Broer Kaup Kupershmidt Research Articles

Hamiltonian and Super-Hamiltonian Extensions Related to Broer-Kaup-Kupershmidt System

Based on the Lie algebra A1, the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra A1 are considered, i.e., the non-semi-simple Lie algebra of 4×4 matrix and the super-Lie algebra of 3×3 matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.

A new General Algebraic Method and its applications to the (2+1)-dimensional Broer–Kaup–Kupershmidt equations

Using a computerized symbolic computation technique, a new method named the Repeated General Algebraic Method (RGAM) is established in this study in order to find exact solutions of Nonlinear Partial Differential Equations (NLPDEs). The new method is validated based on the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) equations. By using the RGAM in various conditions, a number of exact solutions of NLPDEs have been obtained showing potential importance in future physical applications. Also, it is anticipated that the RGAM can be applied to other nonlinear evolution equations in mathematical physics to produce some interesting outcomes.

Finding general and explicit solutions (2 + 1) dimensional Broer–Kaup–Kupershmidt system nonlinear equation by exp-function method

In this work, we implement a relatively new analytical technique, the exp-function method, for solving nonlinear special form of generalized nonlinear (2 + 1) dimensional Broer–Kaup–Kupershmidt equation, which may contain high nonlinear terms. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations which applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and reliability of exp method. It is predicted that exp-function method can be found widely applicable in engineering.

Application of Extended Projective Riccati Equation Method to (2+1)-Dimensional Broer–Kaup–Kupershmidt System

In this paper, extended projective Riccati equation method is presented for constructing more new exact solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the effect of the method, Broer–Kaup–Kupershmidt system is employed and Jacobi doubly periodic solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.

Darboux transformation and multi-soliton solutions for some soliton equations

A class of new spectral problems associated with a Broer–Kaup–Kupershmidt and a Boussinesq–Burgers soliton hierarchy are presented. They are used to generate some new soliton equations. The Darboux transformation for these spectral problems is constructed with the help of a gauge transformation. Finally, by applying the Darboux transformation, some new soliton solutions of these soliton equations are obtained. Some interesting solutions of multi-peak soliton have been found.

On the nontrivial non-travelling wave profiles of nonlinear evolution and wave equations

The overall aim of the present paper is to find and analyze the new non-travelling wave solutions of the nonlinear evolution and wave equations. With the aid of symbolic computation and based on the generalized extended tanh-function method, we propose the newly extended tanh-function expansion algorithm and get many new non-travelling wave solutions of the (2+1)-dimensional Broer–Kaup–Kupershmidt equations. The solutions which we obtain are more abundant than the solutions which the generalized extended tanh-function method gets. At the same time, the solutions contain arbitrary functions which may be helpful to explain some complex phenomena. We also give some figures to describe the property of these solutions. In additions, the method can also be successfully applied to other nonlinear evolution and wave equations.

Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer–Kaup–Kupershmidt equations

Abstract In this Letter, the Exp-function method is used to seek generalized solitonary solutions of Riccati equation. Based on the Riccati equation and its generalized solitonary solutions, new exact solutions with three arbitrary functions of the ( 2 + 1 )-dimensional Broer–Kaup–Kupershmidt equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.

NEW TYPES OF INTERACTIONS BASED ON VARIABLE SEPARATION SOLUTIONS VIA THE GENERAL PROJECTIVE RICCATI EQUATION METHOD

In this paper, first, the general projective Riccati equation method (PREM) is applied to derive variable separation solutions of (2 + 1)-dimensional systems. By further studying, we find that these variable separation solutions obtained by PREM, which seem independent, actually depend on each other. A common formula with some arbitrary functions is obtained to describe suitable physical quantities for some (2 + 1)-dimensional models such as the generalized Nizhnik–Novikov–Veselov system, Broer–Kaup–Kupershmidt equation, dispersive long wave system, Boiti–Leon–Pempinelli model, generalized Burgers model, generalized Ablowitz–Kaup–Newell–Segur system and Maccari equation. The universal formula in Tang, Lou, and Zhang [2] can be simplified to the common formula in the present paper. Second, this method is successfully generalized to (1 + 1)-dimensional systems, such as coupled integrable dispersionless equations, shallow water wave equation, Boiti system and negative KdV model, and is able to obtain another common formula to describe suitable physical fields or potentials of these (1 + 1)-dimensional models, which is similar to the one in (2 + 1)-dimensional systems. Finally, based on the common formula for (2 + 1)-dimensional systems and by selecting appropriate multivalued functions, elastic and inelastic interactions among special dromion, special peakon, foldon and semi-foldon are investigated. Furthermore, the explicit phase shifts for all the local excitations offered by the common formula have been given, and are applied to these novel interactions in detail.

Variable separation approach by means of the mapping method

A mapping method is used to obtain some new types of variable separation excitations, with two arbitrary functions, of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. From the variable separation solution and by selecting appropriate functions, a class of the (2+1)-dimensional non-propagating solitary waves are found. Moreover, two types of chaotic patterns (chaotic line soliton pattern and chaotic–chaotic pattern) are discussed.

The Fission, Fusion and Annihilation of Solitons of the (2+1)-Dimensional Broer-Kaup-Kupershmidt System

By means of an improved mapping approach, a series of excitations of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific fission, fusion and annihilation phenomena of solitons are also obtained. - PACS numbers: 05.45.Yv, 03.65.Ge.

New travelling wave solutions for combined KdV-mKdV equation and (2+1)-dimensional Broer-Kaup-Kupershmidt system

Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg–de Vries (KdV) and modified KdV (mKdV) equation and (2+1)-dimensional Broer–Kaup–Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.

Semifoldons with fusion and fission properties of (2 + 1)-dimensional nonlinear system

In this paper, by virtue of q-deformed hyperbolic functions, the variable separation solution with two variable separated functions of the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equation (BKKE) is obtained. Based on this variable separation solution, soliton fusion and fission phenomena among semifoldons are investigated.

New exact travelling wave solutions of two nonlinear physical models

In this paper, an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely, the quantum Zakharov equations and the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions.The exact solutions are obtained which include new soliton-like solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and its applications are promising.

Variational approach to the Broer–Kaup–Kupershmidt equation

Abstract Using He's semi-inverse method, a variational principle for the Broer–Kaup–Kupershmidt equation is established.

Generalized method and new exact wave solutions for (2 + 1)-dimensional Broer–Kaup–Kupershmidt system

In this paper, a new algebra method, i.e., generalized elliptic equation rational expansion method is devised to uniformly construct a series of exact solutions for nonlinear partial differential equations. Compared with most existing Tanh methods, the proposed method not only recover some known solutions, but also find some new and general solutions. The efficiency of the method can be demonstrated on the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system. As a result, we obtain many new types of solutions include rational formal solitary wave solutions, rational formal triangular periodic wave solutions, rational formal Jacobi and Weierstrass double period solutions and rational wave solutions.

Further improved extended Fan sub-equation method and new exact solutions of the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations

In this paper, a further improved extended Fan sub-equation method is proposed by taking a more general transformation to seek more types of exact solutions of non-linear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations to illustrate the validity and advantages of this method. As a result, many new and more general non-travelling wave and coefficient function solutions are obtained including soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions, Weierstrass elliptic doubly-like periodic solutions.

Special conditional similarity reductions and exact solutions of the (2 + 1)-dimensional VCBKK system

In this paper, we present a method of special conditional similarity reductions for nonlinear partial differential equations. As an concrete example of its applications in physical problems, we apply this method to the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system with variable coefficients (VCBKK), which has the extensive physics background, and obtain abundant exact solutions derived from some reduction equations.

A new generalization of variable coefficients algebraic method for solving nonlinear evolution equations

Abstract In this paper, based on a new intermediate transformation, a more general variable coefficient algebraic method is proposed. The efficiency of the method is demonstrated on the Broer–Kaup–Kupershmidt equations. As a result, several new families of exact solutions of physical interest are obtained. The method can be applied to other nonlinear evolution equations in mathematical physics.

Modified extended tanh-function method and its application on nonlinear physical equations

By means of computerized symbolic computation and a modified extended tanh-function method for constructing multiple traveling wave solutions of nonlinear physical equations is presented and implemented in a computer algebraic system. Applying this method, we consider some of nonlinear equations of special interest in physics namely, the Broer–Kaup–Kupershmidt, nonlinear coupled plasma, and coupled-nonlinear reaction–diffusion equations. As a results, we can successfully recover the previously known solitary wave solutions that had been found by other sophisticated methods. The method is straightforward and concise, and it can also be applied to other nonlinear equations in physics.

A new extended Riccati equation rational expansion method and its application

In this paper, a new extended Riccati equation rational expansion method is suggested to constructing multiple exact solutions for nonlinear evolution equations. The validity and reliability of the method is tested by its application to the dispersive long wave system and the Broer–Kaup–Kupershmidt system. The method can be applied to other nonlinear evolution equations in mathematical physics.