New exact solutions and fractal patterns of generalized Broer–Kaup system via a mapping approach

With the help of an extended mapping approach, a new type of variable separation solution with two arbitrary functions of the (2 + 1)-dimensional generalized Broer–Kaup (GBK) system is derived. Based on the derived solitary wave excitation, we reveal some regular fractal and stochastic fractal patterns in the (2 + 1)-dimensional GBK system.     

Solitons with fission and fusion behaviors in a variable coefficient Broer–Kaup system

Using an extended homogeneous balance approach and a linear variable separation method, a general variable separation excitation of the (2 + 1)-dimensional variable coefficient Broer–Kaup (VCBK) system is derived. Based on the derived solution, we reveal soliton fission and fusion phenomena in the (2 + 1)-dimensional soliton system.     

Construction of new exact rational form non-travelling wave solutions to the (2 + 1)-dimensional generalized Broer-Kaup system

With the aid of symbolic computation Maple, several new families of rational form variable separation solutions with three arbitrary functions to the (2 + 1)-dimensional generalized Broer-Kaup system are derived by using an improved mapping approach and a variable separation approach. These solutions include rational solitary wave solutions, periodic wave solutions and rational wave solutions. The properties of the novel localized excitation are revealed by some figures.     

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.     

Study on the behavior of oscillating solitons using the (2+1)-dimensional nonlinear system

By means of an extended homogeneous balance method and a variable separation hypothesis, a broad general variable separation solution with three specific arbitrary functions of the nonlinear (2+1)-dimensional Broer-Kaup (BK) equations was derived. Based on the derived solution, a number of abundant oscillating solitons, such as dromion, multi-dromion, solitoff, ring, multi-lump and so on, have been revealed in this study by selecting appropriate functions of the general variable separation solution.     

推广的（2＋1）维浅水波方程的精确解及其混沌行为

利用Riccati方程映射法和变量分离法,得到了推广的（2＋1）维浅水波系统的变量分离解（包括孤波解、周期波解和有理函数解）.根据得到的孤波解,构造出了方程的单孤子和双孤子结构,研究了孤子的混沌行为.     

New variable separation solutions and nonlinear phenomena for the (2+1)-dimensional modified Korteweg–de Vries equation

Variable separation approach, which is a powerful approach in the linear science, has been successfully generalized to the nonlinear science as nonlinear variable separation methods. The (2 + 1)-dimensional modified Korteweg–de Vries (mKdV) equation is hereby investigated, and new variable separation solutions are obtained by the truncated Painlevé expansion method and the extended tanh-function method. By choosing appropriate functions for the solution involving three low-dimensional arbitrary functions, which is derived by the truncated Painlevé expansion method, two kinds of nonlinear phenomena, namely, dromion reconstruction and soliton fission phenomena, are discussed.     

New (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources derived by the nonlinear variable separation method

By using the Hirota’s bilinear transformation method and direct variable separation assumption, a new (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources is derived for the first time. Correspondingly, a nonlinear variable separation solution included two lower-dimensional arbitrary functions is obtained.     

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.     

Periodic structures based on variable separation solution of the (2+1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, firstly, a new mapping method is used to obtain the variable separation solutions, with two arbitrary functions, of the (2+1)-dimensional Boiti–Leon–Pempinelli equation. From the variable separation solution and by selecting appropriate functions, some novel Jacobian elliptic wave structure and periodic wave evolutional behaviors are investigated.     

Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system

Starting from the extended tanh-function method based on mapping method, the variable separation solutions of the (2 + 1)-dimensional breaking soliton system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the derived variable separation solution, chaotic behaviors, i.e. periodic solution with chaotic behavior and chaotic peaked and compact line solitons, are investigated.     

New interaction property of (2 + 1)-dimensional localized excitations from Darboux transformation

Using the binary Darboux transformation for the (2 + 1)-dimensional dispersive long wave equation, the “universal” variable separable formula is extended in a different way. From the extended formula, much more abundant localized excitations with arbitrary boundary conditions for the dispersive long wave equation can be obtained. The results obtained via the multi-linear variable separation approach are only a special case of the first step binary Darboux transformation. Two special interacting solutions are explicitly given. Especially, one of the examples exhibits a new interacting phenomenon: a localized solitary wave (dromion) can force an extended wave (solitoff) go back.     

关于广义Beltrami方程组

该文考虑广义Beltrami方程组D~tf(x)H(x)Df(x)=J(x,f)~(2/n)G(x).(*)利用能量和变分方法,在矩阵H(x),G(x)∈S(n)满足一致椭圆型条件下,得到了(*)式所满足的齐次散度型椭圆方程DivA(x,Df(x))=0,并得到了(*)式的分量函数的弱单调性和Caccioppoli不等式.     

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.     

Integral estimates of lengths of level lines of rational functions and zolotarev’s problem

Integral estimates of lengths of level lines (lemniscates) of rational functions of a complex variable are obtained. These estimates are related to the problem of separation of compact sets by rational functions and to Zolotarev’s problem.     

On two types of discrete implications

This paper deals with two kinds of implications defined from t-norms, t-conorms and strong negations on a finite chain L: those defined through the expressions I(x, y) = S(N(x), T(x, y)) and I(x, y) = S(T(N(x), N(y)), y). They are called QL-implications and NQL-implications respectively. We mainly study those QL- and NQL-implications derived from smooth t-norms and smooth t-conorms. It is characterized when functions defined in these ways are implication functions, and their analytical expressions are given. It is proved that both kinds of implications agree. Some additional properties are studied like contrapositive symmetry, the exchange principle and others. In particular, it is proved that contrapositive symmetry holds if and only if S is the only Archimedean t-conorm on L, and T jointly with its N-dual t-conorm satisfy the Frank equation. Finally, some QL- and NQL-implications are also derived from non-smooth t-norms or non-smooth t-conorms and many examples are given showing that in this non-smooth case, QL- and NQL-implications remain strongly connected.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Nonlinear variable separation solutions of some mKdV-type equations and the unified construction method

From the viewpoint of inverse problem research, by using the Hirota’s bilinear transformation method and direct variable separation ansatz, some mKdV-type equations with nonlinear variable separation solutions are unified constructed. These exact special solutions include several lower-dimensional arbitrary functions which can be used to express various soliton structures.     

Multisoliton excitations for the Kadomtsev–Petviashvili equation and the coupled Burgers equation

In high dimensions, there are abundant coherent soliton excitations. From the variable separation approach of the celebrated Kadomtsev–Petviashvili(KP) equation and the coupled Burgers equation, several new multisoliton excitations are derived and some novel features or interesting behaviors of these structures are revealed.     

Painlevé analysis,lump-kink solutions and localized excitation solutions for the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is investigated. The integrability test is performed yielding a positive result. Through the Painlevé–Bäcklund transformation, we derive four types of lump-kink solutions composed of two quadratic functions and N exponential functions. It is shown that fission and fusion interactions occur in the lump-kink solutions. Furthermore, a new variable separation solution with two arbitrary functions is obtained, the localized excitations including lumps, dromions and periodic waves are analyzed by some graphs.