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International Journal of Differential Equations
Volume 2011, Article ID 451420, 11 pages
http://dx.doi.org/10.1155/2011/451420
Research Article

Extended Jacobi Elliptic Function Expansion Method to the ZK-MEW Equation

School of Mathematics, Jiaying University, Meizhou, Guangdong 514015, China

Received 3 May 2011; Accepted 25 June 2011

Academic Editor: Wen Xiu Ma

Copyright © 2011 Weimin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

1. Introduction

It is one of the most important tasks to seek the exact solutions of nonlinear equation in the study of the nonlinear equations. Up to now, many powerful methods have been developed such as inverse scattering transformation [1], Backlund transformation [2], Hirota bilinear method [3], homogeneous balance method [4], extended tanh-function method [5], Jacobi elliptic function expansion method [6] and Ma’s transformed rational function method [7].

Recently [8], an extended-tanh method is used to establish exact travelling wave solution of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. In this paper, an extended Jacobi elliptic function expansion method is employed to construct some new exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation.

As known, the Zakharov-Kuznetsov (ZK) equation are given by 𝑢𝑡+𝑎𝑢𝑢𝑥+𝑢𝑥𝑥𝑥𝑥+𝑢𝑦𝑦𝑢=0,(1.1)𝑡+𝑎𝑢𝑢𝑥+2𝑢𝑥=0,(1.2) where 2=𝜕2𝑥+𝜕2𝑦+𝜕2𝑧 is the isotropic Laplacian. The ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [9]. In [9], the ZK equation is solved by the sine-cosine and the tanh-function methods. In [10], the numbers of solitary waves, periodic waves, and kink waves of the modified Zakharov-Kuznetsov equation are obtained.

The regularized long wave (RLW) equation given by 𝑢𝑡+𝑢𝑥+12𝑢2𝑥𝑢𝑥𝑥𝑡=0,<𝑥<+,𝑡>0,(1.3) appears in many physical applications and has been studied in [11]. Gardner et al. [12] solved the equal width equation by a Petrov Galerkin method using quadratic B-spline spatial finite-elements.

The modified equal width (MEW) equation given by𝑢𝑡+3𝑢2𝑢𝑥𝛽𝑢𝑥𝑥𝑡=0,(1.4) has been discussed in [11]. The MEW equation is related to the RLW equation. This equation has solitary waves with both positive and negative amplitudes. The two-dimensional ZK-MEW equation which first appeared in [13] is given by 𝑢𝑡𝑢+𝑎3𝑥+𝑏𝑢𝑥𝑡+𝑟𝑢𝑦𝑦𝑥=0,(1.5) where 𝑢=𝑢(𝑥,𝑦,𝑡), 𝑎,𝑏,𝑟 are constants. In [13], some exact solutions of the ZK-MEW equation (1.5) was obtained by using the tanh and sine-cosine methods. More detailed description for ZK-MEW equation (1.5) the reader can find in paper [13]. In this paper, we will give some new solutions of Jacobi elliptic function type of ZK-MEW equation by using an extended Jacobi elliptic function method.

The remainder of the paper is organized as follows. In Section 2, we briefly describe the extended Jacobi elliptic function expansion method. In Section 3, we apply this method to ZK-MEW equation to construct exact solutions. Finally, some conclusions are given in Section 4.

2. The Extended Jacobi Elliptic Function Expansion Method

In this section, the extended Jacobi elliptic function expansion method is proposed in [14]. Consider a given nonlinear wave equation, say in two variables𝐹𝑢,𝑢𝑥,𝑢𝑡,𝑢𝑥𝑥,𝑢𝑥𝑡,𝑢𝑡𝑡,=0.(2.1) We make the transformation𝑢=𝑢(𝑥,𝑡)=𝑈(𝜉),𝜉=𝑘(𝑥𝑐𝑡),(2.2) where 𝑘 is a constant to be determined later. Then (2.1) reduceds to a nonlinear ordinary differential equation (ODE) under(2.2)𝐺𝑈,𝑈𝜉,𝑈𝜉𝜉,𝑈𝜉𝜉𝜉,=0.(2.3) By the extended Jacobi elliptic function expansion method, introduce the following ansatz 𝑈(𝜉)=𝑁𝑗=𝑀𝑎𝑗𝑌𝑗(𝜉),(2.4) where 𝑀,𝑁,𝑎𝑗(𝑗=𝑀,,𝑁) are constants to be determined later, 𝑌 is an Jacobi elliptic function, namely, 𝑌=𝑌(𝜉)=𝑠𝑛𝜉=𝑠𝑛(𝜉,𝑚)or𝑐𝑛(𝜉,𝑚)or𝑑𝑛(𝜉,𝑚), 𝑚(0<𝑚<1) is the modulus of Jacobian elliptic functions. Positive integer 𝑀,𝑁 can be determined by balancing the highest-order linear term with the nonlinear term in (2.3). After this, substituting (2.4) into (2.3), we can obtain a system of algebraic equations for 𝑎𝑗(𝑗=𝑀,,𝑁). Solving the above-mentioned equations with the Mathematica Software, then 𝑎𝑗(𝑗=𝑀,,𝑁) can be determined. Substituting these obtained results into (2.4), then a general form of Jacobi elliptic function solution of (2.1) can be given.

3. ZK-MEW Equation

In this section, we employ the extended Jacobi elliptic function expansion method to ZK-MEW equation that is given by (1.5). The transformation 𝑢=𝑈(𝜉),𝜉=𝜆(𝑥+𝜇𝑦𝜎𝑡) converts (1.5) into ODE𝜎𝑈𝜉𝑈+𝑎3𝜉+𝑟𝜆2𝜇2𝑏𝜆2𝜎𝑈𝜉𝜉𝜉=0.(3.1) Integrating (3.1) and setting the constant of integration to zero, we obtain𝑈𝜎𝑈+𝑎3+𝑟𝜆2𝜇2𝑏𝜆2𝜎𝑈𝜉𝜉=0.(3.2) Substituting (2.4) into (3.2) to balance 𝑈3 with 𝑈𝜉𝜉, we find 𝑀=𝑁=1. Thus, the solution admits in the form𝑈(𝜉)=𝑎1𝑌1(𝜉)+𝑎0+𝑎1𝑌(𝜉),(3.3) where 𝑎1,𝑎0,𝑎1 are constants to be determined later.

Notice that𝑑(𝑠𝑛𝜉)𝑑𝜉=𝑠𝑛𝜉=𝑐𝑛𝜉𝑑𝑛𝜉,𝑑(𝑐𝑛𝜉)𝑑𝑑𝜉=𝑐𝑛𝜉=𝑠𝑛𝜉𝑑𝑛𝜉,(𝑑𝑛𝜉)𝑑𝜉=𝑑𝑛𝜉=𝑚𝑠𝑛𝜉𝑐𝑛𝜉,𝑐𝑛2𝜉=1𝑠𝑛2𝜉,𝑑𝑛2𝜉=1𝑚𝑠𝑛2𝜉.(3.4)

3.1. The Case of 𝑌=𝑌(𝜉)=𝑠𝑛𝜉=𝑠𝑛(𝜉,𝑚)

Substituting 𝑌=𝑌(𝜉)=𝑠𝑛𝜉=𝑠𝑛(𝜉,𝑚) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for 𝑎1,𝑎0,𝑎1, and 𝜆 of the following form:2𝜆2𝑟𝜇2𝑎𝑏𝜎1+𝑎𝑎31=0,3𝑎𝑎21𝑎0=0,𝑎1(1+𝑚)𝑟𝜆2𝜇2𝜎+𝑏(1+𝑚)𝜆2𝜎2𝑎𝑎20+3𝑎𝑎21𝑎1=0,𝜎𝑎0+𝑎𝑎30+6𝑎𝑎1𝑎0𝑎1=0,(1𝑚)𝑟𝜆2𝜇2𝑎1+1+𝑏(1+𝑚)𝜆2𝜎𝑎1+3𝑎𝑎20𝑎1+3𝑎𝑎1𝑎21=0.(3.5) Solving the system of the algebraic equations with the aid of Mathematica we can distinguish two cases, namely the following.

Case 1. 𝑎1=0,𝑎0=0,𝑎1=2𝜎𝑎(1+𝑚),𝜆=𝜎(1+𝑚)𝑏𝜎𝑟𝜇2.(3.6)

Case 2. 𝑎1=𝜎(1+𝑚)𝑎,𝑎1=𝜎𝑎(1+𝑚),𝑎0=0,𝜆=𝜎2(1+𝑚)𝑏𝜎𝑟𝜇2.(3.7)

Substituting (3.6), (3.7) into (3.3), respectively, yield, the following solutions of ZK-MEW equation:𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎(1+𝑚)𝑠𝑛1𝜎(1+𝑚)𝑏𝜎𝑟𝜇2,(𝑥+𝜇𝑦𝜎𝑡),𝑚(3.8)𝑢(𝑥,𝑦,𝑡)=𝜎(1+𝑚)𝑎𝑠𝑛𝜎2(1+𝑚)𝑏𝜎𝑟𝜇2(+𝑥+𝜇𝑦𝜎𝑡),𝑚𝜎𝑎(1+𝑚)𝑠𝑛1𝜎2(1+𝑚)𝑏𝜎𝑟𝜇2.(𝑥+𝜇𝑦𝜎𝑡),𝑚(3.9) Notice that 𝑚1, 𝑠𝑛𝜉tanh𝜉, and 𝑚0, 𝑠𝑛𝜉sin𝜉, we can obtain solitary wave solutions and sin-wave solutions from (3.8) and (3.9), respectively,𝑢(𝑥,𝑦,𝑡)=𝜎𝑎coth𝜎2𝑏𝜎𝑟𝜇2,(𝑥+𝜇𝑦𝜎𝑡)(3.10a)𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎sin1𝜎𝑏𝜎𝑟𝜇2(𝑥+𝜇𝑦𝜎𝑡),(3.10b)𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎tanh𝜎2𝑏𝜎𝑟𝜇2+(𝑥+𝜇𝑦𝜎𝑡)𝜎2𝑎coth𝜎2𝑏𝜎𝑟𝜇2(,𝑥+𝜇𝑦𝜎𝑡)(3.11a)𝑢(𝑥,𝑦,𝑡)=𝜎𝑎sin𝜎2𝑏𝜎𝑟𝜇2+(𝑥+𝜇𝑦𝜎𝑡)𝜎𝑎sin1𝜎2𝑏𝜎𝑟𝜇2(.𝑥+𝜇𝑦𝜎𝑡)(3.11b)

Here we only give the graph of (3.8) (see Figure 1) and (3.10a) (see Figure 2) and the other graphs of equations are similar to discussing.

fig1
Figure 1: (a) The graph of (3.8) with 𝑎=𝑟=𝜎=1,𝑏=2,𝑚=0.5 at 𝑡=1, and (b) is its plane when 𝑦=0.
fig2
Figure 2: (a) The graph of (3.10a) with 𝑎=𝑟=𝜎=1,𝑏=2 at 𝑡=1 and (b) is its plane when 𝑦=0 (the case is the same to (3.8) when 𝑚=1).
3.2. The Case of 𝑌=𝑌(𝜉)=𝑑𝑛𝜉=𝑑𝑛(𝜉,𝑚)

The analysis proceeds of this case is as for Section 3.1. Substituting 𝑌=𝑌(𝜉)=𝑑𝑛𝜉=𝑑𝑛(𝜉,𝑚) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for 𝑎1,𝑎0,𝑎1 and 𝜆 of the following form:2(1+𝑚)𝜆2𝑟𝜇2𝑎𝑏𝜎1+𝑎𝑎31=0,3𝑎𝑎21𝑎0=0,𝑎1(2+𝑚)𝑟𝜆2𝜇2+𝜎+2𝑏𝜆2𝜎𝑏𝑚𝜆2𝜎3𝑎𝑎20+3𝑎𝑎21𝑎1=0,𝜎𝑎0+𝑎𝑎30+6𝑎𝑎1𝑎0𝑎1=0,(2𝑚)𝑟𝜆2𝜇2𝑎1+1+𝑏(2+𝑚)𝜆2𝜎𝑎1+3𝑎𝑎20𝑎1+3𝑎𝑎1𝑎21𝑎=0,12𝜆2𝑟𝜇2+𝑏𝜎+𝑎𝑎21=0.(3.12) Solving the system of the algebraic equations with the aid of Mathematica we can distinguish three cases, namely, The following.

Case 1. 𝑎1=0,𝑎0=0,𝑎1=2𝜎𝑎(2𝑚),𝜆=𝜎(2𝑚)𝑟𝜇2𝑏𝜎.(3.13)

Case 2. 𝑎1=2(1𝑚)𝜎𝑎261𝑚𝑚,𝑎0=0,𝑎1=2𝜎𝑎26,1𝑚𝑚𝜆=𝜎261𝑚𝑚𝑏𝜎𝑟𝜇2.(3.14)

Case 3. 𝑎1=2(1𝑚)𝜎𝑎(2𝑚),𝑎0=0,𝑎1=0,𝜆=𝜎(2𝑚)𝑟𝜇2𝑏𝜎.(3.15)

Substituting (3.13), (3.14), and (3.15) into (3.3), respectively, yields the following solutions of ZK-MEW equation:𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎(2𝑚)𝑑𝑛𝜎(2𝑚)𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),𝑚,(3.16)𝑢(𝑥,𝑦,𝑡)=2(1𝑚)𝜎𝑎261𝑚𝑚𝑑𝑛1𝜎261𝑚𝑚𝑏𝜎𝑟𝜇2+(𝑥+𝜇𝑦𝜎𝑡),𝑚2𝜎𝑎261𝑚𝑚𝑑𝑛𝜎261𝑚𝑚𝑏𝜎𝑟𝜇2(,𝑥+𝜇𝑦𝜎𝑡),𝑚(3.17)𝑢(𝑥,𝑦,𝑡)=2(1𝑚)𝜎𝑎(2𝑚)𝑑𝑛1𝜎(2𝑚)𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),𝑚.(3.18) Notice that 𝑚1, 𝑑𝑛𝜉sech𝜉, thus we can obtain solitary wave of solutions of ZK-MEW equation from (3.16) and (3.17), respectively,𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎sech𝜎±𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡).(3.19) Here we only give the graph of (3.17) (see Figure 3) and (3.19) (see Figure 4) and the other graphs of equations are similar to discussing.

fig3
Figure 3: (a) The graph of (3.17) with 𝑎=𝑟=𝜎=1,𝑏=2,𝑚=0.99 at 𝑡=1, and (b) is its plane when 𝑦=0.
fig4
Figure 4: (a) The solitary wave graph of (3.19) with 𝑎=𝑟=𝜎=1,𝑏=2 at 𝑡=1, and (b) is its plane when 𝑦=0 (the case is the same to (3.17) when 𝑚=1).
3.3. The Case of 𝑌=𝑌(𝜉)=𝑐𝑛𝜉=𝑐𝑛(𝜉,𝑚)

The analysis proceeds of this case is as for Sections 3.1 and 3.2. Substituting 𝑌=𝑌(𝜉)=𝑐𝑛𝜉=𝑐𝑛(𝜉,𝑚) and (3.3) into (3.2), making use of (3.4), we obtain a system of algebraic equations, for 𝑎1,𝑎0,𝑎1, and 𝜆 of the following form:2(1+𝑚)𝜆2𝑟𝜇2𝑎𝑏𝜎1+𝑎𝑎31=0,3𝑎𝑎21𝑎0𝑎=0,1(1+2𝑚)𝑟𝜆2𝜇2𝜎+𝑏𝜎(12𝑚)𝜆2+3𝑎𝑎20+3𝑎𝑎21𝑎1=0,𝜎𝑎0+𝑎𝑎30+6𝑎𝑎1𝑎0𝑎1=0,(1+2𝑚)𝑟𝜆2𝜇2𝑎1𝜎𝑎1+𝑏𝜎(12𝑚)𝜆2𝑎1+3𝑎𝑎20𝑎1+3𝑎𝑎21𝑎1=0,3𝑎𝑎21𝑎0𝑎=0,12𝑚𝜆2𝑟𝜇2+𝑏𝜎+𝑎𝑎21=0.(3.20) Solving the system of the algebraic equations (3.20) with the aid of Mathematica, we can distinguish three cases, namely, the following.

Case 1. 𝑎0=0,𝑎1=2𝑚𝜎𝑎(1+2𝑚),𝑎1=0,𝜆=𝜎(1+2𝑚)𝑟𝜇2𝑏𝜎.(3.21)

Case 2. 𝑎0=0,𝑎1=2(1𝑚)𝜎𝑎(1+2𝑚),𝑎1=0,𝜆=𝜎(1+2𝑚)𝑟𝜇2𝑏𝜎.(3.22)

Case 3. 𝑎0=𝜎𝑎,𝑎1=2𝑚𝜎𝑎(12𝑚),𝑎1=0,𝜆=2𝜎(12𝑚)𝑟𝜇2𝑏𝜎.(3.23)

Substituting (3.21), (3.22), and (3.23) into (3.3), respectively, yield the following solutions of ZK-MEW equation:𝑢(𝑥,𝑦,𝑡)=2𝑚𝜎𝑎(1+2𝑚)𝑐𝑛𝜎(1+2𝑚)𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),𝑚,(3.24)𝑢(𝑥,𝑦,𝑡)=2(1𝑚)𝜎𝑎(1+2𝑚)𝑐𝑛1𝜎(1+2𝑚)𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),𝑚,(3.25)𝑢(𝑥,𝑦,𝑡)=𝜎𝑎2𝑚𝜎𝑎(12𝑚)𝑐𝑛2𝜎(12𝑚)𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),𝑚.(3.26) Especially, when 𝑚1, 𝑐𝑛𝜉sech𝜉 and 𝑚0, 𝑐𝑛𝜉cos𝜉, thus we can obtain solutions of ZK-MEW equation from (3.24) and(3.25)𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎sech𝜎𝑟𝜇2𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡),(3.27)𝑢(𝑥,𝑦,𝑡)=2𝜎𝑎cos1𝜎𝑟𝜇2+𝑏𝜎(𝑥+𝜇𝑦𝜎𝑡).(3.28)

Remark 3.1. In these solutions, (3.10a), (3.19) and (3.27) have been obtain in [8], the others solutions are new solutions for the ZK-MEW equation.

4. Conclusions

The extended Jacobi elliptic function expansion method was directly and effectively employed to find travelling wave solutions of the nonlinear ZK-MEW equation. Using the method, we found some new solutions of Jacobi elliptic function type that were not obtained by the sine-cosine method, the extended tanh-method, the mapping method, and other methods. In the limiting case of the Jacobi elliptic function (namely, modulus setting 0 or 1), we also obtained the solutions of sin-type, cos-tye, tanh-type, and sech-type. The extended Jacobi elliptic function expansion method can be applied to some other nonlinear equation and gives more solutions.

The ZK-MEW equation was first appeared in Wazwaz’s paper [13] in 2005. To my acknowledge, its many properties, such as integrability, Lax pairs, and multisoliton solutions, have not been studied. The study of these properties is a very signification work and is our task research in the future.

Acknowledgments

The author would like to thank the anonymous referees for helpful suggestions that served to improve the paper. This paper is supported by Jiaying University (2011KJZ01).

References

  1. M. J. Ablowitz and H. Segur, Soliton and the Inverse Scattering Transformation, SIAM, Philadelphia, Pa, USA, 1981.
  2. L. Tian and J. Yin, “Stability of multi-compacton solutions and Bäcklund transformation in K(m,n,1),” Chaos, Solitons and Fractals, vol. 23, no. 1, pp. 159–169, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. R. Hirota, “Exact solution of the korteweg—de vries equation for multiple Collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. View at Publisher · View at Google Scholar
  4. M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, no. 1–5, pp. 67–75, 1996. View at Google Scholar
  5. E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters. A, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters. A, vol. 289, no. 1-2, pp. 69–74, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. W.-X. Ma and J.-H. Lee, “A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1356–1363, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. M. Inc, “New exact solutions for the ZK-MEW equation by using symbolic computation,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 508–513, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A.-M. Wazwaz, “Exact solutions with solitons and periodic structures for the Zakharov-Kuznetsov (ZK) equation and its modified form,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 6, pp. 597–606, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. X. Zhao, H. Zhou, Y. Tang, and H. Jia, “Travelling wave solutions for modified Zakharov-Kuznetsov equation,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 634–648, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A.-M. Wazwaz, “The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 2, pp. 148–160, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein, “Simulations of the EW undular bore,” Communications in Numerical Methods in Engineering, vol. 13, no. 7, pp. 583–592, 1997. View at Google Scholar
  13. A.-M. Wazwaz, “Exact solutions for the ZK-MEW equation by using the tanh and sine-cosine methods,” International Journal of Computer Mathematics, vol. 82, no. 6, pp. 699–708, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. H. Zhang, “Extended Jacobi elliptic function expansion method and its applications,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 5, pp. 627–635, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH