AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/189486 189486 Research Article Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System Cai Huixian Pan Chaohong Liu Zhengrong Tang Chun-Lei Department of Mathematics South China University of Technology Guangzhou 510640 China scut.edu.cn 2014 772014 2014 29 03 2014 28 05 2014 7 7 2014 2014 Copyright © 2014 Huixian Cai et al. 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.

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov system u t + ( v m ) x = 0 , v t + a ( v n ) x x x + b u x v + c u v x = 0 called D ( m , n ) system. We reveal some interesting bifurcation phenomena as follows. (1) For D ( 2 , 1 ) system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) For D ( 1 , 2 ) system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) For D ( 2 , 2 ) system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

1. Introduction

The D ( m , n ) system [1, 2] is read as (1) u t + ( v m ) x = 0 , v t + a ( v n ) x x x + b u x v + c u v x = 0 , where a , b , c , m , and n are constants. Through some transformations, Xie and Yan  got some compacton and solitary pattern solutions of system (1) with m = n - 1 , including (2) v a ( ξ ) = [ 2 n ( λ - c d 0 ) ( n + 1 ) a α m cos 2 ( ( n - 1 ) α m 2 n ξ ) ] 1 / ( n - 1 ) , u a ( ξ ) = v a n - 1 ( ξ ) λ + d 0 , (3) v b ( ξ ) = [ 2 n ( c d 0 - λ ) ( n + 1 ) a α m sinh 2 ( ( n - 1 ) α m 2 n ξ ) ] 1 / ( n - 1 ) , u b ( ξ ) = v b n - 1 ( ξ ) λ + d 0 , where λ 0 , d 0 are constants, (4) ξ = x - λ t , α m = b m + c a λ ( m + 1 ) . Deng et al. , by using the Weierstrass elliptic function method, presented many solutions of system (1) with n = m + 1 . It also includes the above solutions (2) and (3). Zhang et al.  showed some solutions of system (1) under the special parameters via employing the bifurcation method. By means of the complete discrimination system for polynomial method, many solutions of system (1) were acquired in .

In , the system (5) u t + ( v 2 ) x = 0 , v t + a v x x x + b u x v + c u v x = 0 was introduced. Wang  gave recursion, Hamiltonian, symplectic and cosymplectic operator, roots of symmetries, and scaling symmetry for system (5). Wazwaz , by using the tanh method and the sine-cosine method, obtained many solutions with compact and noncompact structures of system (5), including (6) v c ( ξ ) = λ - 6 2 b + c sech ( λ a ξ ) , u c ( ξ ) = 1 λ v c 2 ( ξ ) , ( λ a > 0 ) , v d ( ξ ) = λ - 3 2 b + c tanh ( - λ 2 a ξ ) , u d ( ξ ) = 1 λ v d 2 ( ξ ) , ( λ a < 0 ) , v e ( ξ ) = λ - 3 2 b + c coth ( - λ 2 a ξ ) , u e ( ξ ) = 1 λ v e 2 ( ξ ) , ( λ a < 0 ) . Biazar and Ayati  obtained some solutions of system (5) through Exp-function method and modification of Exp-function method. Zhang et al. , by employing the complex method, gained all meromorphic exact solutions of system (5). Applying the auxiliary equation method, some exact solutions of system (5) were given in . El-Wakil and Abdou  got some new exact solutions of system (5) by means of modified extended tanh-function method.

The other generalized Drinfel'd-Sokolov system  (7) u t + a 1 u u x + b 1 u x x x + c 0 ( v σ ) x = 0 , v t + a 2 u v x + b 2 v x x x = 0 is considered in . In , many exact solutions of system (7) were obtained. Clearly, system (1) and system (7) are two different systems.

In this paper, we are interested in system (1). We study the bifurcations of nonlinear waves for system (1).

Under the transformations (8) ξ = x - λ t , u ( x , t ) = ψ ( ξ ) , v ( x , t ) = φ ( ξ ) , system (1) is reduced to (9) λ ψ - ( φ m ) = 0 , - λ φ + a ( φ n ) ′′′ + b ψ φ + c ψ φ = 0 . Integrating the first equation of system (9) once, we have (10) ψ = 1 λ φ m + d , where d is an integral constant. Substituting (10) into the second equation of system (9) and integrating it once yield the following equation: (11) g + ( c d - λ ) φ + b m + c λ ( m + 1 ) φ m + 1 + a [ n ( n - 1 ) φ n - 2 ( φ ) 2 + n φ n - 1 φ ′′ ] = 0 , where g is another integral constant. Letting φ ( ξ ) = y , we obtain the planar system (12) d φ d ξ = y , d y d ξ = - n ( n - 1 ) φ n - 2 y 2 + α m φ m + 1 + β φ + γ n φ n - 1 , where α m is given in (4) and (13) β = c d - λ a , γ = g a . Letting n φ n - 1 d τ = d ξ , system (12) becomes (14) d φ d τ = n φ n - 1 y , d y d τ = - n ( n - 1 ) φ n - 2 y 2 - α m φ m + 1 - β φ - γ , which is called P ( m , n ) system. Clearly, systems (12) and (14) possess the same first integral (15) φ n ( n 2 φ n - 2 y 2 + α m m + n + 1 φ m + 1 + β n + 1 φ + γ n ) = h .

Employing (14) and (15), we reveal some interesting bifurcation phenomena listed in the above abstract.

In Section 2, we will consider D ( 2,1 ) system. Firstly, we will show that the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves. Secondly, we will demonstrate that the kink waves can be bifurcated from the smooth solitary waves and the singular waves. In Section 3, we will consider D ( 1,2 ) system. Firstly, we will confirm that the compactons can be bifurcated from the smooth solitary waves. Secondly, we will clarify that the peakons can be bifurcated from the solitary waves and the singular cusp waves. In Section 4,   D ( 2,2 ) system will be considered. We will verify that the solitary waves can be bifurcated from the smooth periodic waves and the periodic singular waves. A short conclusion will be given in Section 5.

2. The Bifurcations of Solitary Waves and Kink Waves for <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M39"> <mml:mi>D</mml:mi> <mml:mo mathvariant="bold">(</mml:mo> <mml:mn>2,1</mml:mn> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula> System

When m = 2 and n = 1 , system (1) becomes D ( 2,1 ) system: (16) u t + ( v 2 ) x = 0 , v t + a v x x x + b u x v + c u v x = 0 .

We will reveal two kinds of interesting bifurcation phenomena to system (16). The first phenomenon is that fractional solitary waves can be bifurcated from two types of smooth periodic waves: trigonometric periodic waves and elliptic periodic waves. The second phenomenon is that the kink waves can be bifurcated from the smooth solitary waves and the singular waves. We state these results and give proof as follows.

Note that P ( 2,1 ) system is read as (17) d φ d τ = y , d y d τ = - α 2 φ 3 - β φ - γ , where (18) α 2 = 2 b + c 3 a λ , and β , γ are given in (13). When 27 α 2 γ 2 > - 4 β 3 , let (19) φ 01 = 1 α 2 ( Ω 18 ) 1 / 3 - β ( 2 3 Ω ) 1 / 3 , φ 02 = β ( 1 - 3 i ) ( 1 12 Ω ) 1 / 3 - 1 + 3 i 2 α 2 ( Ω 18 ) 1 / 3 , φ 03 = β ( 1 + 3 i ) ( 1 12 Ω ) 1 / 3 - 1 - 3 i 2 α 2 ( Ω 18 ) 1 / 3 , Ω = 12 α 2 3 β 3 + 81 α 2 4 γ 2 - 9 α 2 2 γ , (20) γ 1 ( β ) = 2 3 3 ( - β 3 α 2 ) 1 / 2 .

Proposition 1.

For given α 2 > 0 and β < 0 , if   - γ 1 ( β ) < γ < 0 , then φ 01 , φ 02 , and φ 03 are real and system (16) has two types of special periodic wave solutions which become the fractional solitary wave solution (21) v k ( ξ ) = - 3 β α 2 9 + 2 β ξ 2 9 - 6 β ξ 2 , u k ( ξ ) = 1 λ v k 2 ( ξ ) + d , when γ - γ 1 ( β ) + 0 . These two types of special periodic wave solutions are as follows.

Trigonometric periodic wave solution: (22) v 1 ( ξ ) = φ 03 + 2 A 1 Δ 1 cos ( η 1 ξ ) - B 1 , u 1 ( ξ ) = 1 λ v 1 2 ( ξ ) + d ,

where (23) A 1 = ( φ 1 - φ 03 ) ( φ 03 - φ 2 ) , B 1 = φ 1 + φ 2 - 2 φ 03 , Δ 1 = B 1 2 + 4 A 1 , η 1 = - A 1 α 2 2 , φ 1 = - φ 03 + - 2 β α 2 - 2 φ 03 2 , φ 2 = - φ 03 - - 2 β α 2 - 2 φ 03 2 .

Elliptic periodic wave solution: (24) v 2 ( ξ ) = φ 3 ( φ 4 - φ 6 ) + φ 6 ( φ 3 - φ 4 ) s n 2 ( η 2 ξ , k 1 ) ( φ 4 - φ 6 ) + ( φ 3 - φ 4 ) s n 2 ( η 2 ξ , k 1 ) , u 2 ( ξ ) = 1 λ v 2 2 ( ξ ) + d ,

where (25) η 2 = α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) , k 1 2 = ( φ 3 - φ 4 ) ( φ 5 - φ 6 ) ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) , φ 3 = 1 2 ( - φ 5 - φ 6 + - 8 β α 2 - 3 φ 5 2 - 2 φ 5 φ 6 - 3 φ 6 2 ) , φ 4 = 1 2 ( - φ 5 - φ 6 - - 8 β α 2 - 3 φ 5 2 - 2 φ 5 φ 6 - 3 φ 6 2 ) , φ 5 = - 1 3 φ 6 + ( 2 β + 2 3 α 2 φ 6 2 ) ( 1 4 θ 2 ) 1 / 3 ( 1 + i 3 ) - 1 6 α 2 ( θ 2 2 ) 1 / 3 ( 1 - i 3 ) , φ 03 * < φ 6 < φ 03 , φ 03 * = - φ 02 - - 2 β α 2 - 2 φ 02 2 , θ 1 = - 108 α 2 2 γ - 36 α 2 2 β φ 6 - 20 α 2 3 φ 6 3 , θ 2 = θ 1 + 4 ( 6 α 2 β + 2 α 2 2 φ 6 2 ) 3 + θ 1 2 . For the varying process, see Figure 1.

The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of v 1 ( ξ ) and v 2 ( ξ ) when α 2 > 0 , β < 0 , and γ - γ 1 ( β ) + 0 where α 2 = 4 and β = - 9 .

γ = - γ 1 ( β ) + 1 0 - 1

γ = - γ 1 ( β ) + 1 0 - 4

γ = - γ 1 ( β ) + 1 0 - 8

γ = - γ 1 ( β ) + 1 0 - 1

γ = - γ 1 ( β ) + 1 0 - 4

γ = - γ 1 ( β ) + 1 0 - 8

Proposition 2.

For given α 2 > 0 and β < 0 , if 0 < γ < γ 1 ( β ) , then φ 01 , φ 02 , and φ 03 are real and system (16) has two types of special periodic wave solutions which become a fractional solitary wave solution: (26) v y ( ξ ) = - 3 β α 2 2 β ξ 2 + 9 6 β ξ 2 - 9 , u y ( ξ ) = 1 λ v y 2 ( ξ ) + d , when γ γ 1 ( β ) - 0 . These two types of special periodic wave solutions are as follows.

Trigonometric periodic wave solution: (27) v 3 ( ξ ) = φ 01 + 2 A 2 B 2 - Δ 2 cos ( η 3 ξ ) , u 3 ( ξ ) = 1 λ v 3 2 ( ξ ) + d ,

where (28) A 2 = ( φ 7 - φ 01 ) ( φ 01 - φ 8 ) , B 2 = 2 φ 01 - φ 7 - φ 8 , Δ 2 = B 2 2 + 4 A 2 , η 3 = - A 2 α 2 2 , φ 7 = - φ 01 + - 2 β α 2 - 2 φ 01 2 , φ 8 = - φ 01 - - 2 β α 2 - 2 φ 01 2 .

Elliptic periodic wave solution: (29) v 4 ( ξ ) = φ 12 ( φ 9 - φ 11 ) + φ 9 ( φ 11 - φ 12 ) sn 2 ( η 4 ξ , k 2 ) ( φ 9 - φ 11 ) + ( φ 11 - φ 12 ) sn 2 ( η 4 ξ , k 2 ) , u 4 ( ξ ) = 1 λ v 4 2 ( ξ ) + d ,

where (30) η 4 = α 2 8 ( φ 9 - φ 11 ) ( φ 10 - φ 12 ) , k 2 2 = ( φ 9 - φ 10 ) ( φ 11 - φ 12 ) ( φ 9 - φ 11 ) ( φ 10 - φ 12 ) , φ 01 * = - φ 02 + - 2 β α 2 - 2 φ 02 2 , φ 01 < φ 9 < φ 01 * , θ 3 = - 20 v 9 3 α 2 3 - 36 v 9 α 2 2 β - 108 α 2 2 γ , θ 4 = θ 3 + 4 ( 2 v 9 2 α 2 2 + 6 α 2 β ) 3 + θ 3 2 , φ 10 = - φ 9 3 - ( 2 3 φ 9 2 α 2 + 2 β ) ( 2 θ 4 ) 1 / 3 + 1 3 α 2 ( θ 4 2 ) 1 / 3 , φ 11 = 1 2 ( - φ 9 - φ 10 + - 3 φ 9 2 - 2 φ 9 φ 10 - 3 φ 10 2 - 8 β α 2 ) , φ 12 = 1 2 ( - φ 9 - φ 10 - - 3 φ 9 2 - 2 φ 9 φ 10 - 3 φ 10 2 - 8 β α 2 ) . For the varying process, see Figure 2.

The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of v 3 ( ξ ) and v 4 ( ξ ) when α 2 > 0 , β < 0 , and γ γ 1 ( β ) - 0 where α 2 = 4 and β = - 9 .

γ = γ 1 ( β ) - 1 0 - 1

γ = γ 1 ( β ) - 1 0 - 4

γ = γ 1 ( β ) - 1 0 - 8

γ = γ 1 ( β ) - 1 0 - 1

γ = γ 1 ( β ) - 1 0 - 4

γ = γ 1 ( β ) - 1 0 - 8

Proposition 3.

For given α 2 < 0 and β > 0 , if - γ 1 ( β ) < γ < γ 1 ( β ) , then system (16) has four nonlinear wave solutions which become two kink wave solutions: (31) v δ ± ( ξ ) = ± - β α 2 tanh ( β 2 ξ ) , u δ ( ξ ) = 1 λ v δ 2 ( ξ ) + d , when γ 0 . These four nonlinear wave solutions are as follows: (32) v 5 ( ξ ) = 4 A 3 ( φ 01 - δ e - η 5 ξ ) - φ 01 ( δ e - η 5 ξ - B 3 ) 2 4 A 3 - ( δ e - η 5 ξ - B 3 ) 2 , u 5 ( ξ ) = 1 λ v 5 2 ( ξ ) + d , v 6 ( ξ ) = 4 A 3 ( φ 01 - δ e η 5 ξ ) - φ 01 ( δ e η 5 ξ - B 3 ) 2 4 A 3 - ( δ e η 5 ξ - B 3 ) 2 , u 6 ( ξ ) = 1 λ v 6 2 ( ξ ) + d , v 7 ( ξ ) = 4 A 4 ( φ 03 - μ e η 6 ξ ) - φ 03 ( μ e η 6 ξ - B 4 ) 2 4 A 4 - ( μ e η 6 ξ - B 4 ) 2 , u 7 ( ξ ) = 1 λ v 7 2 ( ξ ) + d , v 8 ( ξ ) = 4 A 4 ( φ 03 - μ e - η 6 ξ ) - φ 03 ( μ e - η 6 ξ - B 4 ) 2 4 A 4 - ( μ e - η 6 ξ - B 4 ) 2 ,    u 8 ( ξ ) = 1 λ v 8 2 ( ξ ) + d , where (33) δ = 8 - β α 2 , A 3 = 6 φ 01 2 + 2 β α 2 , η 5 = - A 3 α 2 2 , B 3 = 4 φ 01 , μ = - 8 - β α 2 , A 4 = 6 φ 03 2 + 2 β α 2 , η 6 = - A 4 α 2 2 , B 4 = 4 φ 03 . These four nonlinear wave solutions possess the following properties.

If 0 < γ < γ 1 ( β ) , then v 5 ( ξ ) and v 6 ( ξ ) represent two solitary waves which tend to two kink waves v δ ± ( ξ ) (see Figure 3) when γ 0 + 0 .

If - γ 1 ( β ) < γ < 0 , then v 5 ( ξ ) and v 6 ( ξ ) represent two singular waves which tend to two kink waves v δ ± ( ξ ) (see Figure 4) when γ 0 - 0 .

If 0 < γ < γ 1 ( β ) , then v 7 ( ξ ) and v 8 ( ξ ) represent two singular waves which tend to two kink waves v δ ± ( ξ ) (see Figure 5) when γ 0 + 0 .

If - γ 1 ( β ) < γ < 0 , then v 7 ( ξ ) and v 8 ( ξ ) represent two solitary waves which tend to two kink waves v δ ± ( ξ ) (see Figure 6) when γ 0 - 0 .

Two kink waves are bifurcated from two peak-solitary waves. The varying process for the figures of v 5 ( ξ ) and v 6 ( ξ ) when α 2 < 0 , β > 0 , and γ 0 + 0 where α 2 = - 2 and β = 9 .

γ = 1 × 1 0 - 1

γ = 1 × 1 0 - 4

γ = 1 × 1 0 - 8

Two kink waves are bifurcated from two singular waves. The varying process for the figures of v 5 ( ξ ) and v 6 ( ξ ) when α 2 < 0 , β > 0 , and γ 0 - 0 where α 2 = - 2 and β = 9 .

γ = - 1 × 1 0 - 1

γ = - 1 × 1 0 - 4

γ = - 1 × 1 0 - 8

Two kink waves are bifurcated from two singular waves. The varying process for the figures of v 7 ( ξ ) and v 8 ( ξ ) when α 2 < 0 , β > 0 , and γ 0 + 0 where α 2 = - 2 and β = 9 .

γ = 1 × 1 0 - 1

γ = 1 × 1 0 - 4

γ = 1 × 1 0 - 8

Two kink waves are bifurcated from two valley-solitary waves. The varying process for the figures of v 7 ( ξ ) and v 8 ( ξ ) when α 2 < 0 , β > 0 , and γ 0 - 0 where α 2 = - 2 and β = 9 .

γ = - 1 × 1 0 - 1

γ = - 1 × 1 0 - 4

γ = - 1 × 1 0 - 8

The Derivations of Propositions 13. According to the qualitative theory, we obtain the bifurcation phase portraits of system (17) as in Figure 7. Employing some orbits in Figure 7, we derive the results of Propositions 13 as follows.

The bifurcation phase portraits of system (17).

α 2 > 0

α 2 < 0

(1) When α 2 > 0 , β < 0 , and - γ 1 ( β ) < γ < 0 (as in Figure 7(a)), the closed curves l 1 and l 2 possess the following expressions: (34) l 1 : y 2 = α 2 2 ( φ - φ 03 ) 2 ( φ 1 - φ ) ( φ - φ 2 ) h h h h h h h h h h h h h h h ( φ 03 < φ 2 φ φ 1 ) , l 2 : y 2 = α 2 2 ( φ 3 - φ ) ( φ - φ 4 ) ( φ - φ 5 ) ( φ - φ 6 ) h h h h h h h h h h h h h h h h h ( φ 6 < φ 5 < φ 4 φ φ 3 ) . Substituting (34) into d φ / d ξ = y and integrating them along l 1 and l 2 , respectively, it follows that (35) φ φ 1 d s ( s - φ 03 ) ( φ 1 - s ) ( s - φ 2 ) = α 2 | ξ | ( along    l 1 ) , φ φ 3 d s ( φ 3 - s ) ( s - φ 4 ) ( s - φ 5 ) ( s - φ 6 ) = α 2 | ξ | ( along    l 2 ) . Completing the integrals above and solving the equations for φ , we obtain the solutions v 1 ( ξ ) , u 1 ( ξ ) (see (22)) and v 2 ( ξ ) , u 2 ( ξ ) (see (24)).

(2) When α 2 > 0 , β < 0 , and 0 < γ < γ 1 ( β ) (as in Figure 7(a)), the closed curves l 3 and l 4 possess the following expressions: (36) l 3 : y 2 = α 2 ( φ 01 - φ ) 2 ( φ 7 - φ ) ( φ - φ 8 ) h h h h h h h h h h h h h h h ( φ 8 φ φ 7 < φ 01 ) , l 4 : y 2 = α 2 ( φ 9 - φ ) ( φ 10 - φ ) ( φ 11 - φ ) ( φ - φ 12 ) h h h h h h h h h h h h h h h h i ( φ 12 φ φ 11 < φ 10 < φ 9 ) . Substituting (36) into d φ / d ξ = y and integrating them along l 3 and l 4 , respectively, it follows that (37) φ 8 φ d s ( φ 01 - s ) ( φ 7 - s ) ( s - φ 8 ) = α 2 | ξ | ( along    l 3 ) , φ 12 φ d s ( φ 9 - s ) ( φ 10 - s ) ( φ 11 - s ) ( s - φ 12 ) = α 2 | ξ | ( along    l 4 ) . Completing the integrals above and solving the equations for φ , we obtain the solutions v 3 ( ξ ) , u 3 ( ξ ) (see (27)) and v 4 ( ξ ) , u 4 ( ξ ) (see (29)).

When γ - γ 1 ( β ) + 0 , it follows that φ 2 , φ 03 , and φ 03 * tend to φ 2 * = - - β / 3 α 2 and φ 1 , φ 3 tend to φ 1 * = - 3 β / α 2 . Further, we have (38) φ 4 φ 2 * , φ 5 φ 2 * , φ 6 φ 2 * , A 1 = ( φ 1 - φ 03 ) ( φ 03 - φ 2 ) 0 , B 1 = φ 1 + φ 2 - 2 φ 03 φ 1 * - φ 2 * , η 1 = - A 1 α 2 2 0 , η 2 = α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) 0 , Δ 1 = B 1 2 + 4 A 1 = B 1 + 2 A 1 B 1 + , cos ( η 1 ξ ) = 1 - ( η 1 ξ ) 2 2 ! + ( η 1 ξ ) 4 4 ! + , sn ( η 2 ξ , k 1 ) = η 2 ξ - ( 1 + k 1 2 ) ( η 2 ξ ) 3 3 ! + . Thus, we have (39) lim γ - γ 1 + 0 v 1 ( ξ ) = lim γ - γ 1 + 0 φ 03 + 2 A 1 Δ 1 cos ( η 1 ξ ) - B 1 = lim γ - γ 1 + 0 φ 03 + 2 ( A 1 + o ( A 1 2 ) ) × ( ( B 1 + 2 A 1 B 1 + o ( A 1 2 ) ) ( 1 + α 2 A 1 ξ 2 4 + o ( A 1 2 ) ) × ( 1 + α 2 A 1 ξ 2 4 + o ( A 1 2 ) ) - B 1 ) - 1 = lim γ - γ 1 + 0 φ 03 + 2 A 1 + o ( A 1 2 ) ( 2 A 1 / B 1 ) + ( α 2 A 1 B 1 ξ 2 / 4 ) + o ( A 1 2 ) = φ 2 * + 8 ( φ 1 * - φ 2 * ) α 2 ( φ 1 * - φ 2 * ) 2 ξ 2 + 8 = - 3 β α 2 9 + 2 β ξ 2 9 - 6 β ξ 2 = v k ( ξ ) ( see ( 21 ) ) , lim γ - γ 1 + 0 v 2 ( ξ ) = lim γ - γ 1 + 0 φ 3 ( φ 4 - φ 6 ) + φ 6 ( φ 3 - φ 4 ) s n 2 ( η 2 ξ , k 1 ) ( φ 4 - φ 6 ) + ( φ 3 - φ 4 ) s n 2 ( η 2 ξ , k 1 ) = lim γ - γ 1 + 0 ( + o ( φ 4 - φ 6 ) α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ] 2 ) - 1 ( [ α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ξ + o ( φ 4 - φ 6 ) ] 2 φ 3 ( φ 4 - φ 6 ) + φ 6 ( φ 3 - φ 4 ) h × [ α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ξ + o ( φ 4 - φ 6 ) α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ] 2 ) × ( [ α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ξ + o ( φ 4 - φ 6 ) ] 2 ( φ 4 - φ 6 ) + ( φ 3 - φ 4 ) × [ α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ξ + o ( φ 4 - φ 6 ) α 2 8 ( φ 3 - φ 5 ) ( φ 4 - φ 6 ) ] 2 ) - 1 ) = lim γ - γ 1 + 0 ( × ( φ 4 - φ 6 ) α 2 ξ 2 + o ( ( φ 4 - φ 6 ) 3 / 2 ) ) - 1 ( ( φ 4 - φ 6 ) α 2 ξ 2 + o ( ( φ 4 - φ 6 ) 3 / 2 ) 8 φ 3 ( φ 4 - φ 6 ) + φ 6 ( φ 3 - φ 4 ) ( φ 3 - φ 5 ) × ( φ 4 - φ 6 ) α 2 ξ 2 + o ( ( φ 4 - φ 6 ) 3 / 2 ) ) × ( ( φ 4 - φ 6 ) α 2 ξ 2 + o ( ( φ 4 - φ 6 ) 3 / 2 ) 8 ( φ 4 - φ 6 ) + ( φ 3 - φ 4 ) ( φ 3 - φ 5 ) × ( φ 4 - φ 6 ) α 2 ξ 2 + o ( ( φ 4 - φ 6 ) 3 / 2 ) ) - 1 ) = 8 φ 1 * + φ 2 * ( φ 1 * - φ 2 * ) 2 α 2 ξ 2 8 + ( φ 1 * - φ 2 * ) 2 α 2 ξ 2 = - 3 β α 2 9 + 2 β ξ 2 9 - 6 β ξ 2 = v k ( ξ ) ( see ( 21 ) ) .

Similarly, we can prove the limit property of v 3 ( ξ ) and v 4 ( ξ ) when γ γ 1 ( β ) - 0 .

(3) When α 2 < 0 , β > 0 , and - γ 1 ( β ) < γ < γ 1 ( β ) (as in Figure 7(b)), the curve connecting with ( φ 01 , 0 ) embraces the expression (40) y 2 = - α 2 2 ( φ - φ 01 ) 2 ( φ 2 + 2 φ 01 φ + 3 φ 01 2 + 2 β α 2 ) . Substituting (40) into d φ / d ξ = y and integrating it, we have (41) p φ d s ( s - φ 01 ) 2 ( s 2 + 2 φ 01 s + 3 φ 01 2 + ( 2 β / α 2 ) ) = - α 2 2 ξ , where (42) p = φ 01 - - β α 2 24 α 2 φ 01 2 + 8 β α 2 φ 01 2 - 8 φ 01 - α 2 β + 9 β . Completing the integral above and solving the equation for φ , we obtain (43) v 5 ( ξ ) = 4 A 3 ( φ 01 - δ e - η 5 ξ ) - φ 01 ( δ e - η 5 ξ - B 3 ) 2 4 A 3 - ( δ e - η 5 ξ - B 3 ) 2 . From v 5 ( ξ ) , we get (44) v 6 ( ξ ) = v 5 ( - ξ ) = 4 A 3 ( φ 01 - δ e η 5 ξ ) - φ 01 ( δ e η 5 ξ - B 3 ) 2 4 A 3 - ( δ e η 5 ξ - B 3 ) 2 .

(4) When α 2 < 0 , β > 0 , and - γ 1 ( β ) < γ < γ 1 ( β ) (as in Figure 7(b)), the curve connecting with ( φ 03 , 0 ) embraces the expression (45) y 2 = - α 2 2 ( φ 03 - φ ) 2 ( φ 2 + 2 φ 03 φ + 3 φ 03 2 + 2 β α 2 ) . Substituting (45) into d φ / d ξ = y and integrating it, we have (46) q φ d s ( φ 03 - s ) 2 ( s 2 + 2 φ 03 s + 3 φ 03 2 + ( 2 β / α 2 ) ) = - α 2 2 ξ , where (47) q = φ 03 + - β α 2 24 α 2 φ 03 2 + 8 β α 2 φ 03 2 + 8 φ 03 - α 2 β + 9 β . Completing the integral above and solving the equation for φ , we obtain (48) v 7 ( ξ ) = 4 A 4 ( φ 03 - μ e η 6 ξ ) - φ 03 ( μ e η 6 ξ - B 4 ) 2 4 A 4 - ( μ e η 6 ξ - B 4 ) 2 . From v 7 ( ξ ) , we get (49) v 8 ( ξ ) = v 7 ( - ξ ) = 4 A 4 ( φ 03 - μ e - η 6 ξ ) - φ 03 ( μ e - η 6 ξ - B 4 ) 2 4 A 4 - ( μ e - η 6 ξ - B 4 ) 2 .

Note that when γ 0 , it follows that φ 01 φ 01 * * = - - β / α 2 , A 3 - 4 β / α 2 , B 3 - 4 - β / α 2 , and η 5 - 2 β . Thus, we have (50) lim γ 0 v 5 ( ξ ) = lim γ 0 4 A 3 ( φ 01 - δ e - η 5 ξ ) - φ 01 ( δ e - η 5 ξ - B 3 ) 2 4 A 3 - ( δ e - η 5 ξ - B 3 ) 2 = lim γ 0 δ φ 01 e - 2 η 5 ξ + ( 4 A 3 - 2 φ 01 B 3 ) e - η 5 ξ δ e - 2 η 5 ξ - 2 B 3 e - η 5 ξ = δ - α 2 β e - 2 β ξ - 8 β δ α 2 e - 2 β ξ - 8 - α 2 β = - β α 2 e β / 2 ξ - e - β / 2 ξ e β / 2 ξ + e - β / 2 ξ = - β α 2 tanh ( β 2 ξ ) = v δ + ( ξ ) ( see ( 31 ) ) .

Similarly, we can also get v 6 ( ξ ) v δ - ( ξ ) , v 7 ( ξ ) v δ + ( ξ ) , and v 8 ( ξ ) v δ - ( ξ ) when γ 0 . These complete the derivations of Propositions 13.

3. The Bifurcations of Compactons and Peakons for <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M247"> <mml:mi>D</mml:mi> <mml:mo mathvariant="bold">(</mml:mo> <mml:mn>1,2</mml:mn> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula> System

When m = 1 and n = 2 , system (1) becomes (51) u t + v x = 0 , v t + a ( v 2 ) x x x + b u x v + c u v x = 0 .

We will reveal two kinds of interesting bifurcation phenomena to system (51). The first phenomenon is that the smooth solitary waves can turn into the compactons. The second phenomenon is that the peakons can be bifurcated from the singular cusp waves and the solitary waves. The concrete results are stated as follows.

Note that P ( 1,2 ) system is read as (52) d φ d τ = 2 φ y , d y d τ = - 2 y 2 - α 1 φ 2 - β φ - γ , where (53) α 1 = b + c 2 a λ , and β , γ are given in (13).

For fixed α 1 , put (54) γ 2 ( β ) = 2 β 2 9 α 1 , γ 3 ( β ) = β 2 4 α 1 .

Proposition 4.

For given α 1 > 0 and β 0 , if 0 < γ < γ 3 ( β ) , then system (51) has a family of solitary wave solutions which become the compacton solution (55) v o ( ξ ) = { - 4 β 3 α 1 cos 2 ( α 1 4 ξ ) , | ξ | < 2 π α 1 , 0 , | ξ | 2 π α 1 , u o ( ξ ) = 1 λ v o ( ξ ) + d , when γ 0 + 0 . The solitary wave solutions are as follows.

(1) When β < 0 , the solitary wave solutions possess the expressions v 9 ( ξ ) = φ ( ξ ) and u 9 ( ξ ) = ( 1 / λ ) v 9 ( ξ ) + d , and φ ( ξ ) is given by the implicit function (56) α 1 | ξ | - π 2 = arcsin ( r 1 + r 3 - 2 φ r 1 - r 3 ) - r 2 A 5 ln | Φ 1 | , where (57) r 1 = - 1 3 α 1 ( 3 α 1 r 2 + 2 β - 6 α 1 β r 2 + 4 β 2 ) , r 2 = - 1 2 α 1 ( β + β 2 - 4 α 1 γ ) , r 3 = - 1 3 α 1 ( 3 α 1 r 2 + 2 β + 6 α 1 β r 2 + 4 β 2 ) , A 5 = ( r 1 - r 2 ) ( r 2 - r 3 ) , B 5 = r 1 + r 3 - 2 r 2 , Φ 1 = ( r 1 - r 3 ) ( φ - r 2 ) 2 A 5 ( r 1 - φ ) ( φ - r 3 ) + B 5 ( φ - r 2 ) + 2 A 5 .

(2) When β > 0 , the solitary wave solutions possess the expressions v 10 ( ξ ) = φ ( ξ ) and u 10 ( ξ ) = ( 1 / λ ) v 10 ( ξ ) + d , and φ ( ξ ) is given by the implicit function (58) π - α 1 | ξ | 2 = arcsin ( r 6 + r 4 - 2 φ r 6 - r 4 ) + r 5 A 6 ln | Φ 2 | , where (59) r 4 = - 1 3 α 1 ( 3 α 1 r 5 + 2 β + 6 α 1 β r 5 + 4 β 2 ) , r 5 = - 1 2 α 1 ( β - β 2 - 4 α 1 γ ) , r 6 = - 1 3 α 1 ( 3 α 1 r 5 + 2 β - 6 α 1 β r 5 + 4 β 2 ) , A 6 = ( r 6 - r 5 ) ( r 5 - r 4 ) , B 6 = 2 r 5 - r 6 - r 4 , Φ 2 = 2 A 6 ( r 6 - φ ) ( φ - r 4 ) + B 6 ( r 5 - φ ) + 2 A 6 ( r 6 - r 4 ) ( r 5 - φ ) . For the varying process, see Figure 8.

The solitary waves turn into the compactons. The varying process for the figures of v 9 ( ξ ) and v 10 ( ξ ) when α 1 > 0 , β 0 , and γ 0 + 0 where α 1 = 2 .

β = - 9 , γ = 1 × 1 0 - 1

β = - 9 , γ = 1 × 1 0 - 4

β = - 9 , γ = 1 × 1 0 - 8

β = 9 , γ = 1 × 1 0 - 1

β = 9 , γ = 1 × 1 0 - 4

β = 9 , γ = 1 × 1 0 - 8

Proposition 5.

For given α 1 < 0 and β 0 , if γ 3 ( β ) < γ < γ 2 ( β ) , then system (51) has two types of nonlinear wave solutions which tend to the peakon solution (60) v m ( ξ ) = 2 β 3 α 1 ( e - | α 1 | / 4 | ξ | - 1 ) , u m ( ξ ) = 1 λ v m ( ξ ) + d , when γ γ 2 ( β ) - 0 . For the varying process, see Figures 9 and 10. These two types of nonlinear wave solutions are singular cusp wave solutions and solitary wave solutions of the following expressions.

(1) When β < 0 , the singular cusp wave solutions possess the expression (61) v 11 ( ξ ) = [ ( β 3 α 1 - γ 2 α 1 ) - ( β 3 α 1 + γ 2 α 1 ) e - | α 1 | / 4 | ξ | ] × ( e | α 1 | / 4 | ξ | - 1 ) , u 11 ( ξ ) = 1 λ v 11 ( ξ ) + d , and the solitary wave solutions possess the expressions v 12 ( ξ ) = φ ( ξ ) and u 12 ( ξ ) = ( 1 / λ ) v 12 ( ξ ) + d , and φ ( ξ ) is given by the implicit function (62) | ξ | = - 4 | α 1 | ( ln | 2 Φ 3 - M 1 r 7 - r 8 | h h h h h h h h + r 9 A 7 ln | 2 A 7 Φ 3 + B 7 ( r 9 - φ ) + 2 A 7 ( r 7 - r 8 ) ( r 9 - φ ) | ) , where (63) Φ 3 = ( r 7 - φ ) ( r 8 - φ ) , M 1 = 2 φ - r 7 - r 8 , r 7 = - 1 3 α 1 ( 3 α 1 r 9 + 2 β + 6 α 1 β r 9 + 4 β 2 ) , r 8 = - 1 3 α 1 ( 3 α 1 r 9 + 2 β - 6 α 1 β r 9 + 4 β 2 ) , r 9 = - 1 2 α 1 ( β - β 2 - 4 α 1 γ ) , A 7 = ( r 8 - r 9 ) ( r 7 - r 9 ) , B 7 = r 7 + r 8 - 2 r 9 .

(2) When β > 0 , the singular cusp wave solutions possess the expression (64) v 13 ( ξ ) = [ ( β 3 α 1 - γ 2 α 1 ) - ( β 3 α 1 + γ 2 α 1 ) e | α 1 | / 4 | ξ | ] × ( e - | α 1 | / 4 | ξ | - 1 ) , u 13 ( ξ ) = 1 λ v 13 ( ξ ) + d , and the solitary wave solutions possess the expressions v 14 ( ξ ) = φ ( ξ ) and u 14 ( ξ ) = ( 1 / λ ) v 14 ( ξ ) + d , and φ ( ξ ) is given by the implicit function (65) | ξ | = - 4 | α 1 | ( ln | 2 Φ 4 + M 2 r 11 - r 12 | h - r 10 A 8 ln | 2 A 8 Φ 4 + B 8 ( φ - r 10 ) + 2 A 8 ( r 11 - r 12 ) ( φ - r 10 ) | ) , where (66) Φ 4 = ( φ - r 11 ) ( φ - r 12 ) , M 2 = 2 φ - r 11 - r 12 , r 10 = - 1 2 α 1 ( β + β 2 - 4 α 1 γ ) , r 11 = - 1 3 α 1 ( 3 α 1 r 10 + 2 β + 6 α 1 β r 10 + 4 β 2 ) , r 12 = - 1 3 α 1 ( 3 α 1 r 10 + 2 β - 6 α 1 β r 10 + 4 β 2 ) , A 8 = ( r 10 - r 11 ) ( r 10 - r 12 ) , B 8 = 2 r 10 - r 11 - r 12 .

The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of v 11 ( ξ ) and v 12 ( ξ ) when α 1 < 0 , β < 0 , and γ γ 2 ( β ) - 0 where α 1 = - 2 and β = - 9 .

γ = γ 2 ( β ) - 1 0 - 1

γ = γ 2 ( β ) - 1 0 - 4

γ = γ 2 ( β ) - 1 0 - 8

γ = γ 2 ( β ) - 1 0 - 1

γ = γ 2 ( β ) - 1 0 - 4

γ = γ 2 ( β ) - 1 0 - 8

The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of v 13 ( ξ ) and v 14 ( ξ ) when α 1 < 0 , β > 0 , and γ γ 2 ( β ) - 0 where α 1 = - 2 and β = 9 .

γ = γ 2 ( β ) - 1 0 - 1

γ = γ 2 ( β ) - 1 0 - 4

γ = γ 2 ( β ) - 1 0 - 8

γ = γ 2 ( β ) - 1 0 - 1

γ = γ 2 ( β ) - 1 0 - 4

γ = γ 2 ( β ) - 1 0 - 8

The Derivations of Propositions 4 and 5. According to the qualitative theory, we obtain the bifurcation phase portraits of system (52) as in Figure 11. Through some orbits in Figure 11, we derive the results of Propositions 4 and 5 as follows.

The bifurcation phase portraits of system (52).

α 1 > 0

α 1 < 0

(1) When α 1 > 0 , β < 0 , and 0 < γ < γ 3 ( β ) (as in Figure 11(a)), the homoclinic orbit owns the expression (67) y 2 = α 1 4 φ 2 ( φ - r 2 ) 2 ( r 1 - φ ) ( φ - r 3 ) ( r 3 < r 2 < φ r 1 ) . Substituting (67) into d φ / d ξ = y and integrating it, we have (68) | ξ | = φ r 1 4 s 2 α 1 ( s - r 2 ) 2 ( r 1 - s ) ( s - r 3 ) d s . Completing the integral above, we get (69) | ξ | = 4 α 1 [ arcsin ( r 1 + r 3 - 2 φ r 1 - r 3 ) + π 2 - r 2 A 5 ln | Φ 1 | ] .

(2) When α 1 > 0 , β > 0 , and 0 < γ < γ 3 ( β ) (as in Figure 11(a)), the homoclinic orbit is expressed by (70) y 2 = α 1 4 φ 2 ( r 5 - φ ) 2 ( φ - r 4 ) ( r 6 - φ ) ( r 4 φ < r 5 < r 6 ) . Substituting (70) into d φ / d ξ = y and integrating it, we have (71) | ξ | = r 4 φ 4 s 2 α 1 ( r 5 - s ) 2 ( s - r 4 ) ( r 6 - s ) . Completing the integral above, we get (72) | ξ | = - 4 α 1 [ arcsin ( r 6 + r 4 - 2 φ r 6 - r 4 ) - π 2 + r 5 A 6 ln | Φ 2 | ] .

Note that when β < 0 and γ 0 + 0 , it follows that r 1 r 1 * = - 4 β / 3 α 1 , r 2 0 , and r 3 0 . Further, we have (73) B 5 = r 1 + r 3 - 2 r 2 r 1 * , A 5 = ( r 1 - r 2 ) ( r 2 - r 3 ) 0 , lim γ 0 + 0 r 3 r 2 = lim γ 0 + 0 - 3 α 1 r 2 + 2 β + 6 α 1 β r 2 + 4 β 2 3 α 1 r 2 = lim γ 0 + 0 - 1 3 α 1 r 2 × [ 3 α 1 r 2 + 2 β - 2 β ( 1 + 3 α 1 r 2 4 β + o ( r 2 2 ) ) ] = lim γ 0 + 0 - 1 3 α 1 r 2 ( 3 α 1 r 2 - 3 α 1 r 2 2 + o ( r 2 2 ) ) = - 1 2 , lim γ 0 + 0 r 2 A 5 = lim γ 0 + 0 r 2 ( r 1 - r 2 ) ( r 2 - r 3 ) = lim γ 0 + 0 r 2 r 1 + r 3 - r 2 - r 1 ( r 3 / r 2 ) = 0 ( 3 / 2 ) r 1 * = 0 , lim γ 0 + 0 Φ 1 = lim γ 0 + 0 ( r 1 - r 3 ) ( φ - r 2 ) 2 A 5 ( r 1 - φ ) ( φ - r 3 ) + B 5 ( φ - r 2 ) + 2 A 5 = r 1 * φ r 1 * φ = 1 . Thus, from (74) | ξ | = lim γ 0 + 0 4 α 1 [ arcsin ( r 1 + r 3 - 2 φ r 1 - r 3 ) + π 2 - r 2 A 5 ln | Φ 1 | ] , we have (75) | ξ | = 4 α 1 [ arcsin ( r 1 * - 2 φ r 1 * ) + π 2 ] . Solving (75) for φ , we can get φ = v o ( ξ ) (see (55)), which is the same as (2).

Similarly, from (72), we can also get solution φ v o ( ξ ) when β > 0 and γ 0 + 0 .

(3) When α 1 < 0 , β 0 , and γ 3 ( β ) < γ < γ 2 ( β ) (as in Figure 11(b)), the curves Γ 1 ± and Γ 2 ± own the expression (76) y 2 = - α 1 4 ( φ 2 + 4 β 3 α 1 φ + 2 γ α 1 ) . Substituting (76) into d φ / d ξ = y and integrating it, we have (77) φ 0 d s s 2 + ( 4 β / 3 α 1 ) s + ( 2 γ / α 1 ) = - α 1 4 | ξ | ( β < 0 ) , 0 φ d s s 2 + ( 4 β / 3 α 1 ) s + ( 2 γ / α 1 ) = - α 1 4 | ξ | ( β > 0 ) . Completing the integrals above and solving the equations for φ , we get φ = v 11 ( ξ ) (see (61)) and φ = v 13 ( ξ ) (see (64)).

(4) When α 1 < 0 , β 0 , and γ 3 ( β ) < γ < γ 2 ( β ) (as in Figure 11(b)), two homoclinic orbits can be expressed as (78) y 2 = - α 1 4 φ 2 ( φ - r 9 ) 2 ( r 7 - φ ) ( r 8 - φ ) ( β < 0 ) , (79) y 2 = - α 1 4 φ 2 ( r 10 - φ ) 2 ( φ - r 11 ) ( φ - r 12 ) ( β > 0 ) . Substituting (78) and (79) into d φ / d ξ = y and integrating it, we have (80) | ξ | = φ r 8 4 s 2 | α 1 | ( s - r 9 ) 2 ( r 7 - s ) ( r 8 - s ) d s , | ξ | = r 11 φ 4 s 2 | α 1 | ( r 10 - s ) 2 ( s - r 11 ) ( s - r 12 ) d s . Completing the integrals above, we get (62) and (65).

Note that when β < 0 and γ γ 2 ( β ) - 0 , it follows that γ / 2 α 1 β / 3 α 1 , r 7 0 , r 8 0 , r 9 r 9 * = - 2 β / 3 α 1 , and (81) M 1 = 2 φ - r 7 - r 8 2 φ , B 7 = r 8 + r 7 - 2 r 9 - 2 r 9 * , A 7 = ( r 8 - r 9 ) ( r 7 - r 9 ) - r 9 * , Φ 3 = ( r 7 - φ ) ( r 8 - φ ) - φ . Thus, we have (82) lim γ γ 2 - 0 v 13 ( ξ ) = lim γ γ 2 - 0 [ ( β 3 α 1 - γ 2 α 1 ) h h h h h h h - ( β 3 α 1 + γ 2 α 1 ) e - | α 1 | / 4 | ξ | ] × ( e | α 1 | / 4 | ξ | - 1 ) = 2 β 3 α 1 ( e - | α 1 | / 4 | ξ | - 1 ) = v m ( ξ ) ( see ( 60 ) ) , (83) | ξ | = lim γ γ 2 - 0 - 4 | α 1 | × ( ln | 2 Φ 3 - M 1 r 7 - r 8 | h h h + r 9 A 7 ln | 2 A 7 Φ 3 + B 7 ( r 9 - φ ) + 2 A 7 ( r 7 - r 8 ) ( r 9 - φ ) | ) = lim γ γ 2 - 0 - 4 | α 1 | × ln | ( 2 Φ 3 - M 1 r 7 - r 8 ) ( 2 A 7 Φ 3 + B 7 ( r 9 - φ ) + 2 A 7 ( r 7 - r 8 ) ( r 9 - φ ) ) r 9 / A 7 h h h h h × ( 2 A 7 Φ 3 + B 7 ( r 9 - φ ) + 2 A 7 ( r 7 - r 8 ) ( r 9 - φ ) ) r 9 / A 7 | = - 4 | α 1 | ln | - 4 φ ( r 9 * - φ ) 4 r 9 * φ | = - 4 | α 1 | ln | r 9 * - φ r 9 * | . Solving (78) for φ , we obtain φ = v m ( ξ ) (see (60)).

Similarly, from (65), we can also get φ = v m ( ξ ) (see (60)) when β > 0 and γ γ 2 ( β ) - 0 . These complete the derivations of Propositions 4 and 5.

4. The Bifurcations of Solitary Waves for <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M391"> <mml:mi>D</mml:mi> <mml:mo mathvariant="bold">(</mml:mo> <mml:mn>2,2</mml:mn> <mml:mo mathvariant="bold">)</mml:mo></mml:math> </inline-formula> System

When m = 2 and n = 2 , system (1) becomes (84) u t + ( v 2 ) x = 0 , v t + a ( v 2 ) x x x + b u x v + c u v x = 0 .

We will reveal the interesting bifurcation phenomenon to system (84). That is, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves. The concrete results are stated as follows.

Note that P ( 2,2 ) system is read as (85) d φ d τ = 2 φ y , d y d τ = - 2 y 2 - α 2 φ 3 - β φ - γ , where α 2 is as (18) and β , γ are given in (13). Let (86) r a = 1 3 [ 1 α 2 ( 5 Q 4 ) 1 / 3 - β ( 100 Q ) 1 / 3 ] , r b = β 3 ( 25 2 Q ) 1 / 3 ( 1 - i 3 ) - 1 6 α 2 ( 5 Q 4 ) 1 / 3 ( 1 + i 3 ) , r c = β 3 ( 25 2 Q ) 1 / 3 ( 1 + i 3 ) - 1 6 α 2 ( 5 Q 4 ) 1 / 3 ( 1 - i 3 ) , Q = - 27 α 2 2 γ + 80 α 2 3 β 3 + 729 α 2 4 γ 2 , γ 4 ( β ) = 4 5 27 ( - β 3 α 2 ) 1 / 2 , and let γ 1 ( β ) be as (20).

Proposition 6.

For fixed α 2 and β , system (84) has solitary wave solutions (87) v i ± ( ξ ) = ± - 5 β α 2 [ 2 3 - tanh 2 ( - α 2 β 80 4 ξ ) ] , u i ( ξ ) = 1 λ v i 2 ( ξ ) + d , which can be bifurcated from the following two types of nonlinear wave solutions.

Smooth periodic wave solution: (88) v 15 ( ξ ) = r a - ( r a - r b ) s n 2 ( η 7 ξ , k 3 ) , u 15 ( ξ ) = 1 λ v 15 2 ( ξ ) + d ,

where (89) - γ 4 ( β ) < γ < γ 4 ( β ) , η 7 = α 2 20 ( r a - r c ) , k 3 = r a - r b r a - r c .

Singular periodic wave solutions: (90) v 16 ( ξ ) = ( A 9 + r a ) c n ( η 8 ξ , k 4 ) - A 9 + r a c n ( η 8 ξ , k 4 ) + 1 , u 16 ( ξ ) = 1 λ v 16 2 ( ξ ) + d , ( α 2 > 0 , β < 0 ) , (91) v 17 ( ξ ) = A 9 + r a - ( A 9 - r a ) c n ( η 8 ξ , k 5 ) 1 + c n ( η 8 ξ , k 5 ) , u 17 ( ξ ) = 1 λ v 17 2 ( ξ ) + d , ( α 2 < 0 , β > 0 ) ,

where (92) - γ 1 ( β ) < γ < - γ 4 ( β ) , A 9 2 = ( b 0 - r a ) 2 + a 0 2 , a 0 2 = - 1 4 ( r b - r c ) 2 , b 0 = 1 2 ( r b + r c ) , η 8 = 1 5 A 9 | α 2 | , k 4 2 = A 9 - b 0 + r a 2 A 9 , k 5 2 = A 9 + b 0 - r a 2 A 9 . For the varying process, see Figures 12 and 13.

The solitary wave is bifurcated from the smooth periodic wave and the singular periodic wave. The varying process for the figures of v 15 ( ξ ) and v 16 ( ξ ) when α 2 = 5 , β = - 9 , γ γ 4 ( β ) + 0 , or γ γ 4 ( β ) - 0 .

γ = - γ 4 ( β ) + 1 0 - 1

γ = - γ 4 ( β ) + 1 0 - 4

γ = - γ 4 ( β ) + 1 0 - 8

γ = - γ 4 ( β ) - 1 0 - 1

γ = - γ 4 ( β ) - 1 0 - 4

γ = - γ 4 ( β ) - 1 0 - 8

The solitary wave is bifurcated from the smooth periodic wave and the singular periodic wave. The varying process for the figures of v 15 ( ξ ) and v 17 ( ξ ) when α 2 = - 5 , β = 9 , γ γ 4 ( β ) + 0 , or γ γ 4 ( β ) - 0 .

γ = - γ 4 ( β ) + 1 0 - 1

γ = - γ 4 ( β ) + 1 0 - 4

γ = - γ 4 ( β ) + 1 0 - 8

γ = - γ 4 ( β ) - 1 0 - 1

γ = - γ 4 ( β ) - 1 0 - 4

γ = - γ 4 ( β ) - 1 0 - 8

The Derivations of Proposition 6. According to the qualitative theory, we also obtain the bifurcation phase portraits of system (85) as in Figure 14. Employing some orbits in Figure 14, we derive the results of Proposition 6 as follows.

The bifurcation phase portraits of system (85).

α 2 > 0

α 2 < 0

(1) When α 2 > 0 , β < 0 , and - γ 4 ( β ) < γ < 0 (as in Figure 14(a)), the closed orbit owns the expression (93) y 2 = α 2 5 ( r a - φ ) ( φ - r b ) ( φ - r c ) ( r c < r b φ r a ) . Substituting (93) into d φ / d ξ = y and integrating it, we have (94) φ r a d s ( r a - s ) ( s - r b ) ( s - r c ) = α 2 5 | ξ | . Completing the integral above and solving the equation for φ , we get v 15 ( ξ ) (see (88)).

When α 2 > 0 and - γ 1 ( β ) < γ < - γ 4 ( β ) , r a is real and r b and r c become a pair of conjugate complex in (94). Completing the integral (94) and solving the equation for φ , we get v 16 ( ξ ) (see (90)).

(2) When α 2 < 0 , β > 0 , similarly, we get v 15 ( ξ ) (see (88)) and v 17 ( ξ ) (see (91)).

Note that when α 2 > 0 and γ - γ 4 ( β ) + 0 , it follows that r a r a * = ( 2 / 3 ) - 5 β / α 2 , r b r b * = - ( 1 / 3 ) - 5 β / α 2 , and r c r b * = - ( 1 / 3 ) - 5 β / α 2 . Further, (95) k 3 = r a - r b r a - r c 1 , η 7 = α 2 20 ( r a - r c ) - α 2 β 80 4 . Thus, we have (96) lim γ - γ 4 + 0 v 15 ( ξ ) = lim γ - γ 4 + 0 r a - ( r a - r b ) s n 2 ( η 7 ξ , k 3 ) = r a * - ( r a * - r b * ) tanh 2 ( - α 2 β 80 4 ξ ) = - 5 β α 2 [ 2 3 - tanh 2 ( - α 2 β 80 4 ξ ) ] = v i + ( ξ ) ( see ( 87 ) ) .

If α 2 > 0 and γ - γ 4 ( β ) - 0 , then it follows that r a r a * = ( 2 / 3 ) - 5 β / α 2 , r b r b * = - ( 1 / 3 ) - 5 β / α 2 , and r c r b * = - ( 1 / 3 ) - 5 β / α 2 . Further, it follows that (97) a 0 2 = - 1 4 ( r b - r c ) 2 0 , b 0 = 1 2 ( r b + r c ) r b * , A 9 = ( b 0 - r a ) 2 + a 0 2 r a * - r b * , k 4 2 = A 9 - b 0 + r a 2 A 9 1 , η 8 = 1 5 A 9 α 2 2 - α 2 β 80 4 . Thus, we get (98) lim γ - γ 4 - 0 v 16 ( ξ ) = lim γ - γ 4 - 0 ( A 9 + r a ) cn ( η 8 ξ , k 4 ) - A 9 + r a cn ( η 8 ξ , k 4 ) + 1 = ( 2 r a * - r b * ) sech ( 2 - α 2 β / 80 4 ξ ) + r b * 1 + sech ( 2 - α 2 β / 80 4 ξ ) = ( 2 r a * - r b * ) + r b * [ 2 cosh 2 ( - α 2 β / 80 4 ξ ) - 1 ] 2 cosh 2 ( - α 2 β / 80 4 ξ ) = ( r a * - r b * ) sec h 2 ( - α 2 β 80 4 ξ ) + r b * = - 5 β α 2 [ 2 3 - tanh 2 ( - α 2 β 80 4 ξ ) ] = v i + ( ξ ) ( see ( 87 ) ) .

Similarly, we can also get v 15 ( ξ ) v i - ( ξ ) and v 17 ( ξ ) v i - ( ξ ) when α 2 < 0 , β > 0 , and γ - γ 4 ( β ) ± 0 . Hereto, we have completed all of the derivations.

5. Conclusions

In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we have revealed some interesting bifurcation phenomena of nonlinear waves for the D ( m , n ) system (1). Firstly, for D ( 2,1 ) system, we have pointed out that the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves (see Figures 1 and 2). In the meantime, the kink waves can be bifurcated from the solitary waves and the singular waves (see Figures 36). Secondly, for D ( 1,2 ) system, we have showed that the solitary waves can turn into the compactons (see Figure 8) and the peakons can be bifurcated from the singular cusp waves and the solitary waves (see Figures 9 and 10). Thirdly, for D ( 2,2 ) system, we have confirmed that the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves (see Figures 12 and 13).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (no. 11171115).

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