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Comments on “New types of interactions between solitary waves in (2 + 1)-dimensions”
Chaos, Solitons and Fractals 32 (2007) 388–391 www.elsevier.com/locate/chaos
Discussion
Comments on ‘‘New types of interactions between solitary waves in (2 + 1)-dimensions’’ Chun-Long Zheng *, Zheng-Yi Ma College of Mathematics and Physics, Lishui University, Lishui, Zhejiang 323000, PR China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China Accepted 29 June 2006
Abstract Starting from a special Ba¨cklund transformation, Bai and Zhao [Bai CL, Zhao H. Chaos, Solitons & Fractals. doi: 10.1016/j.chaos.2006.02.005] have claimed that a general variable separation solution of the (2 + 1)-dimensional nonlinear Schro¨dinger equation is derived. Based on the quite universal variable separation solution which is valid for some (2 + 1)-dimensional physical models and by selecting appropriate functions, they have asserted that new types of interactions between the multi-valued solitary waves and the single-valued solitary waves, i.e., the fractal semifolded localized structures, are investigated. We show that several serious wrongs exist in their paper, and say nothing of the conclusion that some new fractal semifolded localized structures for the (2 + 1)-dimensional nonlinear Schro¨dinger equation have been found. Ó 2006 Elsevier Ltd. All rights reserved.
In Ref. [1], Bai et al. have considered the variable separated solution of the following (2 + 1)-dimensional nonlinear Schro¨dinger equation: uxx ¼ uQx iut ;
ð1Þ
uxx
ð2Þ ð3Þ
¼ u Qx þ iut ; Qy ¼ uu : According to the standard truncated Painleve´ expansion, they first obtained the Ba¨cklund transformation g u ¼ ; Q ¼ 2ðln f Þx þ p0 ðx; tÞ; f
ð4Þ
where p0 p0(x, t) is an arbitrary function of {x, t}, then they derived the following simple variable separated equations
q
DOI of original article: 10.1016/j.chaos.2006.02.005. Corresponding author. Address: College of Mathematics and Physics, Lishui University, Lishui, Zhejiang 323000, PR China. Tel.: +86 578 2271126; fax: +86 578 2271250. E-mail address: [email protected] (C.-L. Zheng). *
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.077
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
p1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2a1 a2 px ;
2px pxxx þ
p2xx
þ
q1 ¼
4p2x r2x
þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2a1 a2 qy ; 4rt p2x
þ
4p2x p0x
st ¼ 0; ¼ 0;
389
ð5Þ ð6Þ
pt þ 2px rx ¼ c1 ða2 þ a3 pÞ2 þ c2 ða2 þ a3 pÞ þ a1 a2 a3 ;
ð7Þ
qt ¼ c3 ða1 þ a3 qÞ2 c2 ða1 þ a3 qÞ a1 a2 c1 ;
ð8Þ
where c1, c2, c3 are arbitrary functions of t, f and g are ansatzs f ¼ a1 p þ a2 q þ a3 pq;
g ¼ p1 q1 expðir þ isÞ;
ð9Þ
where a1, a2, a3 are arbitrary constants, the functions p, p1 and r are only functions of {x, t}, and q, q1 and s are only functions of {y, t}. When c1, c2 and c3 are chosen as 1 ½ða1 þ a3 A2 Þ2 A3t þ ða1 a3 þ a23 A2 ÞA1t A1 A23 A2t ; a1 a2 a23 A1 1 c2 ¼ 2 ½2ða1 þ a3 A2 ÞA3t þ a3 A1t ; a3 A1 A3t c3 ¼ 2 ; a3 A1
c1 ¼
ð10Þ ð11Þ ð12Þ
where A1 A1(t), A2 A2(t) and A3 A3(t) are arbitrary functions of t, they obtained via solving Eqs. (6) and (8) p0x ¼ ð4p2x Þ1 ð2px pxxx p2xx 4p2x r2x 4r1 p2x Þ; A1 þ A2 ; q¼ A3 þ F 2 ðyÞ
ð13Þ ð14Þ
where F2 F2(y) is an arbitrary function of y. Finally, the module square of the field u of the (2 + 1)-dimensional nonlinear Schro¨dinger equation reads U ¼ juj2 ¼
2a1 a2 px qy ða1 p þ a2 q þ a3 pqÞ2
;
ð15Þ
where p p(x, t) is an arbitrary function of indicated arguments, and q q(y, t) is shown by Eq. (14). To the above conclusion, we have the following comments: Comment 1. All of the derived results are technical and remarkable, but it is a pity that the deduced process and the derived conclusions are just as the same as in Ref. [2]. Abominably, every symbol in their equation expresses is also identical while Bai et al. have not introduced it in their work. To investigate the interactions among different types of solitary waves in the (2 + 1)-dimensional nonlinear Schro¨dinger system, Bai et al. focused their attention on (2 + 1)-dimensional fractal semifolded localized structures similar to Refs. [3–5]. For example, when p is set as a multi-valued function and q selected as a single-valued function, respectively, Z n px ¼ sech2 ðn tÞ; x ¼ n 1:5 tanhðn tÞ; p ¼ px xn dn; ð16Þ pffiffiffiffiffi ð17Þ q ¼ 1 þ exp½ y 2 ð1 þ cosðln y 2 ÞÞ expðsechtÞ: and a1 = a2 = 1, a3 = 0.2 for the module square of the field u (15), they then said that a new type of fractal semifolded localized excitation looking like a fractal dromion-like loop soliton was obtained. Here, we should make some corrections for their carelessness or other reasons. Comment 2. The formula (15) should be modified as follows: U ¼ juj2 ¼
2ja1 a2 px qy j ða1 p þ a2 q þ a3 pqÞ2
:
ð18Þ
That is, the absolute value of Eq. (15) can not be cancelled since the products of the constants a1, a2 and the functions px, qy are not always positive. Nevertheless, if the authors had considered the real conditions of p1 and q1 in Eq. (5) in their discussion, the formula (15) would still be right.
390
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
1 U
0.5 0 2 1 x0
–1 –2
1.5
1
0 0.5 y
–1 –0.5
–1.5
Fig. 1. A semifolded localized structure, namely a dromion-like loop soliton for the special field U expressed by Eq. (18) with conditions (16) and (17) at time t = 0.
1.2
1.2
1
1
0.8
0.8
U 0.6
U 0.6
0.4
0.4
0.2
0.2
0
0 –0.06 –0.04 –0.02
0 y
0.02
0.04
–0.002 –0.001
0.06
0 y
0.001
0.002
1.2 1 0.8 U0.6 0.4 0.2 0 –0.0001 –5e–05
Fig. 2. Sectional views relate to Fig. 1 with {y 2 [0.000128, 0.000128]}, respectively, at x = 0.
(a)
0 y
5e–05
0.0001
{y 2 [0.0685, 0.0685]},
(b)
{y 2 [0.00295, 0.00295]}
and
(c)
From the above viewpoint, we can readily find or verify that the so-called semifolded localized structure, namely the dromion-like loop soliton depicted in Fig. 1 for the module square of the field u is invalid. The same mistakes also occur in Figs. 2 and 3 were shown in Ref. [1]. The corresponding localized structure for the special field U expressed by Eq. (18) with conditions (16) and (17) at time t = 0 is presented in Fig. 1. The basic concept for fractals is that they contain a large degree of self-similarity. This means they usually possess smaller copies of themselves buried deep within the original, and also have infinite details. Like the coastline problem, the more one zoom in a fractal, the more details (coastline) one get. And this keeps going on forever and ever, you could make a pretty movie of a fractal when you zoom in one or two parts of a fractal. Bai et al. described their semifolded structure with novel properties, namely, it folds as loop soliton in the x direction and localizes as fractal dromion in the y direction. Here let us look their fractal phenomenon again. Based on Eq. (18) with conditions (16) and (17), we show the correct self-similarity localized structures. Fig. 2(a)–(c) presents three
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
391
sectional views of the semifolded localized excitation in the same ranges as Bai’s Fig. 1, namely, {y 2 [0.0685, 0.0685]}, {y 2 [0.00295, 0.00295]} and {y 2 [0.000128, 0.000128]} at x = 0. We say they possess fractal behaviors due to their self-similarity properties, i.e., as the range of variable y gets smaller, the shapes of the special field U have not varieties distinctly.
Acknowledgements The authors are indebted to Profs. J.F. Zhang, J.P. Fang, H.P. Zhu and Dr. W.H. Huang for their helpful suggestions and fruitful discussions. The project was supported by the Natural Science Foundation of China under Grant No. 10272071 and the Natural Science Foundation of Zhejiang Province of China under Grant No. Y604106.
References [1] [2] [3] [4] [5]
Bai CL, Zhao H. Chaos, Solitons & Fractals. doi: 10.1016/j.chaos.2006.02.005. Ruan HY, Chen YX. Acta Phys Sin 2001;50:0586. in Chinese. Zheng CL, Chen LQ. J Phys Soc Jpn 2004;73:293. Zheng CL, Chen LQ, Zhang JF. Phys Lett A 2005;340:402. Zheng CL, Zhu HP, Chen LQ. Chaos, Solitons & Fractals 2005;26:187.
Discussion
Comments on ‘‘New types of interactions between solitary waves in (2 + 1)-dimensions’’ Chun-Long Zheng *, Zheng-Yi Ma College of Mathematics and Physics, Lishui University, Lishui, Zhejiang 323000, PR China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China Accepted 29 June 2006
Abstract Starting from a special Ba¨cklund transformation, Bai and Zhao [Bai CL, Zhao H. Chaos, Solitons & Fractals. doi: 10.1016/j.chaos.2006.02.005] have claimed that a general variable separation solution of the (2 + 1)-dimensional nonlinear Schro¨dinger equation is derived. Based on the quite universal variable separation solution which is valid for some (2 + 1)-dimensional physical models and by selecting appropriate functions, they have asserted that new types of interactions between the multi-valued solitary waves and the single-valued solitary waves, i.e., the fractal semifolded localized structures, are investigated. We show that several serious wrongs exist in their paper, and say nothing of the conclusion that some new fractal semifolded localized structures for the (2 + 1)-dimensional nonlinear Schro¨dinger equation have been found. Ó 2006 Elsevier Ltd. All rights reserved.
In Ref. [1], Bai et al. have considered the variable separated solution of the following (2 + 1)-dimensional nonlinear Schro¨dinger equation: uxx ¼ uQx iut ;
ð1Þ
uxx
ð2Þ ð3Þ
¼ u Qx þ iut ; Qy ¼ uu : According to the standard truncated Painleve´ expansion, they first obtained the Ba¨cklund transformation g u ¼ ; Q ¼ 2ðln f Þx þ p0 ðx; tÞ; f
ð4Þ
where p0 p0(x, t) is an arbitrary function of {x, t}, then they derived the following simple variable separated equations
q
DOI of original article: 10.1016/j.chaos.2006.02.005. Corresponding author. Address: College of Mathematics and Physics, Lishui University, Lishui, Zhejiang 323000, PR China. Tel.: +86 578 2271126; fax: +86 578 2271250. E-mail address: [email protected] (C.-L. Zheng). *
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.077
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
p1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2a1 a2 px ;
2px pxxx þ
p2xx
þ
q1 ¼
4p2x r2x
þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2a1 a2 qy ; 4rt p2x
þ
4p2x p0x
st ¼ 0; ¼ 0;
389
ð5Þ ð6Þ
pt þ 2px rx ¼ c1 ða2 þ a3 pÞ2 þ c2 ða2 þ a3 pÞ þ a1 a2 a3 ;
ð7Þ
qt ¼ c3 ða1 þ a3 qÞ2 c2 ða1 þ a3 qÞ a1 a2 c1 ;
ð8Þ
where c1, c2, c3 are arbitrary functions of t, f and g are ansatzs f ¼ a1 p þ a2 q þ a3 pq;
g ¼ p1 q1 expðir þ isÞ;
ð9Þ
where a1, a2, a3 are arbitrary constants, the functions p, p1 and r are only functions of {x, t}, and q, q1 and s are only functions of {y, t}. When c1, c2 and c3 are chosen as 1 ½ða1 þ a3 A2 Þ2 A3t þ ða1 a3 þ a23 A2 ÞA1t A1 A23 A2t ; a1 a2 a23 A1 1 c2 ¼ 2 ½2ða1 þ a3 A2 ÞA3t þ a3 A1t ; a3 A1 A3t c3 ¼ 2 ; a3 A1
c1 ¼
ð10Þ ð11Þ ð12Þ
where A1 A1(t), A2 A2(t) and A3 A3(t) are arbitrary functions of t, they obtained via solving Eqs. (6) and (8) p0x ¼ ð4p2x Þ1 ð2px pxxx p2xx 4p2x r2x 4r1 p2x Þ; A1 þ A2 ; q¼ A3 þ F 2 ðyÞ
ð13Þ ð14Þ
where F2 F2(y) is an arbitrary function of y. Finally, the module square of the field u of the (2 + 1)-dimensional nonlinear Schro¨dinger equation reads U ¼ juj2 ¼
2a1 a2 px qy ða1 p þ a2 q þ a3 pqÞ2
;
ð15Þ
where p p(x, t) is an arbitrary function of indicated arguments, and q q(y, t) is shown by Eq. (14). To the above conclusion, we have the following comments: Comment 1. All of the derived results are technical and remarkable, but it is a pity that the deduced process and the derived conclusions are just as the same as in Ref. [2]. Abominably, every symbol in their equation expresses is also identical while Bai et al. have not introduced it in their work. To investigate the interactions among different types of solitary waves in the (2 + 1)-dimensional nonlinear Schro¨dinger system, Bai et al. focused their attention on (2 + 1)-dimensional fractal semifolded localized structures similar to Refs. [3–5]. For example, when p is set as a multi-valued function and q selected as a single-valued function, respectively, Z n px ¼ sech2 ðn tÞ; x ¼ n 1:5 tanhðn tÞ; p ¼ px xn dn; ð16Þ pffiffiffiffiffi ð17Þ q ¼ 1 þ exp½ y 2 ð1 þ cosðln y 2 ÞÞ expðsechtÞ: and a1 = a2 = 1, a3 = 0.2 for the module square of the field u (15), they then said that a new type of fractal semifolded localized excitation looking like a fractal dromion-like loop soliton was obtained. Here, we should make some corrections for their carelessness or other reasons. Comment 2. The formula (15) should be modified as follows: U ¼ juj2 ¼
2ja1 a2 px qy j ða1 p þ a2 q þ a3 pqÞ2
:
ð18Þ
That is, the absolute value of Eq. (15) can not be cancelled since the products of the constants a1, a2 and the functions px, qy are not always positive. Nevertheless, if the authors had considered the real conditions of p1 and q1 in Eq. (5) in their discussion, the formula (15) would still be right.
390
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
1 U
0.5 0 2 1 x0
–1 –2
1.5
1
0 0.5 y
–1 –0.5
–1.5
Fig. 1. A semifolded localized structure, namely a dromion-like loop soliton for the special field U expressed by Eq. (18) with conditions (16) and (17) at time t = 0.
1.2
1.2
1
1
0.8
0.8
U 0.6
U 0.6
0.4
0.4
0.2
0.2
0
0 –0.06 –0.04 –0.02
0 y
0.02
0.04
–0.002 –0.001
0.06
0 y
0.001
0.002
1.2 1 0.8 U0.6 0.4 0.2 0 –0.0001 –5e–05
Fig. 2. Sectional views relate to Fig. 1 with {y 2 [0.000128, 0.000128]}, respectively, at x = 0.
(a)
0 y
5e–05
0.0001
{y 2 [0.0685, 0.0685]},
(b)
{y 2 [0.00295, 0.00295]}
and
(c)
From the above viewpoint, we can readily find or verify that the so-called semifolded localized structure, namely the dromion-like loop soliton depicted in Fig. 1 for the module square of the field u is invalid. The same mistakes also occur in Figs. 2 and 3 were shown in Ref. [1]. The corresponding localized structure for the special field U expressed by Eq. (18) with conditions (16) and (17) at time t = 0 is presented in Fig. 1. The basic concept for fractals is that they contain a large degree of self-similarity. This means they usually possess smaller copies of themselves buried deep within the original, and also have infinite details. Like the coastline problem, the more one zoom in a fractal, the more details (coastline) one get. And this keeps going on forever and ever, you could make a pretty movie of a fractal when you zoom in one or two parts of a fractal. Bai et al. described their semifolded structure with novel properties, namely, it folds as loop soliton in the x direction and localizes as fractal dromion in the y direction. Here let us look their fractal phenomenon again. Based on Eq. (18) with conditions (16) and (17), we show the correct self-similarity localized structures. Fig. 2(a)–(c) presents three
C.-L. Zheng, Z.-Y. Ma / Chaos, Solitons and Fractals 32 (2007) 388–391
391
sectional views of the semifolded localized excitation in the same ranges as Bai’s Fig. 1, namely, {y 2 [0.0685, 0.0685]}, {y 2 [0.00295, 0.00295]} and {y 2 [0.000128, 0.000128]} at x = 0. We say they possess fractal behaviors due to their self-similarity properties, i.e., as the range of variable y gets smaller, the shapes of the special field U have not varieties distinctly.
Acknowledgements The authors are indebted to Profs. J.F. Zhang, J.P. Fang, H.P. Zhu and Dr. W.H. Huang for their helpful suggestions and fruitful discussions. The project was supported by the Natural Science Foundation of China under Grant No. 10272071 and the Natural Science Foundation of Zhejiang Province of China under Grant No. Y604106.
References [1] [2] [3] [4] [5]
Bai CL, Zhao H. Chaos, Solitons & Fractals. doi: 10.1016/j.chaos.2006.02.005. Ruan HY, Chen YX. Acta Phys Sin 2001;50:0586. in Chinese. Zheng CL, Chen LQ. J Phys Soc Jpn 2004;73:293. Zheng CL, Chen LQ, Zhang JF. Phys Lett A 2005;340:402. Zheng CL, Zhu HP, Chen LQ. Chaos, Solitons & Fractals 2005;26:187.