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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

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Volume 70, Issue 10

Issues

Some Remarks on the Riccati Equation Expansion Method for Variable Separation of Nonlinear Models

Yu-Peng Zhang
  • School of Sciences, Zhejiang Agriculture and Forestry University, Lin’an, Zhejiang 311300, P.R.China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Chao-Qing Dai
  • Corresponding author
  • School of Sciences, Zhejiang Agriculture and Forestry University, Lin’an, Zhejiang 311300, P.R.China
  • Key Laboratory of Chemical Utilization of Forestry Biomass of Zhejiang Province, Zhejiang A & F University, Lin’an, Zhejiang 311300, P.R.China
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  • Other articles by this author:
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Published Online: 2015-08-19 | DOI: https://doi.org/10.1515/zna-2015-0243

Abstract

Based on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms of (2+1)-dimensional modified Korteweg–de Vries equation are obtained. The following two remarks on the Riccati equation expansion method for variable separation are made: (i) a remark on the equivalence of different solutions constructed by the Riccati equation expansion method. From analysis, we find that these seemly independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations. We should avoid arbitrarily asserting so-called “new” solutions; (ii) a remark on the construction of localised excitation based on variable separation solutions. For two or multi-component systems, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures. We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering and biophysical contexts.

Keywords: (2+1)-Dimensional Modified Korteweg–de Vries Equation; Riccati Equation Expansion Method; Un-physical Structure; Variable Separation

PACS Numbers: 05.45.Yv; 02.30.Jr; 02.30.Ik

1 Introduction

The Korteweg–de Vries (KdV) equation and its relatives become a significant model to describe waves on shallow water surfaces [1], blood waves in blood vessels [2], and ion acoustic waves in a plasma [3], etc. Exact solutions of a modified KdV (mKdV) equation were used to describe wave motions in physical, engineering, and biophysical areas. Many important and effective methods to obtain exact solutions of nonlinear models have been presented such as a self-similar method [4, 5], a mapping approach [6, 7], a direct Bosonisation approach [8], and the Bäcklund transformation method [9] and so on.

Among these methods, the Riccati equation expansion method, which is used to obtain travelling wave solutions, has been extended to obtain variable separation solutions of (2+1)-dimensional nonlinear models [10–12]. Recently, many authors [13–16] pointed out many so-called “new” exact solutions of nonlinear models reported in some literature are not indeed new solutions, and they only possess different forms. Moreover, we have also studied the equivalence of various variable separation solutions derived by the extended tanh-function method [17] and the projective Ricatti equation method [18]. However, up to now, researchers have not studied whether different forms of variable separation solutions constructed by the Riccati equation expansion method are also equivalent.

Based on variable separation solutions, abundant localised structures and their interaction behaviours were investigated [10–12]. However, Ruan [19] has mentioned that one field component for nonlinear models has abundant localised structures, while the other field component of the same equation exists the divergent phenomena for the (2+1)-dimensional nonlinear models.

In this study, we will further focus on some points worthy of remark about localised structures constructed based on variable separation solutions and consider the following (2+1)-dimensional mKdV equation [20, 21]

ut+uxxx3uxuxx2u+3ux34u2+2Avxu+2Auxv=0,ux=vy, (1)(1)

where A is an arbitrary constant. From the study of Zhang and Shen [20], the Painlevé integrable of (1) was studied. In the study of Liang et al. [21], the authors obtained variable separation solutions and discussed dromion reconstruction and soliton fission phenomena.

2 Variable Separation Solutions of (2+1)-Dimensional mKdV Equation

Equation (1) can be rewritten as follows:

4u2ut+4u2uxxx6uuxuxx+3ux3+8Au3vx+8Au2uxv=0,ux=vy. (2)(2)

We assume that (2) has the following ansatz:

u=a0(x,y,t)+i=1m{ai(x,y,t)ϕi[R(x,y,t)]+bi(x,y,t)ϕi[R(x,y,t)]+ci(x,y,t)ϕi1[R(x,y,t)]ϕ[R(x,y,t)]},v=d0(x,y,t)+i=1n{dj(x,y,t)ϕj[R(x,y,t)]+ej(x,y,t)ϕj[R(x,y,t)]+fj(x,y,t)ϕj1[R(x,y,t)]ϕ[R(x,y,t)]}, (3)(3)

where a0, ai, bi, ci, dj, ej, and fj are all functions of {x, y, t}, and ϕ is a solution of the Riccati equation

ϕdϕdR=l0+l1ϕ+l2ϕ2, (4)(4)

whose four types and 27 solutions were listed in [10], and here, we neglect them.

Balancing the highest-order partial differential terms and the highest nonlinear terms in (2), we have m=n=2. Substituting ansatz (3) with (4) into (2), selecting the variable separation form R=p(x, t)+q(y) and eliminating all the coefficients of polynomials of ϕ and l0+l1ϕ(R)+l2ϕ2(R) yields two families of solutions

Family 1

b1=c1=b2=c2=e1=f1=e2=f2=0,a0=3l0l2pxqy2A,a1=3l1l2pxqy2A,a2=3l22pxqy2A,d0=6l1pxxpx23pxx2+(8l0l2+l12)px4+4pxpt+4pxpxxx8Apx2,d1=3l2(l1px2+pxx)2A,d2=3l22px22A, (5)(5)

where pp(x, t) and q=q(y) are arbitrary functions of {x, t} and {y}, respectively.

Family 2

a1=b1=b2=c1=e1=e2=0,a0=3l0l2pxqy4A,a2=3l22pxqy4A,c2=3l23/2pxqy4A,d0=12pxx220l0l2px416pxpt16pxpxxx32Apx2,d1=3l2pxx4A,d2=3l22px24A,f1=3l21/2pxx4A,f2=3l23/2px24A, (6)(6)

with l1=0, pp(x, t), and q=q(y) are arbitrary functions of {x, t} and {y}, respectively.

Family 3

a1=b1=c1=c2=f1=f2=0,a0=3l0l2pxqyA,a2=3l22pxqy2A,b2=3l02pxqy2A,d0=3pxx28l0l2px44pxpt4pxpxxx8Apx2,d1=3l2pxx2A,d2=3l22px22A,e1=3l0pxx2A,e2=3l02px22A, (7)(7)

with l1=0, pp(x, t), and q=q(y) are arbitrary functions of {x, t} and {y}, respectively.

The variable separation solutions of the (2+1)-dimensional mKdV equation can be written as follows:

Family 1

From the above result (5) and some typical solutions of (4) in Zhu [10], we have

Case 1: For l124l0l2>0 and l0l1 ≠ 0 (or l0l2 ≠ 0),

Solution 1

u1=3l0l2pxqy2A+3l1pxqy4A{l1+l124l0l2tanh[l124l0l22(p+q)]}3pxqy8A{l1+l124l0l2tanh[l124l0l22(p+q)]}2,v1=6l1pxxpx23pxx2+(8l0l2+l12)px4+4pxpt+4pxpxxx8Apx2+3(l1px2+pxx)4A×{l1+l124l0l2tanh[l124l0l22(p+q)]}3px28A{l1+l124l0l2tanh[l124l0l22(p+q)]}2, (8)(8)

Solution 2

u2=3l0l2pxqy2A+3l1pxqy4A{l1+l124l0l2coth[l124l0l22(p+q)]}3pxqy8A{l1+l124l0l2coth[l124l0l22(p+q)]}2,v2=6l1pxxpx23pxx2+(8l0l2+l12)px4+4pxpt+4pxpxxx8Apx2+3(l1px2+pxx)4A×{l1+l124l0l2coth[l124l0l22(p+q)]}3px28A{l1+l124l0l2coth[l124l0l22(p+q)]}2, (9)(9)

Case 2: For l124l0l2<0 and l0l1 ≠ 0 (or l0l2 ≠ 0),

Solution 3

u3=3l0l2pxqy2A3l1pxqy4A{l1+4l0l2l12tan[4l0l2l122(p+q)]}3pxqy8A{l1+4l0l2l12tan[4l0l2l122(p+q)]}2,v3=6l1pxxpx23pxx2+(8l0l2+l12)px4+4pxpt+4pxpxxx8Apx23(l1px2+pxx)4A×{l1+4l0l2l12tan[4l0l2l122(p+q)]}3px28A{l1+4l0l2l12tan[4l0l2l122(p+q)]}2, (10)(10)

Solution 4

u4=3l0l2pxqy2A+3l1pxqy4A{l1+4l0l2l12cot[4l0l2l122(p+q)]}3pxqy8A{l1+4l0l2l12cot[4l0l2l122(p+q)]}2,v4=6l1pxxpx23pxx2+(8l0l2+l12)px4+4pxpt+4pxpxxx8Apx2+3(l1px2+pxx)4A×{l1+4l0l2l12cot[4l0l2l122(p+q)]}3px28A{l1+4l0l2l12cot[4l0l2l122(p+q)]}2, (11)(11)

Case 3: For l2 ≠ 0 and l0=l1=0,

Solution 5

u5=3pxqy2A(p+q)2,v5=3pxx24pxpt4pxpxxx8Apx2+3pxx2A(p+q)3px22A(p+q)2, (12)(12)

where pp(x, t) and q=q(y) are arbitrary functions of {x, t} and {y}, respectively.

Family 2

From the above result (6) and some typical solutions of (4) with l1=0 in Dai and Wang [22], we have

Case 1: For l0l2=−1,

Solution 6

u6=3pxqy4A3pxqy4Atanhl0(p+q)[tanhl0(p+q)+isechl0(p+q)],v6=12pxx2+20px416pxpt16pxpxxx32Apx2+3pxx4A[tanhl0(p+q)+isechl0(p+q)]3px24Atanhl0(p+q)[tanhl0(p+q)+isechl0(p+q)], (13)(13)

Solution 7

u7=3pxqy4A3pxqy4Acothl0(p+q)[cothl0(p+q)cschl0(p+q)],v7=12pxx2+20px416pxpt16pxpxxx32Apx2+3pxx4A[cothl0(p+q)cschl0(p+q)]3px24Acothl0(p+q)[cothl0(p+q)cschl0(p+q)], (14)(14)

Case 2: For l0l1=1,

Solution 8

u8=3pxqy4A3pxqy4Atanl0(p+q)[tanl0(p+q)]+secl0(p+q)],v8=12pxx220px416pxpt16pxpxxx32Apx23pxx4A[tanl0(p+q)+secl0(p+q)]3px24Atanl0(p+q)[tanl0(p+q)+secl0(p+q)], (15)(15)

Solution 9

u9=3pxqy4A3pxqy4Acotl0(p+q)[cotl0(p+q)cscl0(p+q)],v9=12pxx220px416pxpt16pxpxxx32Apx2+3pxx4A[cscl0(p+q)cotl0(p+q)]3px24Acotl0(p+q)[cscl0(p+q)cotl0(p+q)], (16)(16)

where pp(x, t) and q=q(y) are arbitrary functions of {x, t} and {y}, respectively. Moreover, functions in solutions (13)–(16) are l-deformed functions [23], whose properties will be recalled, that is,

sinhl(R)=eRleR2,coshl(R)=eR+leR2,tanhl(R)=sinhl(R)coshl(R), sechl(R)=1coshl(R),RC. (17)(17)

It is straightforward to see that the following formulas hold,

(sinhl(R))=coshl(R),(coshl(R))=sinhl(R),coshl2(R)sinhl2(R)=l,(tanhl(R))=lsechl2(R),(sechl(R))=tanhl(R)sechl(R),tanhl2(R)=1lsechl2(R),cothl2(R)=1+lcschl2(R). (18)(18)

Correspondingly, we can define l-deformed triangular functions as follows:

sinl(R)=eiRleiR2i,cosl(R)=eiR+leiR2,tanl(R)=sinl(R)cosl(R),secl(R)=1cosl(R). (19)(19)

They satisfy the following formulas:

(sinl(R))=cosl(R),(cosl(R))=sinl(R),(tanl(R))=lsecl2(R),(secl(R))=tanl(R)secl(R),cosl2(R)+sinl2(R)=l,1+tanl2(R)=lsecl2(R),1+cotl2(R)=lcscl2(R). (20)(20)

Case 3: For l0=0, solution (12) can be obtained.

Family 3

From the above result (7) and some typical solutions of (4) with l1=0 in the study of Dai and Wang [22], we have

Case 1: For l0l2=−1,

u10=3pxqyA3pxqy2A[tanhl02(p+q)+tanhl02(p+q)],v10=3pxx2+8px44pxpt4pxpxxx8Apx2+3pxx2A[tanhl0(p+q)+tanhl01(p+q)]3px22A[tanhl02(p+q)+tanhl02(p+q)], (21)(21)

Case 2: For l0l2=1

u11=3pxqyA3pxqy2A[tanl02(p+q)+tanl02(p+q)],v11=3pxx28px44pxpt4pxpxxx8Apx23pxx2A[tanl0(p+q)+tanl01(p+q)]3px22A[tanl02(p+q)]tanl02(p+q)], (22)(22)

where pp(x, t) and q=q(y) are arbitrary functions of {x, t} and {y} respectively, l-deformed functions have the properties in (17)–(20).

Case 3: For l0=0, we can derive solution (12).

3 Remark on Different Forms of Variable Separation Solutions

By means of the Riccati equation expansion method, we derive 11 kinds of different variable separation solutions of (2+1)-dimensional mKdV equation. The study indicates that these seem independent solutions have different forms, however, they actually depend on each other, and they can transform from one to another via some relations (Tab. 1).

From Table 1, solution (12) can transform into other solutions (8)–(11), (13)–(16), and (21), (22) by re-defining p and q. From this perspective, solutions with different forms are highly likely equivalent, we should avoid arbitrarily asserting so-called “new” solutions.

Table 1

Transformations from solution (12) into other solutions.

4 Remark on the Construction of Localised Excitation Based on Variable Separation Solutions

The authors investigated (2+1)-dimensional localised excitations based on variable separation solutions [11, 12, 17, 18, 21, 22]. However, Ruan [20] has pointed out that abundant (2+1)-dimensional localised structures can be constructed by one field component for nonlinear models, while the divergent phenomena exists for the other field component of the same (2+1)-dimensional nonlinear model. Here, we consider the similar phenomena of other localised excitations based on a variable separation solution (12).

4.1 Interaction Behaviours of Multi-Dromions

Zhou et al. [24] discussed multi-dromion structures based on a variable separation solution. The so-called dromion is a kind of solution exponentially decaying in all directions. Similarly, we can construct this structure based on variable separation solution (12). For the field component u with A=2, when p and q are chosen as follows:

p=k=MM0.4sech(x+5k+t),q=1+0.1l=NN0.4sech(y+5l),M=N=2, (23)(23)

we can obtain the multi-dromion structure shown in Figure 1a. However, when we consider the component v, the un-physical phenomenon will appear in Figure 1b. Therefore, although this multi-dromion structure is interesting for the component u, it lacks its value because the un-physical phenomenon appears for the component v with the same condition.

(a) Multi-dromion structure of the component u at time t=0, and (b) the un-physical phenomenon of the components ν corresponding to (a). All parameters are illustrated in the text.
Figure 1:

(a) Multi-dromion structure of the component u at time t=0, and (b) the un-physical phenomenon of the components ν corresponding to (a). All parameters are illustrated in the text.

In general, the interactions between solitons may be regarded as completely elastic. For instance, if p and q are considered to be

p=k=MM[0.4sech(x+5k+t)+0.6sech(x+5kt)],q=1+0.1l=NN0.4sech(y+5l),M=N=2, (24)(24)

we can obtain the interactions between two multi-dromions for the component u.

Figure 2a–c show interaction behaviours of two multi-dromions of the component u at time t=−20, −10, 20, respectively. These plots indicate that the shapes, amplitudes, and velocities of the two multi-dromions are completely conserved after their interactions. However, when we consider the components v, we find the divergent phenomena. As an example, we exhibit these phenomena corresponding to Figure 2c in Figure 2d. By analysis, this divergent and un-physical phenomenon comes from the appearance of singularity of the component v in solution (12) because of the zero value in the denominators of this term 3pxx24pxpt4pxpxxx8Apx2.

Interaction behaviours between two multi-dromions for the component u at time (a) t=−20, (b) t=−10, and (c) t=20. (d) the un-physical behaviours of the components ν corresponding to (c). All parameters are illustrated in the text.
Figure 2:

Interaction behaviours between two multi-dromions for the component u at time (a) t=−20, (b) t=−10, and (c) t=20. (d) the un-physical behaviours of the components ν corresponding to (c). All parameters are illustrated in the text.

4.2 Annihilation of Dromions

In the study of Qiang et al. [25], authors discussed the annihilation of dromions based on variable separation solution. Here, we can also study this phenomenon based on variable separation solution (12). If functions p and q are chosen as follows:

p=1+sech(x2+t),q=1+sech(y2), (25)(25)

we can study the annihilation of dromions for the physical quantity u of (12) under the condition (25) shown in Figure 3a–d with fixed parameters at time t=−7, 0, 2, 28 and A=2.

Annihilation of dromions for the component u at (a) t=−7, (b) t=0, (c) t=3, and (d) t=28 (e) and (f) the un-physical phenomena for the component ν corresponding to (a) and (d). All parameters are illustrated in the text.
Figure 3:

Annihilation of dromions for the component u at (a) t=−7, (b) t=0, (c) t=3, and (d) t=28 (e) and (f) the un-physical phenomena for the component ν corresponding to (a) and (d). All parameters are illustrated in the text.

In Figure 3a–d, we find that the amplitude and shape of dromions become smaller and smaller after interactions. Finally, they reduce to zero. Similarly, when we consider the component v, the un-physical and divergent phenomenon can be found again in Figure 3e and f. These divergent phenomena also originate from the appearance of singularity because of the zero value in the denominators of this term 3pxx24pxpt4pxpxxx8Apx2.

From the two cases given earlier, we should note these localised structures of (2+1)-dimensional two or multi-component system. In these systems, although we construct localised coherent structures for a special component based on variable separation solutions, these structures lack their value because the un-physical phenomenon appear in the other component for the same model with the same condition. Therefore, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures.

5 Summary

In summary, important results of this paper are given as follows:

  1. Based on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms are constructed.

  2. A remark on the equivalence of different solutions constructed by the Riccati equation expansion method is made. By analysis, we find that these seem independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations (Tab. 1). We should avoid arbitrarily asserting so-called “new” solutions.

  3. A remark on the construction of localised excitation based on variable separation solutions is made. For two or multi-component systems, although we construct localised coherent structures for a special component based on variable separation solutions, these structures lacks their value because the un-physical phenomenon appear in the other component for the same model with the same condition. Therefore, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures.

We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering, and biophysical contexts.

Acknowledgments

This study was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY13F050006), the National Natural Science Foundation of China (Grant No. 11375007), and the Scientific Research and Developed Fund of Zhejiang A & F University (Grant No. 2014FR020). Dr. Chao-Qing Dai is also sponsored by the Foundation of New Century “151 Talent Engineering” of Zhejiang Province of China and Youth Top-notch Talent Development and Training Program of Zhejiang A & F University.

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About the article

Corresponding author: Chao-Qing Dai, School of Sciences, Zhejiang Agriculture and Forestry University, Lin’an, Zhejiang 311300, P.R.China; and Key Laboratory of Chemical Utilization of Forestry Biomass of Zhejiang Province, Zhejiang A & F University, Lin’an, Zhejiang 311300, P.R.China, E-mail: dcq424@126.com


Received: 2015-05-29

Accepted: 2015-07-24

Published Online: 2015-08-19

Published in Print: 2015-10-01


Citation Information: Zeitschrift für Naturforschung A, Volume 70, Issue 10, Pages 835–842, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2015-0243.

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