graphs. The response of the wave solution to different values of wave speed in relation to the wave phase it contains has been described with the help of wave intensity. In addition, the advantages and disadvantages of the layers used in the analytical technique to generate solutions have been discussed. The efficient techniques employed in this research are useful for studying the nonlinear differential equations in one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in plasma.

Under investigation in this work is a generalized ( 2 + 1 ) -dimensional Boussinesq equation. By employing the Bell’s polynomials, bilinear formalism of this generalized ( 2 + 1 ) -dimensional Boussinesq equation is succinctly derived. With the aid of the obtained bilinear formalism, general high-order breather solutions are constructed by using the Hirota’s bilinear method combined with the perturbation expansion. The breathers only periodically propagate along the x -direction. Taking a long-wave limit of the obtained breather solutions and then making further parameter constraints, general smooth rational solutions to the generalized ( 2 + 1 ) -dimensional Boussinesq equation would be succinctly constructed. These smooth rational solutions are high-order lumps and mixed solutions comprising a line rogue wave and lumps. These results exhibit the dynamical behavior of the generalized ( 2 + 1 ) -dimensional nonlinear wave fields.