Broer Kaup Kupershmidt System Research Articles

Nonlocal residual symmetries, N-th Bäcklund transformations and exact interaction solutions for a generalized Broer–Kaup–Kupershmidt system

Nonlocal residual symmetries, N-th Bäcklund transformations and exact interaction solutions for a generalized Broer–Kaup–Kupershmidt system

Some exact solitons to the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system with two different methods

Some exact solitons to the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system with two different methods

Gramian solutions and solitonic interactions of a (2+1)-dimensional Broer–Kaup–Kupershmidt system for the shallow water

PurposeThis paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth.Design/methodology/approachPfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties.FindingsN-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied.Originality/valueShallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

N-solitons and cuspon waves solutions of (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations via hidden symmetries of Lie optimal system

Broer–Kaup–Kupershmidt (BKK) system of partial differential equations in (2 + 1)-dimensions has been investigated and new analytical solutions have benn attained by the aid of discovering hidden symmetries of Lie optimal system. The optimal system of Lie vectors is derived invoking commutator and adjoint tables. Thereafter, BKK system is reduced into a system of ordinary differential equations in two steps. At each step, hidden symmetries are detected. Such symmetries are those which not being inherited by the previous step. Only hidden symmetry vectors are exploited in attaining analytical solutions of BKK equation. Unlike ad-hoc methods, exponential function method and tanh-coth method etc., the solutions are attained throughout symmetry reductions procedures resulting in some new results including N-solitons and cuspon waves. The latter has a very sharp cusp along its crest and has been compred with previous work.

Nonlocal symmetries, consistent Riccati expansion integrability, and their applications of the (2+1)-dimensional Broer–Kaup–Kupershmidt system**Project supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ13A010014) and the National Natural Science Foundation of China (Grant Nos. 11326164, 11401528, 11435005, and 11375090).

For the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion (CRE) is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

Formula omitted]-soliton solutions and localized structures for the [formula omitted]-dimensional Broer–Kaup–Kupershmidt system

In this paper, the ( 2 + 1 ) -dimensional Broer–Kaup–Kupershmidt system is investigated by means of Bäcklund transformation and the Hirota bilinear method. With symbolic computation, the N -soliton solutions with N + 1 arbitrary functions for that system are derived. Based on the derived solutions, some soliton structures, localized structures and special fission interaction behaviour are graphically studied, which might be helpful to understand the propagation processes for nonlinear and dispersive long waves in shallow water.

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.

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.

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.

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.

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 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.

A multiple Riccati equations rational expansion method and novel solutions of the Broer–Kaup–Kupershmidt system

Abstract To construct exact solutions of nonlinear partial differential equation, a multiple Riccati equations rational expansion method (MRERE) is presented and a series of novel solutions of the Broer–Kaup–Kupershmidt system are found. The novel solutions obtained by MRERE method include solutions of hyperbolic (solitary) function and triangular periodic functions appearing at the same time.

New Families of Rational Form Solitary Wave Solutions to(2+1)-Dimensional Broer–Kaup–Kupershmidt System

Taking the (2+1)-dimensionalBroer–Kaup–Kupershmidt system as a simple example, some familiesof rational form solitary wave solutions, triangular periodicwave solutions, and rational wave solutions are constructed byusing the Riccati equation rational expansion method presentedby us. The method can also be applied to solve more nonlinearpartial differential equation or equations.

New exact excitations and soliton fission and fusion for the (2+1)-dimensional Broer–Kaup–Kupershmidt system

With the help of an extended mapping approach, a series of new types of exact excitations with two arbitrary functions of the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system is derived. Based on the derived solitary wave excitation, some specific soliton fission and fusion solutions of the higher-dimensional BKK system are also obtained.

Extended multilinear variable separation approach and multivalued localized excitations for some (2+1)-dimensional integrable systems

The multilinear variable separation approach and the related “universal” formula have been applied to many (2+1)-dimensional nonlinear systems. Starting from the universal formula, abundant (2+1)-dimensional localized excitations have been found. In this paper, the universal formula is extended in two different ways. One is obtained for the modified Nizhnik–Novikov–Veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the (2+1)-dimensional symmetric sine-Gordon system. The other is for the dispersive long wave equation, the Broer–Kaup–Kupershmidt system, the higher order Broer–Kaup–Kupershmidt system, and the Burgers system where arbitrary number of variable separated functions can be involved. Because of the existence of the arbitrary functions in both the original universal formula and its extended forms, the multivalued functions can be used to construct a new type of localized excitations, folded solitary waves (FSWs) and foldons. The FSWs and foldons may be “folded” in quite complicated ways and possess quite rich structures and multiplicate interaction properties.

(2 1)-dimensional compacton solutions with and without completely elastic interaction properties

Taking the (2 + 1)-dimensional Broer–Kaup–Kupershmidt system as a simple example, some special types of (2 + 1)-dimensional compacton solutions are constructed. It is shown that there is quite rich interaction behaviour between two travelling compactons. For some types of compactons, the interactions among them may not be completely elastic. For some others, the interactions are completely elastic. There is no phase shift for the interactions of the (2 + 1)-dimensional compactons discussed in this paper.