A (2 + 1)-dimensional integrable hierarchy and its extending integrable model

A loop algebra is constructed, whose subalgebra is first used to present a Lax pair. By making use of the Tu scheme by Tu Guizhang, a generalized (2 + 1)-dimensional KN hierarchy is worked out. Further, based on the associated relations between the subalgebras in the above loop algebra, an extending integrable model of the generalized (2 + 1)-dimensional KN hierarchy as above is produced.     

Integrable couplings of the (2 + 1)-dimensional dispersive long wave hierarchy and its Hamiltonian structure

A higher loop algebra is constructed, from which the integrable couplings associated with the (2 + 1)-dimensional dispersive long wave hierarchy is obtained with the help of the (2 + 1) zero curvature equation generated from one of reduced equations of the self-dual Yang–Mills equations. Furthermore, the Hamiltonian structure of the integrable couplings is worked out by taking use of the variational identity.     

Multiple soliton solutions for a (2 + 1)-dimensional integrable KdV6 equation

In this work, a completely integrable (2 + 1)-dimensional KdV6 equation is investigated. The Cole-Hopf transformation method combined with the Hirota’s bilinear sense are used to determine two sets of solutions for this equation. Multiple soliton solutions are formally derived to emphasize its complete integrability. Moreover, multiple singular soliton solutions are also developed for this equation. The resonance relation for this equation does not exist.     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings.     

Symmetry reduction and new non-traveling wave solutions of (2 + 1)-dimensional breaking soliton equation

In this paper, the new idea of a combination of Lie group method and homoclinic test technique is first proposed to seek non-traveling wave solutions of (2 + 1)-dimensional breaking soliton equation. The system is reduced to some (1 + 1)-dimensional nonlinear equations by applying the Lie group method and solves reduced equation with homoclinic test technique. Based on this idea and with the aid of symbolic computation, some new explicit solutions of similar systems can be obtained.     

Variable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer–Kaup system

In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Links between some (2 + 1)-dimensional nonlinear evolution equations and Painlevé-II equations

Using the idea of transformation, some links between (2 + 1)-dimensional nonlinear evolution equations and the ordinary differential equations Painlevé-II equations has been illustrated. The Kadomtsev–Petviashvili (KP) equation, generalized (2 + 1)-dimensional break soliton equation and (2 + 1)-dimensional Boussinesq equation are researched. As a result, some new interesting results about these (2 + 1)-dimensional PDEs have been obtained, such as the exact solutions with arbitrary functions, rich rational solutions and the nontrivial Bäcklund transformations have been derived.     

Compacton,peakon and folded localized excitations for the (2 + 1)-dimensional Broer–Kaup system

In high dimensions there are abundant coherent soliton excitations. From the variable separation solutions for the (2 + 1)-dimensional Broer–Kaup system, three kinds of new localized excitations in this system are obtained. Some interesting novel features of these structures are revealed.     

An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions.     

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3 + 1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3 + 1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.     

Soliton-like and periodic form solutions to (2 + 1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE, we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.     

Solitons,chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

Symbolic computation and new families of exact non-travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

The improved tanh function method [Chaos, Solitons & Fractals 2005;24:257] is further improved by constructing new ansatz solution of the considered equation. As its application, the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are considered and abundant new exact non-travelling wave solutions are obtained.     

A generalized AKNS hierarchy,bi-Hamiltonian structure,and Darboux transformation

A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given.     

Invertible linear transformations and the Lie algebras

With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.     

On shifted periodic solutions of two nonlinear equations

Using the linear superposition approach, we find periodic solutions with shifted periods and velocities of the (2 + 1)-dimensional modified Zakharov–Kuznetsov equation and the (3 + 1)-dimensional Kadomtsev–Petviashvili equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure of generating solutions of nonlinear evolution equations is successful as a consequence of some cyclic identities satisfied by the Jacobi elliptic functions which reduce by 2 (or a larger even number) the degree of cyclic homogeneous polynomials in Jacobi elliptic functions.     

Doubly periodic wave and folded solitary wave solutions for (2 + 1)-dimensional higher-order Broer–Kaup equation

A general solution including three arbitrary functions is obtained for the (2 + 1)-dimensional high-order Broer–Kaup equation by means of WTC truncation method. From the general solution, doubly periodic wave solutions in terms of the Jacobian elliptic functions with different modulus and folded solitary wave solutions determined by appropriate multiple valued functions are obtained. Some interesting novel features and interaction properties of these exact solutions and coherent localized structures are revealed.     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

New families of exact soliton-like solutions and dromion solutions to the (2+1)-dimensional higher-order Broer–Kaup equations

Using homogeneous balance method we obtain Bäcklund transformation (BT) and a linear partial differential equation of higher-order Broer–Kaup equations. As a result, new soliton-like solutions and new dromion solution and other exact solutions of (2 + 1)-dimensional higher-order Broer–Kaup equations are given. By analyzing a soliton-like solution, we get some dromions solutions. This method, which can be generalized to some (2 + 1)-dimensional nonlinear evolution equations, is simple and powerful.