Dimensional Broer-Kaup Equations Research Articles

Existence of Classical Solutions for Broer-Kaup Equations

In this paper we investigate the Cauchy problem for one dimensional Broer-Kaup equations for existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.

VARIATIONAL PRINCIPLE FOR (2 + 1)-DIMENSIONAL BROER–KAUP EQUATIONS WITH FRACTAL DERIVATIVES

This paper extends the [Formula: see text]-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered. The variational formulations can help to study the symmetries and find conserved quantities in the fractal space. The obtained variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be generalized to other nonlinear evolution equations with fractal derivatives.

New dynamical behavior of two waves for (2+1)-dimensional Broer-Kaup equation

New dynamical behavior of two waves for (2+1)-dimensional Broer-Kaup equation

New exact solutions of the(2+1)-dimensional Broer-Kaup equation by the consistent Riccati expansion method

In this work, we study the (2+1)-D Broer-Kaup equation. The composite periodic breather wave, the exact composite kink breather wave and the solitary wave solutions are obtained by using the coupled degradation technique and the consistent Riccati expansion method. These results may help us to investigate some complex dynamical behaviors and the interaction between composite non-linear waves in high dimensional models

A new approach for some NLDEs with variable coefficients

This study is related to new soliton solutions compound KdV-Burgers equation, generalized KdV equation and (2+1) dimensional Broer–Kaup equations with variable coefficients. The generalized Kudryashov method (GKM) has been tackled for procuring exact solutions of these equations. By using this method, dark and bright soliton solutions of these equations have been found. Also, some graphical representations have been given to observe the structure of these solutions.

Solitary Wave Solutions of Some Coupled Nonlinear Evolution Equations

In this article, the modified simple equation (MSE) method has been executed to find the traveling wave solutions of the coupled (1+1)-dimensional Broer-Kaup (BK) equations and the dispersive long wave (DLW) equations. The efficiency of the method for finding exact solutions has been demonstrated. It has been shown that the method is direct, effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations. Keywords: MSE method; NLEE; BK equations; DLW equations; Solitary wave solutions. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i2.16671 J. Sci. Res. 6 (2), 273-284 (2014)

Integrability and Solutions of the (2+1)-Dimensional Broer—Kaup Equation with Variable Coefficients

The integrability of the (2+1)-dimensional Broer—Kaup equation with variable coefficients (VCBK) is verified by finding a transformation mapping it to the usual (2+1)-dimensional Broer—Kaup equation (BK). Thus the solutions of the (2+1)-dimensional VCBK are obtained by making full use of the known solutions of the usual (2+1)-dimensional BK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.

Study on the behavior of oscillating solitons using the (2+1)-dimensional nonlinear system

By means of an extended homogeneous balance method and a variable separation hypothesis, a broad general variable separation solution with three specific arbitrary functions of the nonlinear (2+1)-dimensional Broer-Kaup (BK) equations was derived. Based on the derived solution, a number of abundant oscillating solitons, such as dromion, multi-dromion, solitoff, ring, multi-lump and so on, have been revealed in this study by selecting appropriate functions of the general variable separation solution.

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.

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.

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.

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.

A generalized F-expansion method with symbolic computation exactly solving Broer–Kaup equations

In this paper, a generalized F-expansion method is applied to construct exact solutions of the (2 + 1)-dimensional Broer–Kaup equations. As a result, many general exact non-travelling wave and coefficient function solutions are obtained including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions, each of which contains two arbitrary functions. Compared with most existing F-expansion methods, the proposed method gives new and more general exact solutions. More importantly, with the aid of symbolic computation, the method provides a powerful mathematical tool to solve a large many nonlinear partial differential equations.

Symbolic Computation and New Families of Exact Non-travelling WaveSolutions of (2+1)-dimensional Broer–Kaup Equations

The improved tanh function method [Commun. Theor. Phys. (Beijing, China)43 (2005) 585] is further improved by generalizing the ansatz solution of theconsidered equation. As its application, the (2+1)-dimensional Broer–Kaupequations are considered and abundant new exactnon-travelling wave solutions are obtained.

New exact solutions to the (2+1)-dimensional Broer–Kaup equation

Abstract Using the variable coefficient generalized projected Ricatti equation expansion method, we present the explicit solutions of the (2 + 1)-dimensional Broer–Kaup (BK) equation. These solutions include Weierstrass function solutions, solitary wave solutions, soliton-like solutions and trigonometric function solutions. Some of these solutions are found for the first time. Because of the two or three arbitrary functions, rich localized excitations can be found.

Generating Lie Point Symmetry Groups of (2+1)-DimensionalBroer–Kaup Equation via a Simple Direct Method

Using the (2+1)-dimensional Broer–Kaup equation as ansimple example, a new direct method is developed to find symmetrygroups and symmetry algebras and then exact solutions of nonlinearmathematical physical equations.

Using Symbolic Computation to Exactly Solve the Integrable Broer–KaupEquations in (2+1)-Dimensional Spaces

The extended tanh method is further improved by generalizingthe Riccati equation and introducing its twenty seven newsolutions. As its application, the (2+1)-dimensional Broer-Kaupequation is investigated and then its fifty four non-travellingwave solutions have been obtained. The results reported in thispaper show that this method is more powerful than those, such astanh method, extended tanh method, modified extended tanh methodand Riccati equation expansion method introduced in previous literatures.

Variational approach to [formula omitted]-dimensional dispersive long water equations

A variational model is established for ( 2 + 1 ) -dimensional Broer–Kaup equations by the semi-inverse method [Int. J. Turbo Jet-Engines 14 (1) (1997) 23].

New Exact Solutions of Broer-Kaup Equations**The project supported by the Natural Science Foundation of Inner Mongolia under Grant No. 200308020101 and the Key Program of Scientific Research Foundation for Youth of Inner Mongolia Normal University under Grant No. QNZ00111

Based on a known transform, the exact solutions of (2+1)-dimensional Broer–Kaup equations are investigated by using the method of direct integral. A kind of new exact solutions of Broer–Kaup equations are obtained, which contain previous results about solitary wave solutions.

Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a (2 + 1)-Dimensional Broer–Kaup Equations

By means of extended homogeneous balance method and variable separation approach, quite a general excitation of the (2 + 1)-dimensional Broer–Kaup (BK) equations is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, fractal-lump, and multi-peakon excitations of this BK system are found by selecting appropriate functions.