Whitham Broer Kaup Equations Research Articles

New Wave Solutions of Time-Fractional Coupled Boussinesq–Whitham–Broer–Kaup Equation as A Model of Water Waves

The main purpose of this paper is to obtain the wave solutions of conformable time fractional Boussinesq–Whitham–Broer–Kaup equation arising as a model of shallow water waves. For this aim, the authors employed auxiliary equation method which is based on a nonlinear ordinary differential equation. By using conformable wave transform and chain rule, a nonlinear fractional partial differential equation is converted to a nonlinear ordinary differential equation. This is a significant impact because neither Caputo definition nor Riemann–Liouville definition satisfies the chain rule. While the exact solutions of the fractional partial derivatives cannot be obtained due to the existing drawbacks of Caputo or Riemann–Liouville definitions, the reliable solutions can be achieved for the equations defined by conformable fractional derivatives.

Lie Analysis and Novel Analytical Solutions for the Time-Fractional Coupled Whitham–Broer–Kaup Equations

The Lie point symmetries for the fractional Riemann–Liouville system are used to reduce Fractional Whitham–Broer–Kaup (FWBK) equations to nonlinear fractional ordinary differential equations using the prolongation theorem. FWBK describe the propagation of fluid with different dispersive coefficients in shallow waters. Through the conformable derivative, we apply the integrating factors and Riccati–Bernoulli sub-equation methods. New solutions have been generated. Comparison with previous works has been presented. A mathematical modeling for the long water wave motion between two non-mixed fluids was also studied.

On multiple soliton similariton‐pair solutions, conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.

More Solutions of Coupled Whitham–Broer–Kaup Equations

The similarity transformations method has been applied to get invariant solutions of the coupled system of Whitham–Broer–Kaup equations. Seven different classes of similarity solutions are derived by using Lie similarity reduction process. The solutions involve invariants which are mixing of time and space variables. Numerical simulation is performed to accomplish the physical relevance of profiles in each case. Eventually, different explicit solutions have been derived by assuming suitable values of arbitrary constants involved in the infinitesimals.

Symmetry Reduction Related by Nonlocal Symmetry and Explicit Solutions for the Whitham-Broer-Kaup System

The nonlocal symmetry for the Whitham-Broer-Kaup (WBK) equation is obtained by the Painleve analysis. With the introduction of two auxiliary dependent variables, the nonlocal symmetry is localized to the Lie point symmetry. The reciprocal transformation is established by the localization nonlocal symmetry procedure and can be used to construct a link between the WBK equation and the usual Burgers equation. By using the similarity reductions related with the nonlocal symmetry of the Burgers equation, we derive the symmetry invariant solutions for the WBK equation. Furthermore, the power-series solutions are computed by using the power-series theory.

( $$\frac{G^{'}}{G^{2}}$$ G ′ G 2 )-Expansion method: new traveling wave solutions for some nonlinear fractional partial differential equations

In this study, some new traveling wave solutions for fractional partial differential equations (PDEs) have been developed. The time-fractional Burgers equation, fractional biological population model and space-time fractional Whitham Broer Kaup equations have been considered. These equations have significant importance in different areas such as fluid mechanics, determination of birth and death rates and propagation of shallow water waves. The analytical technique ( $$\frac{G^{'}}{G^{2}}$$ ) has been utilized for finding the new traveling wave solutions of the considered fractional PDEs. ( $$\frac{G^{'}}{G^{2}}$$ )-expansion method is a very useful approach and exceptionally helpful as contrast with other analytical methods. The proposed method provides three unique sort of solutions such as hyperbolic, trigonometric and rational solutions. This approach is likewise applicable to other nonlinear fractional models.

Lax pair, infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow water

Under investigation in this paper is a set of the time-dependent Whitham–Broer–Kaup equations, which is used for the shallow water under the Boussinesq approximation. The equations can be transformed into generalized time-dependent coefficient Ablowitz–Kaup–Newell–Segur system via the variable transformation. Lax pair, infinitely-many conservation laws and bilinear forms of the Ablowitz–Kaup–Newell–Segur system are obtained. One-, two- and three-soliton solutions are derived via the Hirota bilinear method. The solitons are physically related to the horizontal velocity field and height that deviates from equilibrium position of the water. Features of the solitons are studied: Soliton amplitude is related to the wave number parameters, while the soliton velocity is related to the wave number parameters and variable coefficient. Interactions between/among the solitons could be elastic or inelastic, determined by the wave number parameters. Interaction property could not be affected by the variable coefficient. Soliton stability is studied via the numerical calculation, which indicates that the solitons could only propagate steadily in a limited time.

On controlled propagation of nonautonomous dispersive coupled Whitham–Broer–Kaup equation in shallow water

In this paper, the dispersive coupled Whitham–Broer–Kaup (DCWBK) equation with time-dependent coefficients describing the propagation of the shallow water waves are obtained. The propagation of solitons and elliptic (or chirped) waves can be manipulated by suitable variations of the dispersion coefficient. Here, controllable transmission of the surface waves for soliton similariton pairs with the snoidal backgrounds is considered. It is found that, when the dispersion coefficient is taken as increasing, the velocity is increasing with the dispersion coefficient increasing. While this holds vice versa for the height of propagation wave.

New solitary solutions of the Gardner equation and Whitham–Broer–Kaup equations by the modified simplest equation method

In this paper, the modified simplest equation method (SEM) is employed to handle the Gardner equation and Whitham-Broer-Kaup (WBK) equations, respectively. Five new solitary solutions are accordingly obtained. The paper proceeds step-by-step with increasing detail about derivation process, first illustrating the algorithm of the modified SEM, which the simplest equation and its exact solution are not pre-defined. Then, new solutions are used to illustrate the connection between the Gardner equation and WBK equations. In addition, new solutions are reconstructed as solutions for the VBEs, ALWEs, the dispersive long wave equations, and the modified KdV equation.

Symmetry analysis and explicit power series solutions of the Boussinesq–Whitham–Broer–Kaup equation

This paper focus on two aspects. Firstly, symmetry analysis is performed for the Boussinesq–Whitham–Broer–Kaup equation which can degenerate to the Boussinesq equation, the Whitham–Broer–Kaup equation and the Broer–Kaup equation. As a byproduct, the similarity reductions and exact solutions of the equation are constructed based on the optimal systems. Secondly, the explicit solutions are considered by the power series method. Moreover, the convergence of the power series solutions are shown. The physical significance of the solutions is considered from the transformation group point of view.

Nonlocal Symmetry, CTE Solvability and Interaction Solutions of Whitham–Broer–Kaup Equations**Supported by the Key Foundation of Anhui Education Bureau under Grant No. KJ2013A028, the 211 Project of Anhhui University under Grant No. J18520104, Scientific Training Project for University Students, National Natural Science Foundation of China under Grant Nos. 11471015, 11571016, and Natural Science Foundation of Anhui Province under Grant No. 1408085MA02

Whitham–Broer–Kaup (WBK) equations in the shallow water small-amplitude regime is hereby under investigation. Nonlocal symmetry and Bäcklund transformation are presented via the truncated Painlevé expansion. This residual symmetry is localised to Lie point symmetry by the properly enlarged system. The finite symmetry transformation of the prolonged system is computed. Based on the CTE method, WBK equations are linearized and new analytic interaction solutions between solitary waves and cnoidal waves are given with the aid of solutions for the linear equation.

Residual symmetries and interaction solutions for the Whitham–Broer–Kaup equation

The nonlocal residual symmetry for the Whitham–Broer–Kaup (WBK) equation is derived by the truncated Painleve analysis. The nonlocal residual symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Based on the consistent tanh expansion method (CTE), some exact interaction solutions among different nonlinear excitations are explicitly presented. Some special interaction solutions are investigated in both analytical and graphical ways.

Traveling wave solution by differential transformation method and reduced differential transformation method

The present study further examines two recent semi-analytic methods, a reduced order of nonlinear differential transformation method (also called RDTM) and differential transformation method along with Pade approximation to discuss Jaulent–Miodek and coupled Whitham–Broer–Kaup equations. The basic ideas of these methods are briefly introduced and performance of the proposed methods for above mentioned equations is evaluated via comparing with exact solution. The results illustrate that the so-called DTM method, unlike RDTM, due to the presence of secular terms (similar to perturbation method), cannot be found practical for nonlinear partial differential equations (particularly in Acoustic and Wave propagation problems) even through utilizing Pade approximation; meanwhile, RDTM method, despite its simplicity and rapid convergence, assured a significant accuracy and great agreement, and thus it is fair to say that nonlinear problems together with Acoustic application which cannot be solved via Analytical methods, can be studied with reduced order of nonlinear differential transformation method.

Exact combined traveling wave solutions and multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations

The homogeneous balance of undetermined coefficients method (HBUCM) is firstly proposed to construct not only the exact traveling wave solutions, three-wave solutions, homoclinic solutions, N-soliton solutions, but also multi-symplectic structures of some nonlinear partial differential equations (NLPDEs). By applying the proposed method to the variant Boussinesq equations (VBEs), the exact combined traveling wave solutions and a multi-symplectic structure of the VBEs are obtained directly. Then, the definition and a multi-symplectic structure of the variant Boussinesq–Whitham–Broer–Kaup type equations (VBWBKTEs) which can degenerate to the VBEs, the Whitham–Broer–Kaup equations (WBKEs) and the Broer–Kaup equations (BKEs) are given in the multi-symplectic sense. The HBUCM is also a standard and computable method, which can be generalized to obtain the exact solutions and multi-symplectic structures for some types of NLPDEs.

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham–Broer–Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated by two aspects. One is analyzing the approximate solutions graphically. The other is comparing the results with those of the Adomian decomposition method (ADM), the variational iteration method (VIM) and the optimal homotopy asymptotic method (OHAM). Illustrative examples reveal that the present technique outperforms the aforementioned methods and can be used as an alternative for solving fractional equations.

Analytical solutions of time–space fractional, advection–dispersion and Whitham–Broer–Kaup equations

In this article, we study time–space fractional advection–dispersion (FADE) equation and time–space fractional Whitham–Broer–Kaup (FWBK) equation that have significant roles in hydrology. We introduce suitable transformations to convert fractional-order derivatives to integer-order derivatives and as a result these equations transform into partial differential equations (PDEs). Then the Lie symmetries and the corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.

The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics

In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.

Solution of coupled Whitham–Broer–Kaup equations using optimal homotopy asymptotic method

In this work, an approximate method called Optimal Homotopy Asymptotic Method (OHAM) is proposed and implemented for the numerical solution of Whitham–Broer–Kaup (WBK) equations with blow-up and periodic solutions. Results obtained through OHAM are compared with the exact as well as with the results available in the literature. It was revealed that only second-order OHAM solutions are sufficient to achieve the desired accuracy in comparison to the higher order solutions of the methods we have made comparison with.

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.

Topological solitons, cnoidal waves and conservation laws of coupled wave equations

In this paper a few coupled wave equations that arise in the dynamics of two-layered shallow water waves in ocean shores and beaches have been studied. The mapping method is applied to extract cnoidal waves and solitary wave solutions to the coupled Korteweg–de Vries (KdV) equation, coupled Boussinesq equation and the coupled Whitham–Broer–Kaup equation. The ansatz method is also applied to obtain topological 1-solution to the coupled KdV equation with power law nonlinearity. The multiplier method then gives a few conserved quantities of the coupled KdV equation.