Dimensional Boussinesq Equation Research Articles

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

Dynamics of resonant soliton, novel hybrid interaction, complex N-soliton and the abundant wave solutions to the (2+1)-dimensional Boussinesq equation

The presented work concerns with some novel solutions of the (2+1)-dimensional Boussinesq equation (BE), which acts as an important model for shallow water wave. Some resonant soliton solutions such as the X-shape soliton (XSS) and Y-shape soliton (YSS) solutions are developed via imposing the resonant conditions on the N-soliton solutions (N-SSs) that developed by the Hirota bilinear approach(HBA). Based on the XSS and YSS solutions, the novel hybrid interactions including the interaction between the 1-soliton and the YSS, the interaction between two YSS solutions are extracted. In addition, the complex N-SSs are also explored and discussed. Finally, the travelling wave solutions including the bright solitary and kinky solitary wave solutions are studied by employing the Bernoulli sub-equation function method (BSEFM). The graphs of the attained solutions are drawn to show the physical properties. The findings of this study are expected to help us apprehend the dynamics of the (2+1)-dimensional BE better.

Construction of bilinear Bäcklund transformation and complexitons for a newer form of Boussinesq equation describing shallow water waves

This research investigates the characteristics and attributes of a new (3+ 1)-dimensional Boussinesq equation that describes shallow water waves in higher dimensions. By utilizing the Hirota bilinear representation, a bilinear Bäcklund transformation is provided for the proposed model to get soliton solutions. Then, the extended transform rational function method is applied to calculate the complexitons type solutions. The results demonstrate various exact solutions with different structures, including periodic, singular, and bright solitons. Comprehensive graphical representations in 2D, 3D, and density plots are provided to highlight the physical properties of these solutions. Our approach is distinguished by the unique nature of the problem and the use of previously untested methods in this context, leading to many new and original optical soliton solutions. These results highlight the effectiveness of the proposed method in tackling nonlinear problems in engineering and the natural sciences, exceeding previous work found in the literature.

Novel insights into the propagation of the generalized Boussinesq equation: Dynamics of bidirectional symmetric solitary waves

In this work, we explore the propagation of bidirectional solitary wave solutions in the generalized (2+1)-dimensional Boussinesq equation. The proposed model was recently introduced by Wazwaz and Kaur as an application to describe wave propagation in various multidimensional physical systems, including shallow water waves and similar phenomena. Via implementing a collection of effective approaches, we extract a diversity physical structures of the proposed model manifesting their propagation in a form of moving simultaneous symmetric bidirectional waves. The proposed methodologies will include the use of sine/cosine-function method, sinh/cosh-function method, tanh–coth expansion approach, and the rational form trial solutions including trigonometric and hyperbolic functions. By conducting graphical analyses, we illustrate the various physical structures and dynamic behaviors of this class of nonlinear partial differential equations. The results of this work are expected to facilitate the study of wave interactions, propagation dynamics, and soliton formation in multidimensional settings.

Phase trajectories and Chaos theory for dynamical demonstration and explicit propagating wave formation

This paper is subjected to study the nonlinear integrable model which is the (3+1)-dimensional Boussinesq equation which has a lot of applications in engineering and modern sciences. To find and examine the analytical exact solitary wave solutions of (3+1)-dimensional Boussinesq equation, a modified generalized exponential rational functional method is exerted. As a result, waves, singular periodic, hyperbolic, and trigonometric type solutions are obtained. These acquired solutions are more innovative and encouraging to researchers in their endeavor to study physical marvels. To illustrate how some selected exact solutions propagate, the graphical representation in 2D, Contour, and 3D of those solutions is provided with various parametric values. The considered equation is additionally transformed into the planar dynamical structure by applying the Galilean transformation. All potential phase portraits of the dynamical system are investigated using the theory of bifurcation. The Hamiltonian function of the dynamical system of differential equations is established to see that, the system is conservative over time. The presentation of energy levels through graphics provides valuable insights, and it demonstrates that the model has solutions that can be expressed in closed form. The periodic, quasi-periodic, and chaotic behaviors of the 2D, 3D, and time series are also observable once the dynamical system is subjected to an external force. Meanwhile, the sensitivity of the derived solutions is carefully examined for a range of initial conditions.

Solitary wave solutions of Camassa–Holm nonlinear Schrödinger and $$(3+1)$$-dimensional Boussinesq equations

Solitary wave solutions of Camassa–Holm nonlinear Schrödinger and $$(3+1)$$-dimensional Boussinesq equations

Dark Localized Waves in Shallow Waters: Analysis within an Extended Boussinesq System

We study dark localized waves within a nonlinear system based on the Boussinesq approximation, describing the dynamics of shallow water waves. Employing symbolic calculus, we apply the Hirota bilinear method to transform an extended Boussinesq system into a bilinear form, and then use the multiple rogue wave method to obtain its dark rational solutions. Exploring the first- and second-order dark solutions, we examine the conditions under which these localized solutions exist and their spatiotemporal distributions. Through the selection of various parameters and by utilizing different visualization techniques (intensity distributions and contour plots), we explore the dynamical properties of dark solutions found: in particular, the first- and second-order dark rogue waves. We also explore the methods of their control. The findings presented here not only deepen the understanding of physical phenomena described by the (1+1)-dimensional Boussinesq equation, but also expand avenues for further research. Our method can be extended to other nonlinear systems, to conceivably obtain higher-order dark rogue waves.

Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

Investigation of the intricate dynamics and a variety of hybrid soliton, solutions of the (3+1)-dimensional Boussinesq equation

This manuscript delves into the captivating domain of the (3+1)-dimensional Boussinesq equation, unraveling its complex dynamics and revealing mesmerizing soliton solutions. Our exploration begins with the application of the well-established traveling wave transformation, transforming the model into a set of ordinary differential equations (ODEs). Navigating through the intriguing intricacies of this system, we delve into bifurcations, chaos, and sensitivity to initial conditions, vividly brought to life through numerical simulations of phase portraits. Additionally, we present and discuss various solitary wave solutions, including singular, bright, breather-like, hybrid singular, rogue wave, and multi-singular periodic waves, vividly illustrating their existence through simulations. The (3+1)-dimensional Boussinesq equation presents rich dynamics with applications spanning fluid dynamics, climate science, optics, engineering, and more. Its soliton solutions and intricate behaviors contribute to advancements in diverse scientific disciplines.

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

Lump and multiple soliton solutions to the new integrable (3+1)-dimensional Boussinesq equation

Collision dynamics of high-order localized waves in a novel [formula omitted]-dimensional integrable Boussinesq model

The pursuit of finding precise solutions for complex nonlinear systems in high dimensions has long been a significant goal in the fields of mathematics and physics. This paper focuses on investigating a novel equation, referred to as the (3+1)-dimensional Boussinesq equation, which accurately describes the behavior of gravity waves on the surface of water. Using the Hirota bilinear method, the study successfully derives multiple-soliton and multi-breather solutions for the Boussinesq equation. Moreover, it presents mixed solutions that combine solitons and breathers under specific parameter conditions. By considering the long wave limit for these solitons, rational and semi-rational solutions for the equation are obtained. These rational solutions encompass line rogue waves and lump waves, while the semi-rational solutions emerge from the interaction between line rogue waves and solitons. Additionally, employing symbolic computation, the paper introduces a class of multiple lumps for the (3+1)-d Boussinesq equation, showcasing intriguing triangular formations and circular patterns. Overall, the findings of this research enhance our understanding of unique nonlinear phenomena and offer valuable theoretical support for future experimental investigations.

Soliton molecules and other diverse wave solutions of the (2 + 1)-dimensional Boussinesq equation for the shallow water

Soliton molecules and other diverse wave solutions of the (2 + 1)-dimensional Boussinesq equation for the shallow water

The (3+1)-dimensional Boussinesq equation: Novel multi-wave solutions

The Boussinesq equation is a partial differential equation that describes the behavior of waves in shallow water. In this paper, we address some new dynamical behaviors to the (3+1)-dimensional Boussinesq equation, which are not constructed beforehand. Various solutions namely: multi-soliton, multi-M-lump, and the hybrid soliton solutions are reported. New explored features of equation are presented graphically to better analyze the gained solutions. For different period of time multi-soliton, multi-lump solutions are plotted. The results have important applications in oceanography, geophysics, fluid dynamics, and also used to study the behavior of waves in complex three-dimensional domains, particularly in situations where the nonlinear effects are strong.

Study on (3+1)-dimensional nonlocal Boussinesq equation: multiple soliton solutions

PurposeThis study aims to explore novel solitary wave solutions of a new (3 + 1)-dimensional nonlocal Boussinesq equation that illustrates nonlinear water dynamics.Design/methodology/approachThe authors use the Painlevé analysis to study its complete integrability in the Painlevé sense.FindingsThe Painlevé analysis demonstrates the compatibility condition for the model integrability with the addition of new extra terms.Research limitations/implicationsThe phase shifts, phase variables and Hirota’s bilinear algorithm are used to furnish multiple soliton solutions.Practical implicationsThe authors also furnish a variety of numerous periodic solutions, kink solutions and singular solutions.Social implicationsThe work formally furnishes algorithms for investigating several physical systems, including plasma physics, optical communications and oceans and seas, among others.Originality/valueThis paper presents an original work using a newly developed Painlevé integrable model, as well as novel and insightful findings.

Doubly periodic solutions and non-topological solitons of [formula omitted] dimension Wazwaz Kaur Boussinesq equation employing Jacobi elliptic function method

In this research communication the Boussinesq like equation namely 2+1− dimensional Wazwaz Kaur Boussinesq equation is studied and gotten their Jacobi elliptic function (doubly periodic) solutions employing the simple algebraic Jacobi elliptic function method. The said method makes the two transformations of nonlinear partial differential equations to ordinary differential equations to solvable system of algebraic equations. The gotten doubly periodic solutions are modular transformed to show the degeneration into non-topological solitons, compound non-topological singular solitons and complex singular solitons. The three dimensional surfaces are drawn for few elliptic solutions and solitons.

Generalized exponential rational function method to the fractional shallow water wave phenomena

In this study we introduce the fractional shallow water wave phenomena, namely, the new extended (2 + 1)-dimensional fractional Boussinesq equation and in the following we apply the generalized exponential rational function for extracting the soliton solutions to the Ex-FBe (extended (2 + 1)-dimensional fractional Boussinesq equation) by using Maple tools for calculating and Mathematica for graphical behaviors. Many types of exact soliton solutions for the fswwp equation have been derived by the erf method. Physical interpretations of some of the obtained solutions are also included. It can be easily concluded that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

Sensitivity analysis and solitary wave solutions to the (2+1)-dimensional Boussinesq equation in dispersive media

This paper explores the dynamic behavior of the [Formula: see text]-dimensional Boussinesq equation, which is a nonlinear water wave equation used to model wave packets in dispersive media with weak nonlinearity. Specifically, we investigate the equation’s traveling wave solutions using the Riccati equation mapping method. Our results include solitary and soliton solutions, each with their own set of parameter values. To provide a comprehensive understanding of these solutions, we present them in general form and visualize their significance using various graphs, such as 3D, 2D, and contour plots. The computational effort and resulting outcomes highlight the efficacy of our approach, which has the potential to be applied to other nonlinear physical problems in fields such as mathematical physics, engineering, and nonlinear science.

Searching For (2+1)-dimensional nonlinear Boussinesq equation from (1+1)-dimensional nonlinear Boussinesq equation

A novel (2+1)-dimensional nonlinear Boussinesq equation is derived from a (1+1)-dimensional Boussinesq equation in nonlinear Schrödinger type based on a deformation algorithm. The integrability of the obtained (2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the (1+1)-dimensional Boussinesq equation. Because of the effects of the deformation, the (2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multi-valued.