Interactions among different types of solitary waves in (2 + 1)-dimensional system

Chaos, Solitons and Fractals 25 (2005) 481–490 www.elsevier.com/locate/chaos

Interactions among different types of solitary waves in (2 + 1)-dimensional system Cheng-lin Bai a

a,*

, Cheng-jie Bai b, Hong Zhao

a

Physics Science and Information Engineering School, Liaocheng University, Shandong 252059, China b Propagation School, Shandong Normal University, Shandong 250014, China Accepted 23 November 2004 Communicated by Prof. M. Wadati

Abstract Starting from a quite universal formula, which is valid for some quite universal (2 + 1)-dimensional physical models, the interactions among different types of solitary waves like peakons, dromions, and compactons are investigated both analytically and graphically. Some novel features or interesting behaviors are revealed. The results show that the interactions for peakon–antidromion, compacton–antidromion, and antipeakon–compacton may be completely inelastic or not completely elastic. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction In the study of non-linear science, soliton theory plays a very important role and has been applied in almost all the natural sciences especially in all the physics branches such as condensed matter physics, field theory, fluid dynamics, plasma physics and optics, etc. [1]. (1 + 1)-Dimensional solitons and solitary wave solutions have been studied quite well both theoretically and experimentally, for example, dromion (exponentially localized in all directions), compacton and peakon (two types of significant weak solutions). From the symmetry study of the (2 + 1)-dimensional integrable models we know that there exist richer symmetry structures than in lower dimensions [2]. This suggests that the solitary waves structures and the interactions between solitary waves of the (2 + 1)-dimensional non-linear models may have quite rich phenomena that have not yet been revealed. Almost all the previous studies of the interactions among solitary waves especially in higher dimensions are restricted to the same types of localized structures, such as the peakon–peakon and dromion-dromion interactions have been investigated and the results indicate that the former is not completely elastic while the latter is completely elastic, and that for some types of compactons the interactions among them are not completely elastic while for some others the interactions are completely elastic, but the interactions among different types of solitary waves like peakon–antidromion, compacton–antidromion, and antipeakon–compacton are not yet studied too often.

*

Corresponding author. E-mail addresses: [email protected], [email protected] (C.-l. Bai).

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.11.021

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Motivated by these reasons, we take a (2 + 1)-dimensional breaking soliton system ut þ buxxy þ 4buvx þ 4bux v ¼ 0;

ð1Þ

vx
uy ¼ 0;

ð2Þ

where b is an arbitrary constant. Eqs. (1) and (2) describe the interaction of a Riemann wave propagating along the yaxis with a long wave along the x-axis and it seems to have been investigated extensively where overlapping solutions have been derived. Since a detailed physical background of the system has been given in Ref. [3], we neglect the corresponding description. The paper is organized as follows. In Section 2, we apply a variable separation approach to solve the (2 + 1)-dimensional soliton equation and obtain its exact excitation. In Section 3, we analyze interaction properties among different types of coherent soliton structures. A brief discussion and summary is given in the last section.

2. Variable separated solutions for the (2 + 1)-dimensional breaking soliton equation There was a wealth of approaches for finding special solutions of the non-linear partial differential equation (PDE), such as inverse scattering method, Backlund transformation, Darboux transformation, Hirota method. All these methods are described in Ref. [4–7]. Recently, Lou and coworkers [8,9] has proposed a multi-linear variable separation approach (MLVSA) to search for the exact solutions of the higher-dimensional, specially (2 + 1)-dimensional, non-linear PDEs. Now we apply the MLVSA for the (2 + 1)-dimensional breaking soliton equation. According to the standard truncated Painleve expansion, we take the following Backlund transformation to u and v in Eqs. (1) and (2) u¼

M X

ui f i
M ;

v¼

i¼0

N X

vj f j
N ;

ð3Þ

j¼0

where uM and vN are the arbitrary seed solutions of the breaking soliton system. By using the leading term analysis, we obtain M ¼ 2;

N ¼ 2:

ð4Þ

Substituting Eqs. (3) and (4) directly into Eqs. (1) and (2) and considering the fact that the function u2 and v2 are the seed solutions of the original model yield 4 X

P 1i f i
5 ¼ 0;

2 X

i¼0

P 2i f i
3 ¼ 0;

ð5Þ

i¼0

where P1i and P2i are the functions of {uj, vj, f, j = 0, 1} and their derivatives. Because of the complexity of the expression of P1i and P2i, we neglect their concrete forms. Vanishing the leading and subleading terms of Eq. (5), we can determine the functions {uj, vj, j = 0, 1}. Inserting all the results into Eq. (3) and rewriting its form, we can obtain the Backlund transformation 3 u ¼ ðln f Þxx þ u2 ; 2

3 v ¼ ðln f Þxy þ v2 : 2

ð6Þ

For convenience of discussion, we choose the seed solutions u2 and v2 as u2 ¼ u20 ðxÞ;

v2 ¼ 0;

ð7Þ

where u20(x) is an arbitrary function of indicated argument. Now substituting Eq. (6) together with Eq. (7) into Eq. (1) yields a trilinear equation in f while Eq. (2) is satisfied identically under the above transformation, which reads ð4bðu20 fxy Þx þ bf xxxxy þ fxxt Þf 2 þ ðð2f xxx fxy
fxxxx fy
8u20 fx fxy
4f xxxy fx þ 4ðu20 fx Þx fy Þb
2f x fxt
fxx ft Þf þ 2ðð3bf xxy þ 4bu20 fy þ ft Þfx2 þ bðfxxx fy
3f xx fxy Þfx Þ ¼ 0:

ð8Þ

After obtaining the trilinear equation of the original system, we select an appropriate variable separation hypothesis for the function f. For many integrable models, it can be chosen as a modifying HoritaÕs multi-soliton form [10]. In our special case, we take f as a slightly changed as

C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

f ¼ a0 þ a1 pðxÞ þ a2 qðy; tÞ þ a3 pðxÞqðy; tÞ;

483

ð9Þ

where the variable separation function p(x)  p and q(y, t)  q are only the functions of {x} and {y, t}, respectively; a0, a1, a2 and a3 are the arbitrary constants. Inserting the ansatz (9) into Eq. (8) and finishing some tedious calculations, we finally obtain the following two simple variable separated equations 4bu20 px þ bpxxx þ cpx ¼ 0;

ð10aÞ

qt
cqy ¼ 0;

ð10bÞ

where c is an arbitrary constants. It is easy to obtain general solutions of Eq. (10). Since u20(x) is an arbitrary seed solutions, we can view p as an arbitrary functions of {x}, then the seed solution u20(x) can be fixed by Eq. (10a), which reads u20 ¼


bpxxx þ cpx : 4bpx

ð11Þ

As to Eq. (10b), its general solution has the form qðy; tÞ ¼ qðy þ ctÞ;

ð12Þ

where q(y + ct) is an arbitrary function of {y + ct}. Finally, substituting Eq. (9) together with Eqs. (11) and (7) into Eq. (6), we obtain a general excitation of the breaking soliton system u¼

v¼


3ða1 þ a3 qÞ2 p2x 2ða0 þ a1 p þ a2 q þ a3 pqÞ2

þ

3ða1 þ a3 qÞpxx bp þ cpx
xxx ; 2ða0 þ a1 p þ a2 q þ a3 pqÞ 4bpx

3ða0 a3
a1 a2 Þpx qy

ð13Þ

ð14Þ

2ða0 þ a1 p þ a2 q þ a3 pqÞ2

with two arbitrary functions p(x) and q(y + ct). In Ref. [3], it was pointed out that the breaking soliton system possesses some special types of coherent structures for potential v rather than the physical field u, and It is interesting that the expression (14) is valid for many (2 + 1)-dimensional models like the DS equation, NNV system, ANNV equation, BK equation and the HBK equation, etc. [9,11–15]. As the arbitrariness of the functions p and q are included in potential (14), v obviously possesses quite rich structures. For instance, if we select the functions p and q appropriately, we can obtain many kinds of localized solutions like the multi-solitoff solutions, multi-dromion and dromion lattice solutions, multiple ring soliton solutions, peakons, compactons and so on. The properties of peakon–peakon, dromion-dromion, and compacton–compacton interactions were also discussed in the previous papers [9,12–15]. Now we pay our attention to interactions of different types of solitary waves, i.e., the interactions between solitary waves and anti-solitary waves. In order to see the interaction behaviors among them more direct and visually, we investigate some special examples by fixing the arbitrary functions p and q in (14).

3. Interactions among different types of solitary waves in a (2 + 1)-dimensional system 3.1. Completely inelastic interactions We first discuss the peakon–antidromion interaction. As is known, if selecting p and q to be some piecewise smooth functions, then one can derive some multi-peakon excitation [9,11–15]. For instance, when p and q are taken to have the following simple forms, p ¼ c0 þ

M X

d i expðmi x þ x0i Þ;

mi x þ x0i 6 0;

i¼1 M X ð
d i expð
mi x
x0i Þ þ 2d i Þ; p ¼ c0 þ i¼1

ð15Þ mi x þ x0i > 0;

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q¼

N X

ej expðnj y þ wj t þ y 0j Þ;

nj y þ wj t þ y 0j 6 0;

j¼1 N X ð
ej expð
nj y
wj t
y 0j Þ þ 2ej Þ; q¼

ð16Þ nj y þ wj t þ y 0j > 0;

j¼1

where c0, di, mi, ej, nj, wj, x0i and y0j are all arbitrary constants, the solution (14) with Eqs. (15) and (16) becomes a multipeakon solution. The dromion solution that is localized exponentially in all directions can be driven by multiple straight-line and curved-line ghost solitons. As simple choices for the functions p and q one can take, M X

p ¼1þ

exp½K i ðxÞ þ x1i ;

ð17Þ

exp½k j ðy þ wtÞ þ y 1j ;

ð18Þ

i¼1 N X

q¼1þ

j¼1

where Ki, kj, x1i, and y1j are arbitrary constants and M, N are positive integers. Now the key problem is how to construct the peakon and antidromion structures for the physical quantity v simultaneously since there may exist peakons and antidromions at the same time in the real natural phenomena. According to the above ideas, if we take p and q to have the following simple forms, N X p ¼1þ exp½k j ðxÞ þ x1j ; ð19Þ j¼1

q¼1þ

M X

exp½K i ðy þ btÞ þ y 1i  þ

i¼1

8 P P > > > < c0 þ d i expðmi y
bi t þ y 0i Þ mi y
bi t þ y 0i 6 0; i¼1

P > P > > : c0 þ ð
d i expð
mi y þ bi t
y 0i Þ þ 2d i Þ mi y
bi t þ y 0i > 0;

ð20Þ

i¼1

where M, N, and P are positive integers, then we may obtain a multi-peakons–antidromions solution for the physical quantity v. For simplicity, we choose M ¼ N ¼ P ¼ 1; K 1 ¼
k 1 ¼
1; b ¼ 1; x11 ¼ y 11 ¼ 0; 1 a0 ¼ a1 ¼ a2 ¼ a3 ¼ 1; d 1 ¼ m1 ¼ 1; b1 ¼ 1=2; c0 ¼ 0; 2

y 01 ¼ 5

ð21Þ

and obtain a combined localized coherent structure depicted in Fig. 1. From Fig. 1, we can see that the peakon–antidromion interaction is completely inelastic, which is very similar to the completely inelastic collisions between two classical particles. Along the same line of argument and performing a similar analysis, when p and q are taken as the following simple forms, p p ¼ c0 ; x 6 x0i
; 2k i M X p p ðbi sinðk i ðx
x0i ÞÞ þ bi Þ; x0i
< x 6 x0i þ ; p ¼ c0 þ 2k 2k i i i¼1 p ¼ c0 þ

M X

2bi ;

x > x0i þ

i¼1

q ¼ 0; q¼

y þ wj t 6 y 0j


p ; 2k i

p ; 2lj

N X ðcj sinðlj ðy þ wj t
y 0j ÞÞ þ cj Þ; j¼1

q¼

N X j¼1

2cj ;

ð22Þ

y þ wj t > y 0j þ

p ; 2lj

y 0j


p p < y þ wj t 6 y 0j þ ; 2lj 2lj

ð23Þ

C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

485

Fig. 1. The evolution of peakon–antidromion interaction as seen in the physical quantity v with the conditions (19)–(21) at times (a) t =
40, (b) t =
24, (c) t =
12, (d) t = 0, (e) t = 20, and (f) t = 106, respectively.

where c0, bi, ki, cj, lj, wj, x0i and y0j all are arbitrary constants, then the solution (14) with (22) and (23) becomes a multicompacton solution. Combining Eqs. (15) and (16), and taking p and q to have the following forms, 8P < Nj¼1 ej expðnj x þ x0j Þ; nj x þ x0j 6 0; p ¼ PN : nj x þ x0j > 0; j¼1 ð
ej expð
nj x
x0j Þ þ 2ej Þ;

ð24Þ

8 c0 ; y þ bi t 6 y 0i
2kp i ; > > > M < X PP exp½K i ðy þ btÞ þ y 1i  þ c0 þ i¼1 ðbi sinðk i ðy þ bi t
y 0i ÞÞ þ bi Þ; q¼1þ > > i¼1 > : c þ PP 2b ; y þ b t > y þ p ; 0 i i 0i i¼1 2k i

y 0i
2kp i < y þ bi t 6 y 0i þ 2kp i ; ð25Þ

where M, N, and P are positive integers, then we may construct antipeakon and compacton structures simultaneously for the physical quantity v. For convenience, we fix the related parameters as M ¼ N ¼ P ¼ 1; k 1 ¼ 1;

b1 ¼
1;

K 1 ¼ 1; y 01 ¼ 0;

b ¼ 2;

y 11 ¼ 0;

e1 ¼ n1 ¼ 1;

c0 ¼ 15; x01 ¼ 5

1 a0 ¼ a1 ¼ a2 ¼ a3 ¼ 1; 2

b1 ¼ 2; ð26Þ

and find that the antipeakon–compacton interaction is also completely inelastic. The corresponding plot is depicted in Fig. 2. According to the above ideas, if we take p and q to have the following forms, p ¼1þ

N X j¼1

exp½k j ðxÞ þ x1j ;

ð27Þ

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C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

Fig. 2. The temporal evolution of antipeakon–compacton interaction for the physical quantity v with the conditions (24)–(26) at times (a) t =
12, (b) t =
6, (c) t = 0, (d) t = 1, (e) t = 2, (f) t = 8, and (g) t = 106, respectively.

8 c ; y þ bi t 6 y 0i
2kp i ; > > > 0 M < X PP q¼1þ exp½K i ðy þ btÞ þ y 1i  þ c0 þ i¼1 ðbi sinðk i ðy þ bi t
y 0i ÞÞ þ bi Þ; > > i¼1 P > : c0 þ Pi¼1 2bi ; y þ bi t > y 0i þ 2kp i ;

y 0i
2kp i < y þ bi t 6 y 0i þ 2kp i ; ð28Þ

where M, N, and P are positive integers, then we may construct compacton and antidromion structures simultaneously for the physical quantity v. For convenience we take M ¼ N ¼ P ¼ 1; b1 ¼ 2;

b1 ¼
1;

K 1 ¼ k 1 ¼ 1; y 11 ¼ 5;

b ¼ 2;

x11 ¼ 0;

c0 ¼ 15;

1 a0 ¼ a1 ¼ a2 ¼ a3 ¼ 1; 2

y 01 ¼ 0:

ð29Þ

Then we derive a combined localized coherent structure depicted in Fig. 3. 3.2. Non-completely elastic interactions It is interesting to mention that though the above choices lead to completely inelastic interaction behaviors for the (2 + 1)-dimensional solutions, one can also derive some combined localized coherent structures with non-completely elastic interaction behaviors by selecting p and q appropriately. One of simple choices of the combined localized coherent structures of a multi-antidromions–peakons with non-completely elastic interaction behavior is

C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

487

Fig. 3. The time evolution of a antidromion–compacton interaction as seen in the physical quantity v with the conditions (27)–(29) at times (a) t =
12, (b) t =
6, (c) t = 0, (d) t = 1, (e) t = 2, (f) t = 8, and (g) t = 106, respectively.

p¼

N X j¼1

8P < Qj¼1 ej expðnj x þ x0j Þ; nj x þ x0j 6 0; C j exp½Lj ðxÞ þ x0j  þ P : Q ð
e expð
n x
x Þ þ 2e Þ; n x þ x > 0; j j 0j j j 0j j¼1

q ¼ c0 þ

M X

( PP Bi tanh½K i ðy þ btÞ þ y 0i  þ

i¼1

i¼1 d i

PP

expðmi y
bi t þ y 0i Þ;

i¼1 ð
d i

ð30Þ

mi y
bi t þ y 0i 6 0;

expð
mi y þ bi t
y 0i Þ þ 2d i Þ;

ð31Þ

mi y
bi t þ y 0i > 0;

where c0, Bi, Ki, b, Cj, Lj, di, mi, bi, ej, nj, x0j, and y0i are all arbitrary constants, and M, N, P, Q are positive integers. The physical quantity v (14) with Eqs. (30) and (31) becomes a combined multi-antidromion and multi-peakon solution with non-completely elastic interaction behavior. For convenience, we can fix the related parameters as c0 ¼ 10; B1 ¼ 1=2; K 1 ¼ 1=2; e1 ¼ n1 ¼ 1; y 01 ¼ 5; x01 ¼ 5

b ¼ 1=2;

C 1 ¼ L1 ¼ 1;

d 1 ¼ m1 ¼ 1;

b1 ¼ 1=2; ð32Þ

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C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

Fig. 4. The evolution of a peakon–antidromion interaction in the physical quantity v with the conditions (30)–(32) at times (a) t =
6, (b) t = 0, (c) t = 10, (d) t = 20, and (e) t = 26, respectively.

Fig. 5. The evolution of a compacton–antidromion interaction for the physical quantity v with the conditions (33)–(35) at times (a) t =
16, (b) t =
8, (c) t = 0, (d) t = 8, and (e) t = 16, respectively.

and obtain a combined localized coherent structure with non-completely elastic interaction behavior as displayed in Fig. 4.

C.-l. Bai et al. / Chaos, Solitons and Fractals 25 (2005) 481–490

489

Similarly, another simple selection of the combined localized coherent structures of antidromion and compacton for the (2 + 1)-dimensional soliton system may be. 8 > 0; x 6 x0j
2lpj ; Q N < X X> cj coscj þ1 ðlj ðx
x0j ÞÞ; x0j
2lpj < x 6 x0j þ 2lpj ; p¼ C j exp½Lj ðxÞ þ x0j  þ ð33Þ > > j¼1 : j¼1 p 0; x > x0j þ ; 2lj

q ¼ c0 þ

M X i¼1

8 0; y þ bi t 6 y 0i
2kp i ; P > < X bi cosai þ1 ðk i ðy þ bi t
y 0i ÞÞ; Bi tanh½K i ðy þ btÞ þ y 0i  þ > : i¼1 0; y þ bi t > y 0i þ 2kp i ;

y 0i
2kp i < y þ bi t 6 y 0i þ 2kp i ;

ð34Þ

where c0, Bi, Ki, b, Cj, Lj, bi, ki, bi, cj, lj, x0j, and y0i are all arbitrary constants, and M, N, P, Q, ai, cj are positive integers. Then the physical quantity v (14) with Eqs. (33) and (34) becomes a combined multi-antidromion–compacton solution, which also shows a non-completely elastic interaction behavior. When choosing c0 ¼ 10;

B1 ¼ 1=2;

b1 ¼
1;

c1 ¼ 1;

K 1 ¼ 1; c1 ¼ 4;

b ¼ 1=2; l1 ¼ 1;

C 1 ¼ 1;

y 01 ¼ 0;

L1 ¼ 1;

x01 ¼ 0;

b1 ¼ 1;

a1 ¼ 4;

k 1 ¼ 1; ð35Þ

we can derive a combined antidromion–compacton localized coherent structure with non-completely elastic behavior depicted in Fig. 5.

4. Summary Starting from the known variable separated excitations, which describe some quite universal (2 + 1)-dimensional physical model, of a (2 + 1)-dimensional breaking soliton equation, we discuss the interactions among peakons, dromions, and compactons both analytically and graphically, and reveal some novel properties and interesting behaviors: the interactions for peakon–antidromion, compacton–antidromion, and antipeakon–compacton may be completely inelastic or non-completely elastic depending on the specific details of the solutions. This paper is only an introductory attempt to describe these phenomena. Because of the wide applications of the soliton theory, to learn more about the interactions between different types of solitary waves and their applications in reality is worth further study.

Acknowledgment This work is supported by the National Natural Science Foundation of China and the Natural Science Foundation of Shandong in China.

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