The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation

The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by the generalizing Riccati equation mapping method and picking up its new solutions. In order to test the validity of this approach, the (2 + 1)-dimensional Boiti–Leon–Pempinelle equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.     

The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.     

A new generalized algebra method and its application in the (2 + 1) dimensional Boiti–Leon–Pempinelli equation

In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.     

A new compound Riccati equations rational expansion method and its application to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system

In this paper, based on a new general ansätze and symbolic computation, a new compound Riccati equations rational expansion method is proposed. Being concise and straightforward, it is applied to the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov system. It is shown that more complexiton solutions can be found by this new method. The method can be applied to other nonlinear partial differential equations in mathematical physics.     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

Symbolic computation and new families of exact solutions to the (2 + 1)-dimensional dispersive long-wave equations

In this Letter, We present a further generalized algebraic method to the (2 + 1)-dimensional dispersive long-wave equations (DLWS), As a result, we can obtain abundant new formal exact solutions of the equation. The method can also be applied to solve more (2 + 1)-dimensional (or (3 + 1)-dimensional) nonlinear partial differential equations (NPDEs).     

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems

Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems are proposed. The generalized bi-Hamiltonian structures for one of the two hierarchies are derived with the aid of the trace identity. Some explicit solutions of a typical nonlinear evolution equation in the hierarchy are obtained, which include soliton and periodic solutions.     

Chaos, solitons and fractals in (2 + 1)-dimensional KdV system derived from a periodic wave solution

With the help of an extended mapping method and a linear variable separation method, new types of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with two arbitrary functions for (2 + 1)-dimensional Korteweg–de Vries system (KdV) are derived. Usually, in terms of solitary wave solutions and rational function solutions, one can find some important localized excitations. However, based on the derived periodic wave solution in this paper, we find that some novel and significant localized coherent excitations such as dromions, peakons, stochastic fractal patterns, regular fractal patterns, chaotic line soliton patterns as well as chaotic patterns exist in the KdV system as considering appropriate boundary conditions and/or initial qualifications.     

New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.     

Explicit N-fold Darboux transformation and multi-soliton solutions for the (1 + 1)-dimensional higher-order Broer–Kaup system

In this paper, we present a new N-fold Darboux transformations of the (1 + 1)-dimensional higher-order Broer–Kaup (HBK) system with the help of a gauge transformation of the spectral problem. As an application, new explicit (2N − 1)-soliton solutions of the (1 + 1)-dimensional HBK system are obtained. Both the N-fold Darboux transformation and (2N − 1)-soliton solutions can be written explicitly in terms of Vandermonde-like determinants which are remarkable compactness and transparency.     

Exotic localized structures based on variable separation solution of (2 + 1)-dimensional KdV equation via the extended tanh-function method

In this paper, firstly, we obtain the variable separation solutions of (2 + 1)-dimensional KdV equation by the extended tanh-function method (ETM) based on mapping method. Novel localized coherent structures about multi-valued functions, i.e. special dromion, special peakon and foldon, and the interactions among them, are discussed. The interactions between two special dromions and between two special peakons possess novel property, that is, there exists a multi-valued foldon in the process of their collision, which is different from the reported cases in previous literature. Moreover, the explicit phase shifts for all the local excitations offered by the quantity u have been given, and are applied to these novel interactions in detail.     

关于极大强单调算子的不精确邻近点算法的收敛性分析

该文研究集值映象方程0∈T(z)的解的迭代逼近，其中T是极大强单调算子．设{x^k}与{e^k}是由不精确邻近点算法x^{k+1}+c_kT(x^{k+1})> x^k+e^{k+1}生成的序列，满足‖e^{k+1}‖≤η_k‖x^{k+1}_x^k‖, ∑^∞_{k=0}(η_k-1)<+∞且inf_(k≥0) η_k=μ≥1.在适当的限制下证明了，{x^k}收敛到T的一个根当且仅当 lim inf_{k→+∞} d(x^k,Z)=0，其中Z是方程0∈T(z)的解集     

The correct traveling wave solutions for the high-order dispersive nonlinear Schrödinger equation

The high-order dispersive nonlinear Schrödinger equation is considered. The exact solutions were obtained by Zhang et al. [J.L. Zhang, M.L. Wang, X.Z. Li, Phys. Lett. A 357 (2006) 188-195] are analyzed. We can demonstrate that some solutions do not satisfy this equation. To obtain the correct solutions, the F-expansion method is applied to solve it.     

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.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

A new note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

Exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili by Zhang [Huiqun Zhang, A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev-Petviashvili equation, Appl. Math. Comput. 216 (2010) 2771-2777] are considered. To look for “new types of exact solutions travelling wave solutions” of equation Zhang has used the G′/G-expansion method. We demonstrate that there is the general solution for the reduction by Zhang from the (2+1)-dimensional Kadomtsev-Petviashvili equation and all solutions by Zhang are found as partial cases from the general solution.     

Classification of qt=P(x, t, q, qx, qxx)

The classification of q_t = P ( x ,  t ,  q ,  q_x ,  q_xx ) is given by using linearization. Two types of integrable equations which are generalizations of the integrable equations of Fokas and Svinolupov are given. Results are compared with Fokas's symmetry and the Mikhailov–Shabat–Sokolov formal symmetry approaches.     

General energy decay of solutions for a weakly dissipative Kirchhoff equation with nonlinear boundary damping

In this article, we study the weak dissipative Kirchhoff equation \({u_{tt}} - M\left( {\left\| {\nabla u} \right\|_2^2} \right)\Delta u + b\left( x \right){u_t} + f\left( u \right) = 0\), under nonlinear damping on the boundary \(\frac{{\partial u}}{{\partial v}} + \alpha \left( t \right)g\left( {{u_t}} \right) = 0\). We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping. Our result extends and improves some results in the literature such as the work by Zhang and Miao (2010) in which only exponential energy decay is considered and the work by Zhang and Huang (2014) where the energy decay has been not considered.     

Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.