Variant Boussinesq Equations Research Articles

Bifurcation analysis, chaotic structures and wave propagation for nonlinear system arising in oceanography

This study focuses on the variant Boussinesq equation, which is used to model waves in shallow water and electrical signals in telegraph lines based on tunnel diodes. The aim of this study is to find closed-form wave solutions using the extended direct algebraic method. By employing this method, a range of wave solutions with distinct shapes, including shock, mixed-complex solitary-shock, singular,mixed-singular, mixed trigonometric, periodic, mixed-shock singular, mixed-periodic, and mixed-hyperbolic solutions, are attained. To illustrate the propagation of selected exact solutions, graphical representations in 2D, contour, and 3D are provided with various parametric values. The equation is transformed into a planar dynamical structure through the Galilean transformation. By utilizing bifurcation theory, the potential phase portraits of nonlinear and super-nonlinear traveling wave solutions are investigated. The Hamiltonian function of the dynamical system of differential equations is established, revealing the system’s conservative nature over time. The graphical representation of energy levels offers valuable insights and demonstrates that the model has closed-form solutions.

Novel solutions to the coupled KdV equations and the coupled system of variant Boussinesq equations

This article examines the coupled KdV equations and the coupled system of variant Boussinesq equations with beta time derivative and explores their travelling wave solutions. This work explains the evolution of waves with fractional parameter. The simple ansatz approach produces a variety of novel solutions in terms of hyperbolic and periodic functions. Graphical representation of solutions are also presented.

Analysis of parametric effects in the wave profile of the variant Boussinesq equation through two analytical approaches

Abstract The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of the solutions. The obtained solutions and graphical representations visualize the dynamics of the phenomena and build up the mathematical foundation of the wave process in dissipative and dispersive media. It turns out that the IBSEF method and the NAE are powerful and might be used in further works to find novel solutions for other types of nonlinear evolution equations ascending in physical sciences and engineering.

Soliton solutions of some nonlinear evolution equations in shallow water theory

This paper proposes some soliton solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation, a (3+1) dimensional modified Korteweg–de Vries–Zakharov–Kuznetsov and a Variant Boussinesq equations. We presented a exponential function method (EFM) for solving the considered evolution equations. The solitary, cuspon and periodic wave solutions are acquired and presented graphically. The soliton solutions of considered equations play a typical role for expressing kinds of wave transmission in any natural instance, particularly in shallow wave kinetics. The results suggest that EFM is effective and useful technique to handle non-linear engineering problems in oceans.

The first integral method and some nonlinear models

In this article, first integral method [Ref. : Feng. Z., Journal of Physics A: Mathematical and General, 35(2002), 343–349] is used to find exact solutions of some nonlinear partial differential equations. It is applied to find exact solutions to a variant Boussinesq equation, the extended modifed Korteweg-de Vries equation and the Kudryashov–Sinel’shchikov equation. The properties of solutions are then discussed and plotted by using suitable values of the parameters involved. Shock wave-like solutions for variant Boussinesq and extended mKdv equations are found. For the Kudryashov–Sinel’shchikov equation one singular and two exponential solutions are obtained.

Solitary wave solitons to one model in the shallow water waves

The current study utilizes the generalized $$\tan (K(\rho )/2)$$ -expansion method, the generalized $$\tanh $$ - $$\coth $$ method and He’s semi-inverse variational method in constructing various soliton and other solutions to the (2+1)-dimensional coupled variant Boussinesq equations which describes the elevation of water wave surface for slowly modulated shallow water waves in lakes and ocean beaches. The system represents collision of a nonlinear wave propagating along y-axis and long wave along x-axis. The integration mechanism that was adopted is modified direct algebraic method, which extracts different solitons (dark and singular) and combo (dark-singular) solitons for different values of parameters. The existence criteria for solitons production is also established for both Kerr and power nonlinearities. Additionally, some other solutions known as singular periodic and rational are also emerged in the process of derivation

The Revised NIM for Solving the Non-Linear System Variant Boussinesq Equations and Comparison with NIM

The Revised NIM for Solving the Non-Linear System Variant Boussinesq Equations and Comparison with NIM

A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water

For different nonlinear time conformable derivative models, a versatile built-in gadget namely the generalized exp⁡(-φ(ξ))-expansion (GEE) method is devoted to retrieving different categories of new explicit solutions. These models include the time fractional approximate long wave, the time fractional variant-Boussinesq equations and the time fractional Wu-Zhang system of equations. The GEE technique is investigated with the help of fractional complex transform and conformable derivative. As a result, we found four types of exact solutions involving hyperbolic function, periodic function, rational functional, and exponential function solutions. The physical significance of the explored solutions depends on the choice of arbitrary parameter values. Finally, we conclude that the GEE method is effective to establish the explicit new exact solutions rather than exp⁡(-φ(ξ))-expansion method.

New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics

The main objective of this paper is to introduce an analytical study for the water wave solutions of coupled fractional variant Boussinesq equation, which is modelled to investigate the waves in fluid dynamics. Wave transformation in fractional form is applied to convert the original fractional-order nonlinear partial differential equation into another nonlinear ordinary differential equation. The strategy here is to use the unified method to obtain a variety of exact solutions. The unified method works well and reveals distinct exact solutions which are classified into two different types, namely polynomial function and rational function solutions. The results are also depicted graphically for different values of fractional parameter. These findings are highly encouraging and have significant importance for some special physical phenomena in fluid dynamics

Lie symmetry analysis, analytical solutions and conservation laws to the coupled time fractional variant Boussinesq equations

In this paper, under the background of the Riemann–Liouville fractional differential, a series of investigation of the coupled time fractional variant Boussinesq equations was done. Firstly, the Lie point symmetries were obtained by using the Li symmetry method. The symmetry reductions is also derived ulteriorly. Secondly, the power series method and the invariant subspace method are employed to acquire exact solutions of the coupled time fractional variant Boussinesq equations. Finally, conservation laws are well constructed based on the Noether theorem. What is more, dynamic behavior of all these exact solutions of the equation is described with the value of α changing.

EXACT SOLUTIONS OF THE VARIANT BOUSSINESQ EQUATIONS

EXACT SOLUTIONS OF THE VARIANT BOUSSINESQ EQUATIONS

A Novel Approach to a Time-Dependent-Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Under investigation in this paper is a more general time-dependent-coefficient Whitham-Broer-Kaup (tdcWBK) system, which includes some important models as special cases, such as the approximate equations for long water waves, the WBK equations in shallow water, the Boussinesq-Burgers equations, and the variant Boussinesq equations. To construct doubly periodic wave solutions, we extend the generalized F-expansion method for the first time to the tdcWBK system. As a result, many new Jacobi elliptic doubly periodic solutions are obtained; the limit forms of which are the hyperbolic function solutions and trigonometric function solutions. It is shown that the original F-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system, but the novel approach of this paper is valid. To gain more insight into the doubly periodic waves contained in the tdcWBK system, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations.

Some new analytical solutions of the variant Boussinesq equations

In this manuscript, the exact analytical solutions for the variant Boussinesq equations are constructed by utilizing the improved $$\tanh (\phi (\xi )/2)$$ -expansion method (ITEM) and the improved $$(G'{/}G)$$ -expansion method (IGEM). Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the ITEM and IGEM for hyperbolic function, trigonometric function and rational function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.

NEW EXACT SOLUTIONS OF A GENERALISED BOUSSINESQ EQUATION WITH DAMPING TERM AND A SYSTEM OF VARIANT BOUSSINESQ EQUATIONS VIA DOUBLE REDUCTION THEORY

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.

Solitary wave solutions of three special types of Boussinesq equations

In this paper, the \((G'/G)\)-expansion method is employed to construct more general solitary wave solutions of three special types of Boussinesq equation, namely Boussinesq equation, improved Boussinesq equation and variant Boussinesq equation, where the French scientist Joseph Valentin Boussinesq (1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with a small amplitude. Our work is motivated by the fact that the \((G'/G)\)-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.

Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the Generalised Whitham–Broer–Kaup–Like Equations

Abstract In this article, a generalised Whitham–Broer–Kaup–Like (WBKL) equations is investigated, which can describe the bidirectional propagation of long waves in shallow water. The equations can be reduced to the dispersive long wave equations, variant Boussinesq equations, Whitham–Broer–Kaup–Like equations, etc. The Lie symmetry analysis method is used to consider the vector fields and optimal system of the equations. The similarity reductions are given on the basic of the optimal system. Furthermore, the power series solutions are derived by using the power series theory. Finally, based on a new theorem of conservation laws, the conservation laws associated with symmetries of this equations are constructed with a detailed derivation.

Analytical study of solitons for the variant Boussinesq equations

In this paper, two useful methods that are traveling wave hypothesis and auxiliary function method are constructed new periodic solution, kink soliton solution and bell-shaped solitary wave solution of the variant Boussinesq equations. The existence conditions are given. A pair of solitary wave solutions via the traveling wave hypothesis is mainly studied: the waveform change and the nature of the analysis depending on the choice of solitonic parameters. A new physical phenomenon occurring in this system is the waveform of each soliton gets change or recovery under different parameters, which might provide us with a different approach of load information transfer.

Soliton solutions of cubic-quintic nonlinear Schrödinger and variant boussinesq equations by the first integral method

The cubic-quintic nonlinear Schr\{o}dinger equation emerges in models of light propagation in diverse optical media, such as non-Kerr crystals, chalcogenide glasses, organic materials, colloids, dye solutions and ferroelectrics. The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. By using the extended first integral method, we construct exact solutions of a fourth-order dispersive cubic-quintic nonlinear Schr$\ddot{o}$dinger equation and the variant Boussinesq system. The stability analysis for these solutions are discussed.

The classification of the single travelling wave solutions to the variant Boussinesq equations

The discrimination system for the polynomial method is applied to variant Boussinesq equations to classify single travelling wave solutions. In particular, we construct corresponding solutions to the concrete parameters to show that each solution in the classification can be realized.

Exact traveling wave solutions for system of nonlinear evolution equations.

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.