Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

Abstract

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)_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_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 as𝑢_𝑡+𝑣^𝑚_𝑥𝑣=0,_𝑡+𝑎𝑣^𝑛_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_𝑥=0, (1) where 𝑎, 𝑏, 𝑐, 𝑚, and 𝑛 are constants. Through some transformations, Xie and Yan [3] got some compacton and solitary pattern solutions of system (1) with 𝑚=𝑛−1, including 𝑣_𝑎(𝜉)=2𝑛(𝜆−𝑐𝑑₀)(𝑛+1)𝑎𝛼_𝑚cos²(𝑛−1)√𝛼_𝑚𝜉2𝑛^1/(𝑛−1),𝑢_𝑎(𝜉)=𝑣_𝑎^𝑛−1(𝜉)𝜆+𝑑₀,𝑣_𝑏(𝜉)=2𝑛(𝑐𝑑₀−𝜆)(𝑛+1)𝑎𝛼_𝑚sinh²(𝑛−1)√𝛼_𝑚𝜉2𝑛^1/(𝑛−1),𝑢_𝑏(𝜉)=𝑣_𝑏^𝑛−1(𝜉)𝜆+𝑑₀, (2)(3) where 𝜆≠0, 𝑑₀ are constants, 𝜉=𝑥−𝜆𝑡,𝛼_𝑚=𝑏𝑚+𝑐().𝑎𝜆𝑚+1 (4) Deng et al. [4], 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. [5] 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 [6].

In [2], the system 𝑢_𝑡+𝑣²_𝑥𝑣=0,_𝑡+𝑎𝑣_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_𝑥=0 (5) was introduced. Wang [7] gave recursion, Hamiltonian, symplectic and cosymplectic operator, roots of symmetries, and scaling symmetry for system (5). Wazwaz [8], by using the tanh method and the sine-cosine method, obtained many solutions with compact and noncompact structures of system (5), including 𝑣_𝑐(𝜉)=𝜆−62𝑏+𝑐sech𝜆𝑎𝜉,𝑢_𝑐(𝜉)=1𝜆𝑣²_𝑐(𝜉),𝜆𝑎,𝑣>0_𝑑(𝜉)=𝜆−32𝑏+𝑐tanh−𝜆𝜉2𝑎,𝑢_𝑑(𝜉)=1𝜆𝑣²_𝑑(𝜉),𝜆𝑎,𝑣<0_𝑒(𝜉)=𝜆−32𝑏+𝑐coth−𝜆𝜉2𝑎,𝑢_𝑒(𝜉)=1𝜆𝑣²_𝑒(𝜉),𝜆𝑎<0. (6) Biazar and Ayati [9] obtained some solutions of system (5) through Exp-function method and modification of Exp-function method. Zhang et al. [10], 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 [11]. El-Wakil and Abdou [12] got some new exact solutions of system (5) by means of modified extended tanh-function method.

The other generalized Drinfel'd-Sokolov system [13] 𝑢_𝑡+𝑎₁𝑢𝑢_𝑥+𝑏₁𝑢_𝑥𝑥𝑥+𝑐₀𝑣^𝜎_𝑥𝑣=0,_𝑡+𝑎₂𝑢𝑣_𝑥+𝑏₂𝑣_𝑥𝑥𝑥=0 (7) is considered in [14–22]. In [23–31], 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 transformations𝜉=𝑥−𝜆𝑡,𝑢(𝑥,𝑡)=𝜓(𝜉),𝑣(𝑥,𝑡)=𝜑(𝜉), (8)system (1) is reduced to𝜆𝜓^′−𝜑^𝑚′=0,−𝜆𝜑^′+𝑎𝜑^{𝑛′′′}+𝑏𝜓^′𝜑+𝑐𝜓𝜑^′=0. (9) Integrating the first equation of system (9) once, we have 1𝜓=𝜆𝜑^𝑚+𝑑, (10) where 𝑑 is an integral constant. Substituting (10) into the second equation of system (9) and integrating it once yield the following equation: 𝑔+(𝑐𝑑−𝜆)𝜑+𝑏𝑚+𝑐𝜆()𝜑𝑚+1^𝑚+1+𝑎𝑛(𝑛−1)𝜑^𝑛−2𝜑^′2+𝑛𝜑^𝑛−1𝜑^′′=0, (11) where 𝑔 is another integral constant. Letting 𝜑^′(𝜉)=𝑦, we obtain the planar system 𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦𝑛()𝜑𝑑𝜉=−𝑛−1^𝑛−2𝑦²+𝛼_𝑚𝜑^𝑚+1+𝛽𝜑+𝛾𝑛𝜑^𝑛−1, (12) where 𝛼_𝑚 is given in (4) and 𝛽=𝑐𝑑−𝜆𝑎,𝑔𝛾=𝑎. (13) Letting 𝑛𝜑^𝑛−1𝑑𝜏=𝑑𝜉, system (12) becomes 𝑑𝜑𝑑𝜏=𝑛𝜑^𝑛−1𝑦,𝑑𝑦()𝜑𝑑𝜏=−𝑛𝑛−1^𝑛−2𝑦²−𝛼_𝑚𝜑^𝑚+1−𝛽𝜑−𝛾, (14) which is called 𝑃(𝑚,𝑛) system. Clearly, systems (12) and (14) possess the same first integral 𝜑^𝑛𝑛2𝜑^𝑛−2𝑦²+𝛼_𝑚𝜑𝑚+𝑛+1^𝑚+1+𝛽𝛾𝑛+1𝜑+𝑛=ℎ. (15)

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 𝐷(2,1) System

When 𝑚=2 and 𝑛=1, system (1) becomes 𝐷(2,1) system: 𝑢_𝑡+𝑣²_𝑥𝑣=0,_𝑡+𝑎𝑣_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_𝑥=0. (16)

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 𝑑𝜑𝑑𝜏=𝑦,𝑑𝑦𝑑𝜏=−𝛼₂𝜑³−𝛽𝜑−𝛾, (17) where 𝛼₂=2𝑏+𝑐,3𝑎𝜆 (18) and 𝛽, 𝛾 are given in (13). When 27𝛼₂𝛾²>−4𝛽³, let 𝜑₀₁=1𝛼₂Ω18^1/32−𝛽3Ω^1/3,𝜑₀₂=𝛽1−√13𝑖12Ω^1/3−1+√3𝑖2𝛼₂Ω18^1/3,𝜑₀₃=𝛽1+√13𝑖12Ω^1/3−1−√3𝑖2𝛼₂Ω18^1/3,Ω=12𝛼³₂𝛽³+81𝛼⁴₂𝛾²−9𝛼²₂𝛾,𝛾₁𝛽=23√3−𝛽³𝛼₂^1/2. (19)(20)

Proposition 1. For given 𝛼₂>0 and 𝛽<0, if  −𝛾₁(𝛽)<𝛾<0, then 𝜑₀₁, 𝜑₀₂, and 𝜑₀₃ are real and system (16) has two types of special periodic wave solutions which become the fractional solitary wave solution 𝑣_𝑘(𝜉)=−3𝛽𝛼₂9+2𝛽𝜉²9−6𝛽𝜉²,𝑢_𝑘(𝜉)=1𝜆𝑣²_𝑘(𝜉)+𝑑, (21) when 𝛾→−𝛾₁(𝛽)+0. These two types of special periodic wave solutions are as follows.(1)Trigonometric periodic wave solution: 𝑣₁(𝜉)=𝜑₀₃+2𝐴₁Δ₁cos𝜂₁𝜉−𝐵₁,𝑢₁(𝜉)=1𝜆𝑣²₁(𝜉)+𝑑, (22)where 𝐴₁=𝜑₁−𝜑₀₃𝜑₀₃−𝜑₂,𝐵₁=𝜑₁+𝜑₂−2𝜑₀₃,Δ₁=𝐵²₁+4𝐴₁,𝜂₁=−𝐴₁𝛼₂2,𝜑₁=−𝜑₀₃+−2𝛽𝛼₂−2𝜑²₀₃,𝜑₂=−𝜑₀₃−−2𝛽𝛼₂−2𝜑²₀₃. (23)(2)Elliptic periodic wave solution: 𝑣₂(𝜉)=𝜑₃𝜑₄−𝜑₆+𝜑₆𝜑₃−𝜑₄𝑠𝑛²𝜂₂𝜉,𝑘₁𝜑₄−𝜑₆+𝜑₃−𝜑₄𝑠𝑛²𝜂₂𝜉,𝑘₁,𝑢₂(𝜉)=1𝜆𝑣²₂(𝜉)+𝑑, (24)where 𝜂₂=𝛼₂8𝜑₃−𝜑₅𝜑₄−𝜑₆,𝑘²₁=𝜑₃−𝜑₄𝜑₅−𝜑₆𝜑₃−𝜑₅𝜑₄−𝜑₆,𝜑₃=12−𝜑₅−𝜑₆+𝛽−8𝛼₂−3𝜑²₅−2𝜑₅𝜑₆−3𝜑²₆,𝜑₄=12−𝜑₅−𝜑₆−𝛽−8𝛼₂−3𝜑²₅−2𝜑₅𝜑₆−3𝜑²₆,𝜑₅1=−3𝜑₆+22𝛽+3𝛼₂𝜑²₆14𝜃₂^1/31+𝑖√3−16𝛼₂𝜃₂2^1/31−𝑖√3,𝜑^∗₀₃<𝜑₆<𝜑₀₃,𝜑^∗₀₃=−𝜑₀₂−−2𝛽𝛼₂−2𝜑²₀₂,𝜃₁=−108𝛼²₂𝛾−36𝛼²₂𝛽𝜑₆−20𝛼³₂𝜑³₆,𝜃₂=𝜃₁+46𝛼₂𝛽+2𝛼²₂𝜑²₆³+𝜃²₁. (25) For the varying process, see Figure 1.

(a) 𝛾=−𝛾1(𝛽)+10−1

(b) 𝛾=−𝛾1(𝛽)+10−4

(c) 𝛾=−𝛾1(𝛽)+10−8

(d) 𝛾=−𝛾1(𝛽)+10−1

(e) 𝛾=−𝛾1(𝛽)+10−4

(f) 𝛾=−𝛾1(𝛽)+10−8

Figure 1 

The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of 𝑣1(𝜉) and 𝑣2(𝜉) when 𝛼2>0, 𝛽<0, and 𝛾→−𝛾1(𝛽)+0 where 𝛼2=4 and 𝛽=−9.

Proposition 2. For given 𝛼₂>0 and 𝛽<0, if 0<𝛾<𝛾₁(𝛽), then 𝜑₀₁, 𝜑₀₂, and 𝜑₀₃ are real and system (16) has two types of special periodic wave solutions which become a fractional solitary wave solution: 𝑣_𝑦(𝜉)=−3𝛽𝛼₂2𝛽𝜉²+96𝛽𝜉²−9,𝑢_𝑦(𝜉)=1𝜆𝑣²_𝑦(𝜉)+𝑑, (26) when 𝛾→𝛾₁(𝛽)−0. These two types of special periodic wave solutions are as follows.(1)Trigonometric periodic wave solution: 𝑣₃(𝜉)=𝜑₀₁+2𝐴₂𝐵₂−Δ₂cos𝜂₃𝜉,𝑢₃(𝜉)=1𝜆𝑣²₃(𝜉)+𝑑, (27)where 𝐴₂=𝜑₇−𝜑₀₁𝜑₀₁−𝜑₈,𝐵₂=2𝜑₀₁−𝜑₇−𝜑₈,Δ₂=𝐵²₂+4𝐴₂,𝜂₃=−𝐴₂𝛼₂2,𝜑₇=−𝜑₀₁+−2𝛽𝛼₂−2𝜑²₀₁,𝜑₈=−𝜑₀₁−−2𝛽𝛼₂−2𝜑²₀₁. (28)(2)Elliptic periodic wave solution: 𝑣₄(𝜉)=𝜑₁₂𝜑₉−𝜑₁₁+𝜑₉𝜑₁₁−𝜑₁₂𝑠𝑛²𝜂₄𝜉,𝑘₂𝜑₉−𝜑₁₁+𝜑₁₁−𝜑₁₂𝑠𝑛²𝜂₄𝜉,𝑘₂,𝑢₄(𝜉)=1𝜆𝑣²₄(𝜉)+𝑑, (29)where 𝜂₄=𝛼₂8𝜑₉−𝜑₁₁𝜑₁₀−𝜑₁₂,𝑘²₂=𝜑₉−𝜑₁₀𝜑₁₁−𝜑₁₂𝜑₉−𝜑₁₁𝜑₁₀−𝜑₁₂,𝜑^∗₀₁=−𝜑₀₂+−2𝛽𝛼₂−2𝜑²₀₂,𝜑₀₁<𝜑₉<𝜑^∗₀₁,𝜃₃=−20𝑣³₉𝛼³₂−36𝑣₉𝛼²₂𝛽−108𝛼²₂𝜃𝛾,₄=𝜃₃+42𝑣²₉𝛼²₂+6𝛼₂𝛽³+𝜃²₃,𝜑₁₀𝜑=−⁹3−23𝜑²₉𝛼₂2+2𝛽𝜃₄^1/3+13𝛼₂𝜃₄2^1/3,𝜑₁₁=12−𝜑₉−𝜑₁₀+−3𝜑²₉−2𝜑₉𝜑₁₀−3𝜑²₁₀𝛽−8𝛼₂,𝜑₁₂=12−𝜑₉−𝜑₁₀−−3𝜑²₉−2𝜑₉𝜑₁₀−3𝜑²₁₀𝛽−8𝛼₂. (30) For the varying process, see Figure 2.

(a) 𝛾=𝛾1(𝛽)−10−1

(b) 𝛾=𝛾1(𝛽)−10−4

(c) 𝛾=𝛾1(𝛽)−10−8

(d) 𝛾=𝛾1(𝛽)−10−1

(e) 𝛾=𝛾1(𝛽)−10−4

(f) 𝛾=𝛾1(𝛽)−10−8

Figure 2 

The solitary wave is bifurcated from two types of periodic waves. The varying process for the figures of 𝑣3(𝜉) and 𝑣4(𝜉) when 𝛼2>0, 𝛽<0, and 𝛾→𝛾1(𝛽)−0 where 𝛼2=4 and 𝛽=−9.

Proposition 3. For given 𝛼₂<0 and 𝛽>0, if −𝛾₁(𝛽)<𝛾<𝛾₁(𝛽), then system (16) has four nonlinear wave solutions which become two kink wave solutions: 𝑣^±_𝛿(𝜉)=±−𝛽𝛼₂tanh𝛽2𝜉,𝑢_𝛿(𝜉)=1𝜆𝑣²_𝛿(𝜉)+𝑑, (31) when 𝛾→0. These four nonlinear wave solutions are as follows: 𝑣₅(𝜉)=4𝐴₃𝜑₀₁−𝛿𝑒^−𝜂5𝜉−𝜑₀₁𝛿𝑒^−𝜂5𝜉−𝐵₃²4𝐴₃−𝛿𝑒^−𝜂5𝜉−𝐵₃²,𝑢₅(𝜉)=1𝜆𝑣²₅(𝜉)𝑣+𝑑,₆(𝜉)=4𝐴₃𝜑₀₁−𝛿𝑒^𝜂5𝜉−𝜑₀₁𝛿𝑒^𝜂5𝜉−𝐵₃²4𝐴₃−𝛿𝑒^𝜂5𝜉−𝐵₃²,𝑢₆(𝜉)=1𝜆𝑣²₆(𝜉)𝑣+𝑑,₇(𝜉)=4𝐴₄𝜑₀₃−𝜇𝑒^𝜂6𝜉−𝜑₀₃𝜇𝑒^𝜂6𝜉−𝐵₄²4𝐴₄−𝜇𝑒^𝜂6𝜉−𝐵₄²,𝑢₇(𝜉)=1𝜆𝑣²₇(𝜉)𝑣+𝑑,₈(𝜉)=4𝐴₄𝜑₀₃−𝜇𝑒^−𝜂6𝜉−𝜑₀₃𝜇𝑒^−𝜂6𝜉−𝐵₄²4𝐴₄−𝜇𝑒^−𝜂6𝜉−𝐵₄²,𝑢₈(𝜉)=1𝜆𝑣²₈(𝜉)+𝑑, (32) where 𝛿=8−𝛽𝛼₂,𝐴₃=6𝜑²₀₁+2𝛽𝛼₂,𝜂₅=−𝐴₃𝛼₂2,𝐵₃=4𝜑₀₁,𝜇=−8−𝛽𝛼₂,𝐴₄=6𝜑²₀₃+2𝛽𝛼₂,𝜂₆=−𝐴₄𝛼₂2,𝐵₄=4𝜑₀₃. (33) These four nonlinear wave solutions possess the following properties. (a)If 0<𝛾<𝛾₁(𝛽), then 𝑣₅(𝜉) and 𝑣₆(𝜉) represent two solitary waves which tend to two kink waves 𝑣^±_𝛿(𝜉) (see Figure 3) when 𝛾→0+0. (b)If −𝛾₁(𝛽)<𝛾<0, then 𝑣₅(𝜉) and 𝑣₆(𝜉) represent two singular waves which tend to two kink waves 𝑣^±_𝛿(𝜉) (see Figure 4) when 𝛾→0−0. (c)If 0<𝛾<𝛾₁(𝛽), then 𝑣₇(𝜉) and 𝑣₈(𝜉) represent two singular waves which tend to two kink waves 𝑣^±_𝛿(𝜉) (see Figure 5) when 𝛾→0+0. (d)If −𝛾₁(𝛽)<𝛾<0, then 𝑣₇(𝜉) and 𝑣₈(𝜉) represent two solitary waves which tend to two kink waves 𝑣^±_𝛿(𝜉) (see Figure 6) when 𝛾→0−0.

(a) 𝛾=1×10−1

(b) 𝛾=1×10−4

(c) 𝛾=1×10−8

Figure 3 

Two kink waves are bifurcated from two peak-solitary waves. The varying process for the figures of 𝑣5(𝜉) and 𝑣6(𝜉) when 𝛼2<0, 𝛽>0, and 𝛾→0+0 where 𝛼2=−2 and 𝛽=9.

(a) 𝛾=−1×10−1

(b) 𝛾=−1×10−4

(c) 𝛾=−1×10−8

Figure 4 

Two kink waves are bifurcated from two singular waves. The varying process for the figures of 𝑣5(𝜉) and 𝑣6(𝜉) when 𝛼2<0, 𝛽>0, and 𝛾→0−0 where 𝛼2=−2 and 𝛽=9.

(a) 𝛾=1×10−1

(b) 𝛾=1×10−4

(c) 𝛾=1×10−8

Figure 5 

Two kink waves are bifurcated from two singular waves. The varying process for the figures of 𝑣7(𝜉) and 𝑣8(𝜉) when 𝛼2<0, 𝛽>0, and 𝛾→0+0 where 𝛼2=−2 and 𝛽=9.

(a) 𝛾=−1×10−1

(b) 𝛾=−1×10−4

(c) 𝛾=−1×10−8

Figure 6 

Two kink waves are bifurcated from two valley-solitary waves. The varying process for the figures of 𝑣7(𝜉) and 𝑣8(𝜉) when 𝛼2<0, 𝛽>0, and 𝛾→0−0 where 𝛼2=−2 and 𝛽=9.

The Derivations of Propositions 1–3. 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 1–3 as follows.

(a) 𝛼2>0

(b) 𝛼2<0

Figure 7 

The bifurcation phase portraits of system (17).

(1) When 𝛼₂>0, 𝛽<0, and −𝛾₁(𝛽)<𝛾<0 (as in Figure 7(a)), the closed curves 𝑙₁ and 𝑙₂ possess the following expressions: 𝑙₁∶𝑦²=𝛼₂2𝜑−𝜑₀₃²𝜑₁−𝜑𝜑−𝜑₂𝜑₀₃<𝜑₂≤𝜑≤𝜑₁,𝑙₂∶𝑦²=𝛼₂2𝜑₃−𝜑𝜑−𝜑₄𝜑−𝜑₅𝜑−𝜑₆𝜑₆<𝜑₅<𝜑₄≤𝜑≤𝜑₃. (34) Substituting (34) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating them along 𝑙₁ and 𝑙₂, respectively, it follows that ^𝜑1_𝜑𝑑𝑠𝑠−𝜑₀₃𝜑₁−𝑠𝑠−𝜑₂=𝛼2𝜉along𝑙₁,_𝜑3𝜑𝑑𝑠𝜑₃−𝑠𝑠−𝜑₄𝑠−𝜑₅𝑠−𝜑₆=𝛼2𝜉along𝑙₂. (35) Completing the integrals above and solving the equations for 𝜑, we obtain the solutions 𝑣₁(𝜉), 𝑢₁(𝜉) (see (22)) and 𝑣₂(𝜉), 𝑢₂(𝜉) (see (24)).

(2) When 𝛼₂>0, 𝛽<0, and 0<𝛾<𝛾₁(𝛽) (as in Figure 7(a)), the closed curves 𝑙₃ and 𝑙₄ possess the following expressions: 𝑙₃∶𝑦²=𝛼2𝜑₀₁−𝜑²𝜑₇−𝜑𝜑−𝜑₈𝜑₈≤𝜑≤𝜑₇<𝜑₀₁,𝑙₄∶𝑦²=𝛼2𝜑₉−𝜑𝜑₁₀−𝜑𝜑₁₁−𝜑𝜑−𝜑₁₂𝜑₁₂≤𝜑≤𝜑₁₁<𝜑₁₀<𝜑₉. (36) Substituting (36) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating them along 𝑙₃ and 𝑙₄, respectively, it follows that ^𝜑_𝜑8𝑑𝑠𝜑₀₁−𝑠𝜑₇−𝑠𝑠−𝜑₈=𝛼2𝜉along𝑙₃,_𝜑𝜑12𝑑𝑠𝜑₉−𝑠𝜑₁₀−𝑠𝜑₁₁−𝑠𝑠−𝜑₁₂=𝛼2𝜉along𝑙₄. (37) Completing the integrals above and solving the equations for 𝜑, we obtain the solutions 𝑣₃(𝜉), 𝑢₃(𝜉) (see (27)) and 𝑣₄(𝜉), 𝑢₄(𝜉) (see (29)).

When 𝛾→−𝛾₁(𝛽)+0, it follows that 𝜑₂, 𝜑₀₃, and 𝜑^∗₀₃ tend to 𝜑^∗₂=−−𝛽/3𝛼₂ and 𝜑₁, 𝜑₃ tend to 𝜑^∗₁=−3𝛽/𝛼₂. Further, we have 𝜑₄⟶𝜑^∗₂,𝜑₅⟶𝜑^∗₂,𝜑₆⟶𝜑^∗₂,𝐴₁=𝜑₁−𝜑₀₃𝜑₀₃−𝜑₂𝐵⟶0,₁=𝜑₁+𝜑₂−2𝜑₀₃⟶𝜑^∗₁−𝜑^∗₂,𝜂₁=−𝐴₁𝛼₂2𝜂⟶0,₂=𝛼₂8𝜑₃−𝜑₅𝜑₄−𝜑₆⟶0,Δ₁=𝐵²₁+4𝐴₁=𝐵₁+2𝐴₁𝐵₁𝜂+⋯,cos₁𝜉𝜂=1−¹𝜉²+𝜂2!¹𝜉⁴𝜂4!+⋯,sn₂𝜉,𝑘₁=𝜂₂𝜉−1+𝑘²₁𝜂₂𝜉³3!+⋯. (38) Thus, we have lim_{𝛾→−𝛾1+0}𝑣₁(𝜉)=lim_{𝛾→−𝛾1+0}𝜑₀₃+2𝐴₁Δ₁cos𝜂₁𝜉−𝐵₁=lim_{𝛾→−𝛾1+0}𝜑₀₃+2𝐴₁+𝑜𝐴²₁×𝐵₁+2𝐴₁𝐵₁+𝑜𝐴²₁×𝛼1+²𝐴₁𝜉²4+𝑜𝐴²₁−𝐵₁⁻¹=lim_{𝛾→−𝛾1+0}𝜑₀₃+2𝐴₁+𝑜𝐴²₁2𝐴₁/𝐵₁+𝛼₂𝐴₁𝐵₁𝜉²/4+𝑜𝐴²₁=𝜑^∗₂+8𝜑^∗₁−𝜑^∗₂𝛼₂𝜑^∗₁−𝜑^∗₂²𝜉²=+8−3𝛽𝛼₂9+2𝛽𝜉²9−6𝛽𝜉²=𝑣_𝑘(𝜉)(see(21)),lim_{𝛾→−𝛾1+0}𝑣₂(𝜉)=lim_{𝛾→−𝛾1+0}𝜑₃𝜑₄−𝜑₆+𝜑₆𝜑₃−𝜑₄sn²𝜂₂𝜉,𝑘₁𝜑₄−𝜑₆+𝜑₃−𝜑₄sn²𝜂₂𝜉,𝑘₁=lim_{𝛾→−𝛾1+0}𝜑₃𝜑₄−𝜑₆+𝜑₆𝜑₃−𝜑₄×𝛼₂8𝜑₃−𝜑₅𝜑₄−𝜑₆𝜉+𝑜𝜑₄−𝜑₆²×𝜑₄−𝜑₆+𝜑₃−𝜑₄×𝛼₂8𝜑₃−𝜑₅𝜑₄−𝜑₆𝜉+𝑜𝜑₄−𝜑₆²⁻¹=lim_{𝛾→−𝛾1+0}8𝜑₃𝜑₄−𝜑₆+𝜑₆𝜑₃−𝜑₄𝜑₃−𝜑₅×𝜑₄−𝜑₆𝛼₂𝜉²+𝑜𝜑₄−𝜑₆^3/2×8𝜑₄−𝜑₆+𝜑₃−𝜑₄𝜑₃−𝜑₅×𝜑₄−𝜑₆𝛼₂𝜉²+𝑜𝜑₄−𝜑₆^3/2−1=8𝜑^∗₁+𝜑^∗₂𝜑^∗₁−𝜑^∗₂²𝛼₂𝜉²8+𝜑^∗₁−𝜑^∗₂²𝛼₂𝜉²=−3𝛽𝛼₂9+2𝛽𝜉²9−6𝛽𝜉²=𝑣_𝑘(𝜉)(see(21)). (39)

Similarly, we can prove the limit property of 𝑣₃(𝜉) and 𝑣₄(𝜉) when 𝛾→𝛾₁(𝛽)−0.

(3) When 𝛼₂<0, 𝛽>0, and −𝛾₁(𝛽)<𝛾<𝛾₁(𝛽) (as in Figure 7(b)), the curve connecting with (𝜑₀₁,0) embraces the expression 𝑦²𝛼=−²2𝜑−𝜑₀₁²𝜑²+2𝜑₀₁𝜑+3𝜑²₀₁+2𝛽𝛼₂. (40) Substituting (40) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑_𝑝𝑑𝑠𝑠−𝜑₀₁²𝑠²+2𝜑₀₁𝑠+3𝜑²₀₁+2𝛽/𝛼₂=−𝛼₂2𝜉, (41) where 𝑝=𝜑₀₁−−𝛽𝛼₂24𝛼₂𝜑²₀₁+8𝛽𝛼₂𝜑²₀₁−8𝜑₀₁−𝛼₂.𝛽+9𝛽 (42) Completing the integral above and solving the equation for 𝜑, we obtain 𝑣₅(𝜉)=4𝐴₃𝜑₀₁−𝛿𝑒^−𝜂5𝜉−𝜑₀₁𝛿𝑒^−𝜂5𝜉−𝐵₃²4𝐴₃−𝛿𝑒^−𝜂5𝜉−𝐵₃². (43) From 𝑣₅(𝜉), we get 𝑣₆(𝜉)=𝑣₅(−𝜉)=4𝐴₃𝜑₀₁−𝛿𝑒^𝜂5𝜉−𝜑₀₁𝛿𝑒^𝜂5𝜉−𝐵₃²4𝐴₃−𝛿𝑒^𝜂5𝜉−𝐵₃². (44)

(4) When 𝛼₂<0, 𝛽>0, and −𝛾₁(𝛽)<𝛾<𝛾₁(𝛽) (as in Figure 7(b)), the curve connecting with (𝜑₀₃,0) embraces the expression 𝑦²𝛼=−²2𝜑₀₃−𝜑²𝜑²+2𝜑₀₃𝜑+3𝜑²₀₃+2𝛽𝛼₂. (45) Substituting (45) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑_𝑞𝑑𝑠𝜑₀₃−𝑠²𝑠²+2𝜑₀₃𝑠+3𝜑²₀₃+2𝛽/𝛼₂=−𝛼₂2𝜉, (46) where 𝑞=𝜑₀₃+−𝛽𝛼₂24𝛼₂𝜑²₀₃+8𝛽𝛼₂𝜑²₀₃+8𝜑₀₃−𝛼₂.𝛽+9𝛽 (47) Completing the integral above and solving the equation for 𝜑, we obtain 𝑣₇(𝜉)=4𝐴₄𝜑₀₃−𝜇𝑒^𝜂6𝜉−𝜑₀₃𝜇𝑒^𝜂6𝜉−𝐵₄²4𝐴₄−𝜇𝑒^𝜂6𝜉−𝐵₄². (48) From 𝑣₇(𝜉), we get 𝑣₈(𝜉)=𝑣₇()=−𝜉4𝐴₄𝜑₀₃−𝜇𝑒^−𝜂6𝜉−𝜑₀₃𝜇𝑒^−𝜂6𝜉−𝐵₄²4𝐴₄−𝜇𝑒^−𝜂6𝜉−𝐵₄². (49)

Note that when 𝛾→0, it follows that 𝜑₀₁→𝜑^∗∗₀₁=−−𝛽/𝛼₂, 𝐴₃→−4𝛽/𝛼₂, 𝐵₃→−4−𝛽/𝛼₂, and 𝜂₅→−2𝛽. Thus, we have lim_𝛾→0𝑣₅(𝜉)=lim_𝛾→04𝐴₃𝜑₀₁−𝛿𝑒^−𝜂5𝜉−𝜑₀₁𝛿𝑒^−𝜂5𝜉−𝐵₃²4𝐴₃−𝛿𝑒^−𝜂5𝜉−𝐵₃²=lim_𝛾→0𝛿𝜑₀₁𝑒^{−2𝜂5𝜉}+4𝐴₃−2𝜑₀₁𝐵₃𝑒^−𝜂5𝜉𝛿𝑒^{−2𝜂5𝜉}−2𝐵₃𝑒^−𝜂5𝜉=𝛿−𝛼₂𝛽𝑒^−2𝛽𝜉−8𝛽𝛿𝛼₂𝑒^−2𝛽𝜉−8−𝛼₂𝛽=−𝛽𝛼₂𝑒^𝛽/2𝜉−𝑒^{−𝛽/2𝜉}𝑒^𝛽/2𝜉+𝑒^{−𝛽/2𝜉}=−𝛽𝛼₂tanh𝛽2𝜉=𝑣⁺_𝛿(𝜉)(see(31)). (50)

Similarly, we can also get 𝑣₆(𝜉)→𝑣⁻_𝛿(𝜉), 𝑣₇(𝜉)→𝑣⁺_𝛿(𝜉), and 𝑣₈(𝜉)→𝑣⁻_𝛿(𝜉) when 𝛾→0. These complete the derivations of Propositions 1–3.

3. The Bifurcations of Compactons and Peakons for 𝐷(1,2) System

When 𝑚=1 and 𝑛=2, system (1) becomes 𝑢_𝑡+𝑣_𝑥𝑣=0,_𝑡+𝑎𝑣²_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_𝑥=0. (51)

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 𝑑𝜑𝑑𝜏=2𝜑𝑦,𝑑𝑦𝑑𝜏=−2𝑦²−𝛼₁𝜑²−𝛽𝜑−𝛾, (52) where 𝛼₁=𝑏+𝑐,2𝑎𝜆 (53) and 𝛽, 𝛾 are given in (13).

For fixed 𝛼₁, put 𝛾₂𝛽=2𝛽²9𝛼₁,𝛾₃𝛽=𝛽²4𝛼₁. (54)

Proposition 4. For given 𝛼₁>0 and 𝛽≠0, if 0<𝛾<𝛾₃(𝛽), then system (51) has a family of solitary wave solutions which become the compacton solution 𝑣_𝑜(𝜉)=−4𝛽3𝛼₁cos²√𝛼₁4𝜉,𝜉<2𝜋√𝛼₁,0,𝜉≥2𝜋√𝛼₁,𝑢_𝑜(𝜉)=1𝜆𝑣_𝑜(𝜉)+𝑑, (55) when 𝛾→0+0. The solitary wave solutions are as follows.
(1) When 𝛽<0, the solitary wave solutions possess the expressions 𝑣₉(𝜉)=𝜑(𝜉) and 𝑢₉(𝜉)=(1/𝜆)𝑣₉(𝜉)+𝑑, and 𝜑(𝜉) is given by the implicit function √𝛼₁𝜉−𝜋2𝑟=arcsin¹+𝑟₃−2𝜑𝑟₁−𝑟₃−𝑟₂𝐴₅lnΦ₁, (56) where 𝑟₁1=−3𝛼₁3𝛼₁𝑟₂+2𝛽−6𝛼₁𝛽𝑟₂+4𝛽²,𝑟₂1=−2𝛼₁𝛽+𝛽²−4𝛼₁𝛾,𝑟₃1=−3𝛼₁3𝛼₁𝑟₂+2𝛽+6𝛼₁𝛽𝑟₂+4𝛽²,𝐴₅=𝑟₁−𝑟₂𝑟₂−𝑟₃,𝐵₅=𝑟₁+𝑟₃−2𝑟₂,Φ₁=𝑟₁−𝑟₃𝜑−𝑟₂2𝐴₅𝑟₁−𝜑𝜑−𝑟₃+𝐵₅𝜑−𝑟₂+2𝐴₅. (57)
(2) When 𝛽>0, the solitary wave solutions possess the expressions 𝑣₁₀(𝜉)=𝜑(𝜉) and 𝑢₁₀(𝜉)=(1/𝜆)𝑣₁₀(𝜉)+𝑑, and 𝜑(𝜉) is given by the implicit function 𝜋−√𝛼₁𝜉2𝑟=arcsin⁶+𝑟₄−2𝜑𝑟₆−𝑟₄+𝑟₅𝐴₆lnΦ₂, (58) where 𝑟₄1=−3𝛼₁3𝛼₁𝑟₅+2𝛽+6𝛼₁𝛽𝑟₅+4𝛽²,𝑟₅1=−2𝛼₁𝛽−𝛽²−4𝛼₁𝛾,𝑟₆1=−3𝛼₁3𝛼₁𝑟₅+2𝛽−6𝛼₁𝛽𝑟₅+4𝛽²,𝐴₆=𝑟₆−𝑟₅𝑟₅−𝑟₄,𝐵₆=2𝑟₅−𝑟₆−𝑟₄,Φ₂=2𝐴₆𝑟₆−𝜑𝜑−𝑟₄+𝐵₆𝑟₅−𝜑+2𝐴₆𝑟₆−𝑟₄𝑟₅.−𝜑 (59) For the varying process, see Figure 8.

(a) 𝛽=−9,𝛾=1×10−1

(b) 𝛽=−9,𝛾=1×10−4

(c) 𝛽=−9,𝛾=1×10−8

(d) 𝛽=9,𝛾=1×10−1

(e) 𝛽=9,𝛾=1×10−4

(f) 𝛽=9,𝛾=1×10−8

Figure 8 

The solitary waves turn into the compactons. The varying process for the figures of 𝑣9(𝜉) and 𝑣10(𝜉) when 𝛼1>0, 𝛽≠0, and 𝛾→0+0 where 𝛼1=2.

Proposition 5. For given 𝛼₁<0 and 𝛽≠0, if 𝛾₃(𝛽)<𝛾<𝛾₂(𝛽), then system (51) has two types of nonlinear wave solutions which tend to the peakon solution 𝑣_𝑚(𝜉)=2𝛽3𝛼₁𝑒^{−|𝛼1|/4|𝜉|},𝑢−1_𝑚(𝜉)=1𝜆𝑣_𝑚(𝜉)+𝑑, (60) when 𝛾→𝛾₂(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 𝑣₁₁(𝜉)=𝛽3𝛼₁−𝛾2𝛼₁−𝛽3𝛼₁+𝛾2𝛼₁𝑒^{−|𝛼1|/4|𝜉|}×𝑒^{|𝛼1|/4|𝜉|},𝑢−1₁₁(𝜉)=1𝜆𝑣₁₁(𝜉)+𝑑, (61) and the solitary wave solutions possess the expressions 𝑣₁₂(𝜉)=𝜑(𝜉) and 𝑢₁₂(𝜉)=(1/𝜆)𝑣₁₂(𝜉)+𝑑, and 𝜑(𝜉) is given by the implicit function 𝜉=−4𝛼₁2lnΦ₃−𝑀₁𝑟₇−𝑟₈+𝑟₉𝐴₇2ln𝐴₇Φ₃+𝐵₇𝑟₉−𝜑+2𝐴₇𝑟₇−𝑟₈𝑟₉,−𝜑 (62) where Φ₃=𝑟₇−𝜑𝑟₈,𝑀−𝜑₁=2𝜑−𝑟₇−𝑟₈,𝑟₇1=−3𝛼₁3𝛼₁𝑟₉+2𝛽+6𝛼₁𝛽𝑟₉+4𝛽²,𝑟₈1=−3𝛼₁3𝛼₁𝑟₉+2𝛽−6𝛼₁𝛽𝑟₉+4𝛽²,𝑟₉1=−2𝛼₁𝛽−𝛽²−4𝛼₁𝛾,𝐴₇=𝑟₈−𝑟₉𝑟₇−𝑟₉,𝐵₇=𝑟₇+𝑟₈−2𝑟₉. (63)
(2) When 𝛽>0, the singular cusp wave solutions possess the expression 𝑣₁₃(𝜉)=𝛽3𝛼₁−𝛾2𝛼₁−𝛽3𝛼₁+𝛾2𝛼₁𝑒^{|𝛼1|/4|𝜉|}×𝑒^{−|𝛼1|/4|𝜉|},𝑢−1₁₃(𝜉)=1𝜆𝑣₁₃(𝜉)+𝑑, (64) and the solitary wave solutions possess the expressions 𝑣₁₄(𝜉)=𝜑(𝜉) and 𝑢₁₄(𝜉)=(1/𝜆)𝑣₁₄(𝜉)+𝑑, and 𝜑(𝜉) is given by the implicit function 𝜉=−4𝛼₁2lnΦ₄+𝑀₂𝑟₁₁−𝑟₁₂−𝑟₁₀𝐴₈2ln𝐴₈Φ₄+𝐵₈𝜑−𝑟₁₀+2𝐴₈𝑟₁₁−𝑟₁₂𝜑−𝑟₁₀, (65) where Φ₄=𝜑−𝑟₁₁𝜑−𝑟₁₂,𝑀₂=2𝜑−𝑟₁₁−𝑟₁₂,𝑟₁₀1=−2𝛼₁𝛽+𝛽²−4𝛼₁𝛾,𝑟₁₁1=−3𝛼₁3𝛼₁𝑟₁₀+2𝛽+6𝛼₁𝛽𝑟₁₀+4𝛽²,𝑟₁₂1=−3𝛼₁3𝛼₁𝑟₁₀+2𝛽−6𝛼₁𝛽𝑟₁₀+4𝛽²,𝐴₈=𝑟₁₀−𝑟₁₁𝑟₁₀−𝑟₁₂,𝐵₈=2𝑟₁₀−𝑟₁₁−𝑟₁₂. (66)

(a) 𝛾=𝛾2(𝛽)−10−1

(b) 𝛾=𝛾2(𝛽)−10−4

(c) 𝛾=𝛾2(𝛽)−10−8

(d) 𝛾=𝛾2(𝛽)−10−1

(e) 𝛾=𝛾2(𝛽)−10−4

(f) 𝛾=𝛾2(𝛽)−10−8

Figure 9 

The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of 𝑣11(𝜉) and 𝑣12(𝜉) when 𝛼1<0, 𝛽<0, and 𝛾→𝛾2(𝛽)−0 where 𝛼1=−2 and 𝛽=−9.

(a) 𝛾=𝛾2(𝛽)−10−1

(b) 𝛾=𝛾2(𝛽)−10−4

(c) 𝛾=𝛾2(𝛽)−10−8

(d) 𝛾=𝛾2(𝛽)−10−1

(e) 𝛾=𝛾2(𝛽)−10−4

(f) 𝛾=𝛾2(𝛽)−10−8

Figure 10 

The peakon is bifurcated from the singular cusp wave and the solitary wave. The varying process for the figures of 𝑣13(𝜉) and 𝑣14(𝜉) when 𝛼1<0, 𝛽>0, and 𝛾→𝛾2(𝛽)−0 where 𝛼1=−2 and 𝛽=9.

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.

(a) 𝛼1>0

(b) 𝛼1<0

Figure 11 

The bifurcation phase portraits of system (52).

(1) When 𝛼₁>0, 𝛽<0, and 0<𝛾<𝛾₃(𝛽) (as in Figure 11(a)), the homoclinic orbit owns the expression 𝑦²=𝛼₁4𝜑²𝜑−𝑟₂²𝑟₁−𝜑𝜑−𝑟₃𝑟₃<𝑟₂<𝜑≤𝑟₁. (67) Substituting (67) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have 𝜉=^𝑟1_𝜑4𝑠²𝛼₁𝑠−𝑟₂²𝑟₁−𝑠𝑠−𝑟₃𝑑𝑠. (68) Completing the integral above, we get 𝜉=4𝛼₁𝑟arcsin¹+𝑟₃−2𝜑𝑟₁−𝑟₃+𝜋2−𝑟₂𝐴₅lnΦ₁. (69)

(2) When 𝛼₁>0, 𝛽>0, and 0<𝛾<𝛾₃(𝛽) (as in Figure 11(a)), the homoclinic orbit is expressed by 𝑦²=𝛼₁4𝜑²𝑟₅−𝜑²𝜑−𝑟₄𝑟₆−𝜑𝑟₄≤𝜑<𝑟₅<𝑟₆. (70) Substituting (70) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have 𝜉=^𝜑_𝑟44𝑠²𝛼₁𝑟₅−𝑠²𝑠−𝑟₄𝑟₆.−𝑠 (71) Completing the integral above, we get 𝜉=−4𝛼₁𝑟arcsin⁶+𝑟₄−2𝜑𝑟₆−𝑟₄−𝜋2+𝑟₅𝐴₆lnΦ₂. (72)

Note that when 𝛽<0 and 𝛾→0+0, it follows that 𝑟₁→𝑟^∗₁=−4𝛽/3𝛼₁, 𝑟₂→0, and 𝑟₃→0. Further, we have 𝐵₅=𝑟₁+𝑟₃−2𝑟₂⟶𝑟^∗₁,𝐴₅=𝑟₁−𝑟₂𝑟₂−𝑟₃⟶0,lim_𝛾→0+0𝑟₃𝑟₂=lim_𝛾→0+0−3𝛼₁𝑟₂+2𝛽+6𝛼₁𝛽𝑟₂+4𝛽²3𝛼₁𝑟₂=lim_𝛾→0+0−13𝛼₁𝑟₂×3𝛼₁𝑟₂+2𝛽−2𝛽1+3𝛼₁𝑟₂𝑟4𝛽+𝑜²₂=lim_𝛾→0+0−13𝛼₁𝑟₂3𝛼₁𝑟₂−3𝛼₁𝑟₂2+𝑜𝑟²₂1=−2,lim_𝛾→0+0𝑟₂𝐴₅=lim_𝛾→0+0𝑟₂𝑟₁−𝑟₂𝑟₂−𝑟₃=lim_𝛾→0+0𝑟₂𝑟₁+𝑟₃−𝑟₂−𝑟₁𝑟₃/𝑟₂=0(3/2)𝑟^∗₁=0,lim_𝛾→0+0Φ₁=lim_𝛾→0+0𝑟₁−𝑟₃𝜑−𝑟₂2𝐴₅𝑟₁−𝜑𝜑−𝑟₃+𝐵₅𝜑−𝑟₂+2𝐴₅=𝑟^∗₁𝜑𝑟^∗₁𝜑=1. (73) Thus, from 𝜉=lim_𝛾→0+04𝛼₁𝑟arcsin¹+𝑟₃−2𝜑𝑟₁−𝑟₃+𝜋2−𝑟₂𝐴₅lnΦ₁, (74) we have 𝜉=4𝛼₁𝑟arcsin^∗1−2𝜑𝑟^∗₁+𝜋2. (75) 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 𝛼₁<0, 𝛽≠0, and 𝛾₃(𝛽)<𝛾<𝛾₂(𝛽) (as in Figure 11(b)), the curves Γ^±₁ and Γ^±₂ own the expression 𝑦²𝛼=−¹4𝜑²+4𝛽3𝛼₁𝜑+2𝛾𝛼₁. (76) Substituting (76) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ⁰_𝜑𝑑𝑠𝑠²+4𝛽/3𝛼₁𝑠+2𝛾/𝛼₁=−𝛼₁4𝜉𝛽<0_𝜑0,𝑑𝑠𝑠²+4𝛽/3𝛼₁𝑠+2𝛾/𝛼₁=−𝛼₁4𝜉𝛽>0. (77) Completing the integrals above and solving the equations for 𝜑, we get 𝜑=𝑣₁₁(𝜉) (see (61)) and 𝜑=𝑣₁₃(𝜉) (see (64)).

(4) When 𝛼₁<0, 𝛽≠0, and 𝛾₃(𝛽)<𝛾<𝛾₂(𝛽) (as in Figure 11(b)), two homoclinic orbits can be expressed as 𝑦²𝛼=−¹4𝜑²𝜑−𝑟₉²𝑟₇−𝜑𝑟₈−𝜑𝛽<0,𝑦²𝛼=−¹4𝜑²𝑟₁₀−𝜑²𝜑−𝑟₁₁𝜑−𝑟₁₂𝛽>0. (78)(79) Substituting (78) and (79) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have 𝜉=^𝑟8_𝜑4𝑠²𝛼₁𝑠−𝑟₉²𝑟₇−𝑠𝑟₈𝜉=−𝑠𝑑𝑠,_𝜑𝑟114𝑠²𝛼₁𝑟₁₀−𝑠²𝑠−𝑟₁₁𝑠−𝑟₁₂𝑑𝑠. (80) Completing the integrals above, we get (62) and (65).

Note that when 𝛽<0 and 𝛾→𝛾₂(𝛽)−0, it follows that 𝛾/2𝛼₁→𝛽/3𝛼₁, 𝑟₇→0, 𝑟₈→0, 𝑟₉→𝑟^∗₉=−2𝛽/3𝛼₁, and 𝑀₁=2𝜑−𝑟₇−𝑟₈𝐵⟶2𝜑,₇=𝑟₈+𝑟₇−2𝑟₉⟶−2𝑟^∗₉,𝐴₇=𝑟₈−𝑟₉𝑟₇−𝑟₉⟶−𝑟^∗₉,Φ₃=𝑟₇−𝜑𝑟₈−𝜑⟶−𝜑. (81) Thus, we have lim_{𝛾→𝛾2−0}𝑣₁₃(𝜉)=lim_{𝛾→𝛾2−0}𝛽3𝛼₁−𝛾2𝛼₁−𝛽3𝛼₁+𝛾2𝛼₁𝑒^{−|𝛼1|/4|𝜉|}×𝑒^{|𝛼1|/4|𝜉|}=−12𝛽3𝛼₁𝑒^{−|𝛼1|/4|𝜉|}−1=𝑣_𝑚(𝜉)(see(60)),𝜉=lim_{𝛾→𝛾2−0}−4𝛼₁×2lnΦ₃−𝑀₁𝑟₇−𝑟₈+𝑟₉𝐴₇2ln𝐴₇Φ₃+𝐵₇𝑟₉−𝜑+2𝐴₇𝑟₇−𝑟₈𝑟₉−𝜑=lim_{𝛾→𝛾2−0}−4𝛼₁2×lnΦ₃−𝑀₁𝑟₇−𝑟₈×2𝐴₇Φ₃+𝐵₇(𝑟₉−𝜑)+2𝐴₇(𝑟₇−𝑟₈)(𝑟₉−𝜑)^𝑟9/𝐴7=−4𝛼₁𝑟ln−4𝜑^∗₉−𝜑4𝑟^∗₉𝜑=−4𝛼₁𝑟ln^∗9−𝜑𝑟^∗₉. (82)(83) 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→𝛾₂(𝛽)−0. These complete the derivations of Propositions 4 and 5.

4. The Bifurcations of Solitary Waves for 𝐷(2,2) System

When 𝑚=2 and 𝑛=2, system (1) becomes 𝑢_𝑡+𝑣²_𝑥𝑣=0,_𝑡+𝑎𝑣²_𝑥𝑥𝑥+𝑏𝑢_𝑥𝑣+𝑐𝑢𝑣_𝑥=0. (84)

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 𝑑𝜑𝑑𝜏=2𝜑𝑦,𝑑𝑦𝑑𝜏=−2𝑦²−𝛼₂𝜑³−𝛽𝜑−𝛾, (85) where 𝛼₂ is as (18) and 𝛽, 𝛾 are given in (13). Let 𝑟_𝑎=131𝛼₂5𝑄4^1/3−𝛽100𝑄^1/3,𝑟_𝑏=𝛽3252𝑄^1/31−𝑖√3−16𝛼₂5𝑄4^1/31+𝑖√3,𝑟_𝑐=𝛽3252𝑄^1/31+𝑖√3−16𝛼₂5𝑄4^1/31−𝑖√3,𝑄=−27𝛼²₂𝛾+80𝛼³₂𝛽³+729𝛼⁴₂𝛾²,𝛾₄𝛽=4√5−𝛽27𝛼³₂^1/2, (86) and let 𝛾₁(𝛽) be as (20).

Proposition 6. For fixed 𝛼₂ and 𝛽, system (84) has solitary wave solutions 𝑣^±_𝑖(𝜉)=±−5𝛽𝛼₂23−tanh²⁴−𝛼₂𝛽𝜉,𝑢80_𝑖(𝜉)=1𝜆𝑣²_𝑖(𝜉)+𝑑, (87) which can be bifurcated from the following two types of nonlinear wave solutions.(1)Smooth periodic wave solution: 𝑣₁₅(𝜉)=𝑟_𝑎−𝑟_𝑎−𝑟_𝑏𝑠𝑛²𝜂₇𝜉,𝑘₃,𝑢₁₅(𝜉)=1𝜆𝑣²₁₅(𝜉)+𝑑, (88)where −𝛾₄𝛽<𝛾<𝛾₄𝛽,𝜂₇=𝛼₂𝑟20^𝑎−𝑟_𝑐,𝑘₃=𝑟_𝑎−𝑟_𝑏𝑟_𝑎−𝑟_𝑐. (89)(2)Singular periodic wave solutions: 𝑣₁₆(𝜉)=𝐴₉+𝑟_𝑎𝑐𝑛𝜂₈𝜉,𝑘₄−𝐴₉+𝑟_𝑎𝑐𝑛𝜂₈𝜉,𝑘₄,𝑢+1₁₆(𝜉)=1𝜆𝑣²₁₆(𝜉)+𝑑,𝛼₂>0,𝛽<0,𝑣₁₇(𝜉)=𝐴₉+𝑟_𝑎−𝐴₉−𝑟_𝑎𝑐𝑛𝜂₈𝜉,𝑘₅1+𝑐𝑛𝜂₈𝜉,𝑘₅,𝑢₁₇(𝜉)=1𝜆𝑣²₁₇(𝜉)+𝑑,𝛼₂<0,𝛽>0, (90)(91)where −𝛾₁𝛽<𝛾<−𝛾₄𝛽,𝐴²₉=𝑏₀−𝑟_𝑎²+𝑎²₀,𝑎²₀1=−4𝑟_𝑏−𝑟_𝑐²,𝑏₀=12𝑟_𝑏+𝑟_𝑐,𝜂₈=15𝐴₉𝛼₂,𝑘²₄=𝐴₉−𝑏₀+𝑟_𝑎2𝐴₉,𝑘²₅=𝐴₉+𝑏₀−𝑟_𝑎2𝐴₉. (92) For the varying process, see Figures 12 and 13.

(a) 𝛾=−𝛾4(𝛽)+10−1

(b) 𝛾=−𝛾4(𝛽)+10−4

(c) 𝛾=−𝛾4(𝛽)+10−8

(d) 𝛾=−𝛾4(𝛽)−10−1

(e) 𝛾=−𝛾4(𝛽)−10−4

(f) 𝛾=−𝛾4(𝛽)−10−8

Figure 12 

The solitary wave is bifurcated from the smooth periodic wave and the singular periodic wave. The varying process for the figures of 𝑣15(𝜉) and 𝑣16(𝜉) when 𝛼2=5, 𝛽=−9, 𝛾→𝛾4(𝛽)+0, or 𝛾→𝛾4(𝛽)−0.

(a) 𝛾=−𝛾4(𝛽)+10−1

(b) 𝛾=−𝛾4(𝛽)+10−4

(c) 𝛾=−𝛾4(𝛽)+10−8

(d) 𝛾=−𝛾4(𝛽)−10−1

(e) 𝛾=−𝛾4(𝛽)−10−4

(f) 𝛾=−𝛾4(𝛽)−10−8

Figure 13 

The solitary wave is bifurcated from the smooth periodic wave and the singular periodic wave. The varying process for the figures of 𝑣15(𝜉) and 𝑣17(𝜉) when 𝛼2=−5, 𝛽=9, 𝛾→𝛾4(𝛽)+0, or 𝛾→𝛾4(𝛽)−0.

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.

(a) 𝛼2>0

(b) 𝛼2<0

Figure 14 

The bifurcation phase portraits of system (85).

(1) When 𝛼₂>0, 𝛽<0, and −𝛾₄(𝛽)<𝛾<0 (as in Figure 14(a)), the closed orbit owns the expression 𝑦²=𝛼₂5𝑟_𝑎−𝜑𝜑−𝑟_𝑏𝜑−𝑟_𝑐𝑟_𝑐<𝑟_𝑏≤𝜑≤𝑟_𝑎. (93) Substituting (93) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝑟𝑎_𝜑𝑑𝑠𝑟_𝑎−𝑠𝑠−𝑟_𝑏𝑠−𝑟_𝑐=𝛼₂5𝜉. (94) Completing the integral above and solving the equation for 𝜑, we get 𝑣₁₅(𝜉) (see (88)).

When 𝛼₂>0 and −𝛾₁(𝛽)<𝛾<−𝛾₄(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 𝑣₁₆(𝜉) (see (90)).

(2) When 𝛼₂<0, 𝛽>0, similarly, we get 𝑣₁₅(𝜉) (see (88)) and 𝑣₁₇(𝜉) (see (91)).

Note that when 𝛼₂>0 and 𝛾→−𝛾₄(𝛽)+0, it follows that 𝑟_𝑎→𝑟^∗_𝑎=(2/3)−5𝛽/𝛼₂, 𝑟_𝑏→𝑟^∗_𝑏=−(1/3)−5𝛽/𝛼₂, and 𝑟_𝑐→𝑟^∗_𝑏=−(1/3)−5𝛽/𝛼₂. Further, 𝑘₃=𝑟_𝑎−𝑟_𝑏𝑟_𝑎−𝑟_𝑐𝜂⟶1,₇=𝛼₂𝑟20^𝑎−𝑟_𝑐⟶⁴−𝛼₂𝛽.80 (95) Thus, we have lim_{𝛾→−𝛾4+0}𝑣₁₅(𝜉)=lim_{𝛾→−𝛾4+0}𝑟_𝑎−𝑟_𝑎−𝑟_𝑏sn²𝜂₇𝜉,𝑘₃=𝑟^∗_𝑎−𝑟^∗_𝑎−𝑟^∗_𝑏tanh²⁴−𝛼₂𝛽𝜉=80−5𝛽𝛼₂23−tanh²⁴−𝛼₂𝛽𝜉80=𝑣⁺_𝑖(𝜉)(see(87)). (96)

If 𝛼₂>0 and 𝛾→−𝛾₄(𝛽)−0, then it follows that 𝑟_𝑎→𝑟^∗_𝑎=(2/3)−5𝛽/𝛼₂, 𝑟_𝑏→𝑟^∗_𝑏=−(1/3)−5𝛽/𝛼₂, and 𝑟_𝑐→𝑟^∗_𝑏=−(1/3)−5𝛽/𝛼₂. Further, it follows that 𝑎²₀1=−4𝑟_𝑏−𝑟_𝑐²𝑏⟶0,₀=12𝑟_𝑏+𝑟_𝑐⟶𝑟^∗_𝑏,𝐴₉=𝑏₀−𝑟_𝑎²+𝑎²₀⟶𝑟^∗_𝑎−𝑟^∗_𝑏,𝑘²₄=𝐴₉−𝑏₀+𝑟_𝑎2𝐴₉𝜂⟶1,₈=15𝐴₉𝛼₂⟶2⁴−𝛼₂𝛽.80 (97) Thus, we get lim_{𝛾→−𝛾4−0}𝑣₁₆(𝜉)=lim_{𝛾→−𝛾4−0}𝐴₉+𝑟_𝑎cn𝜂₈𝜉,𝑘₄−𝐴₉+𝑟_𝑎cn𝜂₈𝜉,𝑘₄=+12𝑟^∗_𝑎−𝑟^∗_𝑏sech2⁴−𝛼₂𝛽/80𝜉+𝑟^∗_𝑏1+sech2⁴−𝛼₂=𝛽/80𝜉2𝑟^∗_𝑎−𝑟^∗_𝑏+𝑟^∗_𝑏2cosh²⁴−𝛼₂𝛽/80𝜉−12cosh²⁴−𝛼₂=𝑟𝛽/80𝜉_∗𝑎−𝑟^∗_𝑏sech²⁴−𝛼₂𝛽𝜉80+𝑟^∗_𝑏=−5𝛽𝛼₂23−tanh²⁴−𝛼₂𝛽𝜉80=𝑣⁺_𝑖(𝜉)(see(87)). (98)

Similarly, we can also get 𝑣₁₅(𝜉)→𝑣⁻_𝑖(𝜉) and 𝑣₁₇(𝜉)→𝑣⁻_𝑖(𝜉) when 𝛼₂<0, 𝛽>0, and 𝛾→−𝛾₄(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 3–6). 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).