Kaup Equations Research Articles

Uniformly constructing combinatorial solutions, combining a rational function with hyperbolic or trigonometric functions, for the (2+1) dimensional Broer–Kaup–Kupershmidt equation

According to two dependent rational solutions to a generalized Riccati equation together with the equation itself, a rational-exponent solution to a nonlinear partial differential equation can be constructed. By selecting different parameter values in the rational-exponent solution, many families of combinatorial solutions combined with a rational function such as hyperbolic functions or trigonometric functions, are rapidly derived. This method is applied to the Whitham–Broer–Kaup equation and a series of combinatorial solutions are obtained, showing that this method is a more concise and efficient approach and can uniformly construct many types of combined solutions to nonlinear partial differential equations.

Extended double Wronskian solutions to the Whitham–Broer–Kaup equations in shallow water

Whitham–Broer–Kaup (WBK) equations describing the propagation of shallow-water waves, with a variable transformation, are transformed into a generalized Ablowitz–Kaup–Newell–Segur system, the bilinear forms of which are obtained via the rational transformations. Employing the matrix extension and symbolic computation, we derive types of solutions of the WBK equations through the selection of different canonical matrices, including solitons, rational solutions, and complexitons. Furthermore, dynamic properties of the solutions are discussed graphically and a novel phenomenon is observed, i.e., the coexistence of the elastic–inelastic interactions without disturbing each other.

Exact solutions of nonlinear evolution equations with variable coefficients using exp-function method

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations with variable coefficients. The proposed technique is tested on the Zakharov–Kuznetsov and (2+1)-dimensional Broer–Kaup equations with variable coefficients. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Obtained results clearly indicate the reliability and efficiency of the proposed exp-function method.

Traveling wave solutions of Whitham–Broer–Kaup equations by homotopy perturbation method

The homotopy perturbation method (HPM) is employed to find the explicit and numerical traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow-up solutions and periodic solutions. The numerical solutions are calculated in the form of convergence power series with easily computable components. The homotopy perturbation method performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.

The improved [formula omitted]-expansion method and its applications to the Broer–Kaup equations and approximate long water wave equations

By introducing a new general ansätze, the improved G ′ G -expansion method is proposed to construct exact solutions of both Broer–Kaup equations and approximate long water wave equations. As a result, some new travelling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. The proposed method is straightforward, concise and effective, and can be applied to other nonlinear evolution equations in mathematical physics.

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the ( G′/ G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.

New exact solutions to breaking soliton equations and Whitham–Broer–Kaup equations

In this paper, by using the improved Riccati equations method, we obtain several types of exact traveling wave solutions of breaking soliton equations and Whitham–Broer–Kaup equations. These explicit exact solutions contain solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions. The method employed here can also be applied to solve more nonlinear evolution equations.

NEW TYPES OF MULTIPLE SOLITARY WAVE SOLUTIONS FOR THE HIGH-ORDER BROER–KAUP EQUATIONS

Using the extended homogeneous balance method, we obtain a new type of multiple solitary wave solutions for the (1+1)-dimensional high-order Broer–Kaup equations and a new type of multiple solitary wave-like solutions for the (2+1)-dimensional high-order BK equations. The method used here is very concise and primary, and can be generalized to deal with other class of nonlinear equation.

Exact solutions of [formula omitted]-dimensional stochastic Broer–Kaup equation

In this Letter, with the help of Hermit transformation, we employ the Riccati equation and its new exact solutions to find new exact solutions of ( 2 + 1 ) -dimensional stochastic Broer–Kaup equation in the white noise environment. As some special examples, some analytic solutions can degenerate into these solutions reported in open literatures.

A generalized [formula omitted]-expansion method and its application to the (2 + 1)-dimensional Broer–Kaup equations

In this paper, a generalized G ′ G -expansion method is used to seek more general exact solutions of the (2 + 1)-dimensional Broer–Kaup equations. As a result, non-travelling wave solutions with three arbitrary functions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. The proposed method is more powerful than Wang et al.’s method in [M.L. Wang, X.Z. Li, J.L. Zhang, The G ′ G -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (2008) 417–423] and it can be used for many other nonlinear evolution equations in mathematical physics.

Evolutions of Wave Patterns in Whitham–Broer–Kaup Equation

Upon investigation of the parameter influence on the structure of WBK equation, transition boundaries are derived. All possible bounded waves as well as the existence conditions are obtained. The evolution of waves with variation of the parameters is discussed in detail, which reveals the bifurcation mechanism between different wave patterns.

Exact Solutions and Coherence Structures of Generalized Breor–Kaup Equations

Using the mapping method based on q-deformed hyperbolic functions, the exact solutions of generalized Breor–Kaup equations are obtained. Based on the solutions, two coherent structures, periodic-branch kink and non-propagating kink, have been obtained. Moreover, one solitonal interaction form, two line solitons interaction on the kink background, has been found.

Darboux transformation and new solutions for the Whitham–Broer–Kaup equations

In this paper, we derive the Lax pair associated with the Whitham–Broer–Kaup equations. Based on the Lax pair obtained, we construct the Darboux transformation with multi-parameters and obtain the one- and multi-soliton solutions. In addition, we qualitatively analyze various types of interaction behavior between two solitary waves along with graphical demonstration, in order to provide valuable information on the shallow water motion.

Periodic Folded Wave Patterns for (2+1)-Dimensional Higher-Order Broer–Kaup Equation

A general solution including three arbitrary functions is obtained for the (2+1)-dimensional higher-order Broer–Kaup equation by means of WTC truncation method. Introducing proper multiple valued functions and Jacobi elliptic functions in the seed solution, special types of periodic folded waves are derived. In long wave limit these periodic folded wave patterns may degenerate into single localized folded solitary wave excitations. The interactions of the periodic folded waves and their degenerated single folded solitary waves are investigated graphically and are found to be completely elastic.

Application of the [formula omitted]-expansion to travelling wave solutions of the Broer–Kaup and the approximate long water wave equations

By using the G ′ G -expansion proposed recently the travelling wave solutions involving parameters of the Broer–Kaup equations and the approximate long water wave equations are found out. The travelling wave solutions are expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters are taken as special values the solitary wave solutions are obtained.

Periodic Semifolded Solitary Waves for (2+1)-Dimensional Variable Coefficient Broer–Kaup System

Applying the extended mapping method via Riccati equation, many exact variable separation solutions for the (2+1)-dimensional variable coefficient Broer–Kaup equation are obtained. Introducing multiple valued function and Jacobi elliptic function in the seed solution, special types of periodic semifolded solitary waves are derived. In the long wave limit these periodic semifolded solitary wave excitations may degenerate into single semifolded localized soliton structures. The interactions of the periodic semifolded solitary waves and their degenerated single semifolded soliton structures are investigated graphically and found to be completely elastic.

Solitary waves for a nonlinear dispersive long wave equation

All the possible traveling wave solutions of Whitham–Broer–Kaup (WBK) equation are investigated in the present paper. By employing phase plane analysis, transition boundaries are derived to divide the parameter space into several regions associated with different types of phase portraits corresponding to different forms of wave solutions. All the exact expressions of bounded wave solutions are obtained as well as their existence conditions. The mechanism of bifurcation between different waves with varying Hamiltonian value has been revealed. It is pointed out that as the periods of two coexisted periodic waves tend to infinity, they may evolve to two solitary waves. Furthermore, when their trajectories pass through the common saddle point, the two solitary waves may merge into a periodic wave, and its amplitude is nearly equal to the sum of the amplitudes of the two solitary wave solutions.

New Multi-soliton Solutions for the (2+1)-Dimensional Kadomtsev–Petviashvili Equation

In this paper, an explicit N-fold Darboux transformation with multi-parameters for both a (1+1)-dimensional Broer–Kaup (BK) equation and a (1+1)-dimensional high-order Broer–Kaup equation is constructed with the help of a gauge transformation of their spectral problems. By using the Darboux transformation and new basic solutions of the spectral problems, 2N-soliton solutions of the BK equation, the high-order BK equation, and the Kadomtsev–Petviashvili (KP) equation are obtained.

A generalized [formula omitted]-expansion method and its applications

In this Letter, the ( G ′ G ) -expansion method [M.L. Wang, X.Z. Li, J.L. Zhang, Phys. Lett. A 372 (2008) 417] is improved and a generalized ( G ′ G ) -expansion method is proposed to seek exact solutions of nonlinear evolution equations. We choose the ( 2 + 1 ) -dimensional Nizhnik–Novikov–Vesselov equation, the ( 2 + 1 ) -dimensional Broer–Kaup equation and the ( 2 + 1 ) -dimensional Kadomstev–Petviashvili equation to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions and coefficient function solutions are obtained.

Application of the variational iteration method to the Whitham–Broer–Kaup equations

Explicit traveling wave solutions including blow-up and periodic solutions of the Whitham–Broer–Kaup equations are obtained by the variational iteration method. Moreover, the results are compared with those obtained by the Adomian decomposition method, revealing that the variational iteration method is superior to the Adomian decomposition method.