On generating (2 + 1)-dimensional hierarchies of evolution equations

Two isospectral problems are constructed with the help of a 6-dimensional Lie algebra. By using the Tu scheme, a (1 + 1)-dimensional expanding integrable couplings of the KdV hierarchy is obtained and the corresponding Hamiltonian structure is established. In addition, the 2-order matrix operators proposed by Tuguizhang are extended to the case where some 4-order matrices are given. Based on the extension, a new hierarchy of 2 + 1 dimensions is obtained by the Hamiltonian operator of the above (1 + 1)-dimensional case and the TAH scheme. The new hierarchy of 2 + 1 dimensions can be reduced to a coupled (2 + 1)-dimensional nonlinear equation and furthermore it can be reduced to the (2 + 1)-dimensional KdV equation which has important physics applications. The Hamiltonian structure for the (2 + 1)-dimensional hierarchy is derived with the aid of an extended trace identity. To the best of our knowledge, generating the (2 + 1)-dimensional equation hierarchies by virtue of the TAH scheme has not been studied in detail except to previous little work by Tu et al.     

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.     

Differential form method for finding symmetries of a (2 + 1)-dimensional Camassa–Holm system based on its Lax pair

In this paper, we use the differential form method to seek Lie point symmetries of a (2 + 1)-dimensional Camassa–Holm (CH) system based on its Lax pair. Then we reduce both the system and its Lax pair with the obtained symmetries, as a result some reduced (1 + 1)-dimensional equations with their new Lax pairs are presented. At last, the conservation laws for the CH system are derived from a direct method.     

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.     

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.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

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.     

A new approach for solutions of the (2 + 1)-dimensional cubic nonlinear Schrödinger equation

A new method to solve the nonlinear evolution equations is presented, which combines the two kind methods – the tanh function method and symmetry group method. To demonstrate the method, we consider the (2 + 1)-dimensional cubic nonlinear Schrödinger (NLS) equation. As a result, some novel solitary solutions of the Schrödinger equation are obtained. And graphs of some solutions are displayed.     

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 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.     

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.     

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.     

Comments on “The generalizing Riccati equation mapping method in nonlinear evolution equation: Application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation”

By means of a so-called generalizing Riccati equation mapping method, Zhu [Zhu S D, Chaos, Solitons & Fractals; 2006. doi:10.1016/j.chaos.2006.10.015] has claimed that abundant new solutions to the (2 + 1)-dimensional Boiti–Leon–Pempinelle (BLP) equation are derived. Based on the derived variable separation solution and by selecting appropriate functions, he has asserted that abundant new non-travelling waves are obtained. We show that the generalizing Riccati equation mapping method is equivalent to the usual mapping approach, and say nothing of the conclusion that many new non-travelling wave solutions have been found.     

Homoclinic breather-wave solutions for Sine–Gordon equation

A new type of two-wave solution, homoclinic breather-wave solution, for (1 + 1)D Sine–Gordon (SG) equation is obtained using the extended homoclinic test method. Moreover, the mechanical feature of two-wave solution is investigated, and the dynamical behavior of solution is exhibited.     

A new Jacobi elliptic function rational expansion method and its application to (1 + 1)-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansätz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.     

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.     

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.     

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.     

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.