Different Kinds of Singular and Nonsingular Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation in the Special Parametric Conditions

Abstract

In this paper, by using the integral bifurcation method, we studied the Kudryashov-Sinelshchikov equation. In the special parametric conditions, some singular and nonsingular exact traveling wave solutions, such as periodic cusp-wave solutions, periodic loop-wave solutions, smooth loop-soliton solutions, smooth solitary wave solutions, periodic double wave solutions, periodic compacton solutions, and nonsmooth peakon solutions are obtained. Further more, the dynamic behaviors of these exact traveling wave solutions are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameters.

1. Introduction

The investigation of the traveling wave solutions for nonlinear evolution equations plays an important role in mathematical physics. For example, the phenomena of wave motion observed in fluid dynamics, plasma, and elastic media are usually described by the solitary wave, kink wave, peakon, compacton, and loop soliton. In fact, lots of physical models have solitary wave solutions, kink wave solutions, peakon solutions, compacton solutions, and loop-soliton solutions. Lots of new traveling wave solutions and phenomena of wave motion are discovered continually by many researchers on investigating exact solutions of different kinds of nonlinear evolution equations. Recently, Kudryashov and Sinelshchikov [1] introduced the following equation: 𝑢_𝑡+𝛼𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝜖𝑢𝑢_𝑥𝑥𝑥−𝛽𝑢_𝑥𝑢_𝑥𝑥−𝜈𝑢_𝑥𝑥−𝛿𝑢𝑢_𝑥𝑥=0, (1) where 𝛼,𝜖,𝛽,𝜈, and 𝛿 are real parameters. Equation (1) describes the pressure waves in the liquid with gas bubbles taking into account the heat transfer and viscosity [1]. We call (1) Kudryashov-Sinelshchikov equation. When 𝜖=𝛽=𝛿=0, (1) becomes classical KdV-Burgers (Burgers-KdV) equation; see [2–7] and the references cited therein. When 𝜖=𝛽=𝜈=𝛿=0, (1) becomes famous KdV equation. Very recently, when parameters 𝜈=𝛿=0, in [8–11], the authors studied the special case of (1) as follows: 𝑢_𝑡+𝛼𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥−𝛽𝑢_𝑥𝑢_𝑥𝑥=0. (2)

In [8], Ryabov found four families of solitary wave solutions of (2) when 𝛽=−3 or 𝛽=−4. In [9], Li and Chen discussed the existence of different kinds of traveling wave solutions by using the approach of dynamical system, according to different phase orbits of the traveling system of (2); twenty-six kinds of exact traveling wave solutions are obtained under the parametric conditions 𝛽=−3,−4 and 𝛽=1,2. In [10], He et al. discussed the bifurcations of phase portraits and investigated exact traveling wave solutions of (2) under 𝛽=−3,1,2. In [11], He et al. investigated periodic loop solutions of (2) and discussed the limit forms of these periodic loop solutions focusing on the case 𝛽=2. In the paper [12], Randrüüt studied (2) under the transformation 𝑏=2+𝛽; in the cases 𝑏<0,𝑏=0, and 𝑏>0 (i.e., 𝛽<−2,𝛽=−2, and 𝛽>−2), Randrüüt obtained some exact traveling wave solutions and discussed their dynamical behaviors; some interesting phenomena of traveling waves are successfully explained. Particularly, when 𝑏>2 (i.e., 𝛽>0), a kind of new periodic wave solution which is called meandering solution was obtained by Randrüüt in this paper. However, many singular and nonsingular exact traveling wave solutions of Kudryashov-Sinelshchikov equation were not obtained yet in these references [8–12].

In this paper, we will study (2) again by using the integral bifurcation method [13, 14]; the integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system [15–19] and auxiliary equation method; it is easily combined with some transformations [20, 21] and useful for many nonlinear partial differential equations (PDEs) including some higher order equations of KdV type, such as the higher order KdV equation of neglecting the highest order infinitesimal term [22]. So, by using this method, we will obtain some new traveling wave solutions of (1), which are different from the results in [8–12]. Some interesting phenomena will be presented.

The rest of this paper is organized as follows: in Section 2, we will derive two-dimensional planar system which is equivalent to (2) and give its first integral equation. In Section 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions of (2) and discuss their dynamic properties.

2. The 2-Dimensional Planar Dynamical System of (2)

Making a transformation 𝑢(𝑡,𝑥)=𝜙(𝜉)+𝜆 with 𝜉=𝑘𝑥−𝑐𝑡, (2) can be reduced to the following ordinary differential equation: −𝑐𝜙^′+𝛼𝜙+𝜆𝜙^′+𝜙^′′′−𝜙+𝜆𝜙^′′′−𝛽𝜙^′𝜙^′′=0. (3) Integrating (3) with respect to 𝜉, we have 1−𝜆−𝜙𝜙^′′()1=𝑔+𝑐−𝛼𝜆𝜙−2𝛼𝜙²+12𝛽𝜙^′2, (4) where 𝑔 is an integral constant. Let 𝑑𝜙/𝑑𝜉=𝑦. Thus, (4) can be rewritten as the following 2-dimensional planar system: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=()()𝑑𝜉𝑔+𝑐−𝛼𝜆𝜙−1/2𝛼𝜙²+(1/2)𝛽𝑦²,1−𝜆−𝜙 (5) which is a singular system. The system (5) has the following first integral: 𝐻𝜙,𝑦≡−1+𝜆+𝜙^𝛽𝑦²+𝑎₀+𝑎₁𝜙+𝑎₂𝜙²𝛽𝛽+1𝛽+2=ℎ, (6) where ℎ is integral constant and 𝑎₀=2𝑔(𝛽+1)(𝛽+2)−2(𝜆−1)[𝑐(𝛽+2)−𝛼𝜆(𝛽+1)−𝛼], 𝑎₁=2(𝑐−𝛼𝜆)𝛽²+2𝛽[2𝑐−𝛼(𝜆+1)], and 𝑎₂=−𝛼𝛽(𝛽+1). Particularly, when 𝛽₁=1, (6) becomes 𝐻≡×𝑦𝜙,𝑦−1+𝜆+𝜙₂+13𝛼𝜙³+(𝑐−𝛼)𝜙²+2𝑔𝜙=ℎ. (7) And when 𝛽=2, (6) becomes 𝐻𝜙,𝑦≡−1+𝜆+𝜙²𝑦²−14𝛼𝜙⁴+2𝑐−3𝛼𝜆+𝛼3𝜙³+𝑔+(𝑐−𝛼𝜆)(−1+𝜆)𝜙²+2𝑔(−1+𝜆)𝜙=ℎ. (8) Obviously, (7) is very simpler than (8), that is, the case of 𝛽=1 is very simpler than the case of 𝛽=2. Thus we will only discuss the case of 𝛽=2; the case of 𝛽=1 can be similarly discussed, we omit the discussions of case 𝛽=1 in this paper.

It is easy to find that the derivative 𝑑𝑦/𝑑𝜉 in the right side of (5) is not defined when 1−𝜆−𝜙=0. Therefore, we make a transformation as follows: 𝑑𝜉=1−𝜆−𝜙𝑑𝜏, (9) where 𝜏 is a parameter. Under the transformation (9) and 𝛽=2, (5) becomes the following regular system: 𝑑𝜙=𝑑𝜏1−𝜆−𝜙𝑦,𝑑𝑦()1𝑑𝜏=𝑔+𝑐−𝛼𝜆𝜙−2𝛼𝜙²+𝑦². (10) Obviously, (8) can be rewritten as follows: 𝑦²=1𝜙+𝜆−1²×𝛼4𝜙⁴−2(𝑐−𝛼𝜆)−𝛼(𝜆−1)3𝜙³−𝑔+(𝑐−𝛼𝜆)(𝜆−1)𝜙²−2𝑔(𝜆−1)𝜙+ℎ. (11)

3. Exact Traveling Wave Solutions of (2)

In this section, we will investigate exact traveling wave solutions of (2) by using its first integral equation (11).

Case  1. When 𝜆=(2𝑐+𝛼)/3𝛼,𝑔=0, (11) can be reduced to 𝑦=±𝑃𝜙⁴+𝑄𝜙²+𝑅,𝜙+𝜆−1 (12) where 𝑃=(1/4)𝛼,𝑄=−8(𝑐−𝛼)²/9𝛼, and 𝑅=ℎ. Substituting (12) into the first equation of (10) yields 𝑑𝜙𝑑𝜏=±𝑃𝜙⁴+𝑄𝜙²+𝑅. (13) By using (13) and (9), we will obtain different kinds of exact traveling wave solutions of (2); see the following discussion.(i)If −10<𝑐<−4−2√2 or −4+3√2<𝑐<2 and 𝛼=−4,𝑚=(1/3)−(1/2)𝑐²−4𝑐+10, ℎ=(−1/18)(𝑐²+8𝑐−2), then 𝑃=−1,𝑄=2−𝑚², and 𝑅=𝑚²−1. Under these conditions, by using (13) and (9), we obtain a periodic wave solution of (2) as follows: 𝑢=𝜆+dn(𝜏,𝑚),𝜉=(1−𝜆)𝜏−arcsin(sn(𝜏,𝑚)). (14) The profile of solution (14) is a periodic loop wave, which is shown in Figure 1. (ii)If −2−√34/2<𝑐<−2+√34/2 and 𝑚=(1/34)1462−136𝑐±102√−2𝑐²−8𝑐+9,  𝛼=(2/17)(4𝑐−9)±(6/17)√−2𝑐²−8𝑐+9, then 𝑃=1−𝑚²,𝑄=2−𝑚² and 𝑅=1. Under these conditions, by using (13) and (9), we obtain a periodic wave solution of (2) as follows: (),()1𝑢=𝜆+sc𝜏,𝑚𝜉=1−𝜆𝜏−2√1−𝑚²()+√lndn𝜏,𝑚1−𝑚²dn(𝜏,𝑚)−√1−𝑚². (15)(iii)If −∞<𝑐<4−3√2 or 4+3√2<𝑐<∞ and 𝑚=(1/3)(1/2)𝑐²−4𝑐−1,𝛼=4,  ℎ=(1/18)(𝑐²−8𝑐−2), then 𝑃=1,𝑄=−(1+𝑚²) and 𝑅=𝑚². Under these conditions, by using (13) and (9), we obtain a periodic wave solution of (2) as follows: (),()()𝑢=𝜆+ns𝜏,𝑚𝜉=1−𝜆𝜏−lnsn𝜏,𝑚()().cn𝜏,𝑚+dn𝜏,𝑚 (16)(iv)If 4−3√2<𝑐<4+3√2 and 𝑚=(1/6)√−𝑐²+8𝑐−2,𝛼=4, ℎ=(1/1296)(𝑐²−8𝑐+34)(𝑐²−8𝑐−2), then 𝑃=1,𝑄=2𝑚²−1, 𝑅=−𝑚²+𝑚⁴. Under these conditions, by using (13) and (9), we obtain a periodic wave solution of (2) as follows: (),()()𝑢=𝜆+ds𝜏,𝑚𝜉=1−𝜆𝜏−lnsn𝜏,𝑚().1+cn𝜏,𝑚 (17)(v) If 1−3√2/2<𝑐<−1/2 or 5/2<𝑐<1+3√2/2 and 𝑚=(1/3)√−4𝑐²+8𝑐+14,𝛼=1,  ℎ=(1/81)(2𝑐²−4𝑐−7)², then 𝑃=1/4,𝑄=(1/2)(𝑚²−2), and 𝑅=(1/4)𝑚⁴. Under these conditions, by using (13) and (9), we obtain two periodic wave solutions of (2) as follows: ()(),()()𝑢=𝜆+ns𝜏,𝑚+ds𝜏,𝑚𝜉=1−𝜆𝜏−lnsn𝜏,𝑚()()()cn𝜏,𝑚+dn𝜏,𝑚−lnsn𝜏,𝑚(),()(),()()1+cn𝜏,𝑚𝑢=𝜆+ns𝜏,𝑚−ds𝜏,𝑚𝜉=1−𝜆𝜏−lnsn𝜏,𝑚()()()cn𝜏,𝑚+dn𝜏,𝑚+lnsn𝜏,𝑚().1+cn𝜏,𝑚 (18)(19)

Figure 1 

The profile of periodic loop-wave solution (14) for 𝛼=−4,𝑐=0.5.

The waveforms of solution (19) are transformable; the smooth periodic wave becomes the non-smooth periodic cusp wave as parameter 𝑐 varies; their profiles are shown in Figures 2(a) and 2(b). In fact, this non-smooth periodic cusp wave is a limit form of smooth periodic wave.

(a) Smooth periodic wave

(b) Nonsmooth periodic cusp wave

Figure 2 

The profiles of periodic wave solutions (19) for (a) 𝛼=1,𝑐=2.7, (b) 𝛼=1,𝑐=2.5001.

Case  2. When 𝑔=0,ℎ=0, (11) can be reduced to 𝑦=𝐴𝜙²+𝐵𝜙³+𝐶𝜙⁴,𝜙+𝜆−1 (20) where 𝐴=(𝑐−𝛼𝜆)(𝜆−1),𝐵=𝛼𝜆−(1/3)(2𝑐+𝜆), and 𝐶=(1/4)𝛼. Substituting (20) into the first equation of (10) yields 𝑑𝜙𝑑𝜏=±𝐴𝜙²+𝐵𝜙³+𝐶𝜙⁴. (21) By using (21) and (9), we will obtain different kinds of soliton solutions; see the following discussions.(i)When 𝐴>0,𝐶>0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆−𝐴𝐵sech²𝐴/2𝜏𝐵²−𝐴𝐶1+𝜖tanh𝐴/2𝜏²,()𝜏+2𝜉=1−𝜆𝜖√𝐶tanh⁻¹√𝐴𝐶𝐵√1+𝜖tanh𝐴2𝜏, (22) where 𝜏 is a parameter. The solution (22) contains three transformable wave-forms as the parameter 𝑐 varies; these three wave-forms are dark loop soliton, dark peakon, and smooth-dark soliton, which are shown in Figures 3(a), 3(b), and 3(c). (ii)When 𝐴>0,𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆−𝐴𝐵sech²√𝐴/2𝜏𝐵²−𝐴𝐶1+𝜖tanh√𝐴/2𝜏²,()2𝜉=1−𝜆𝜏+𝜖√√−𝐶×arctan−𝐴𝐶𝐵√1+𝜖tanh𝐴2𝜏, (23) where 𝜏 is a parameter. The solution (23) contains two wave-forms, the dark loop soliton, and smooth-bright soliton which are shown in Figure 4. (iii)When 𝐴<0,𝐶>0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆+𝐴𝐵csch²√𝐴/2𝜏𝐵²−𝐴𝐶1+𝜖coth√𝐴/2𝜏²,()2𝜉=1−𝜆𝜏+𝜖√𝐶×tanh⁻¹√𝐴𝐶𝐵√1+𝜖coth𝐴2𝜏, (24)(iv)When 𝐴<0,𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆+𝐴𝐵csch²√𝐴/2𝜏𝐵²−𝐴𝐶1+𝜖coth√𝐴/2𝜏²,()2𝜉=1−𝜆𝜏+𝜖√√−𝐶×arctan−𝐴𝐶𝐵√1+𝜖coth𝐴2𝜏. (25)The solution (25) contains four transformable wave-forms as the parameter 𝑐 varies; these four wave-forms are broken wave, broken soliton, dark loop soliton, and smooth-bright soliton, which are shown in Figures 5(a), 5(b), 5(c), and 5(d). (v)When 𝐴>0,Δ>0, and 𝐶>0, by using (21) and (9), we obtain a soliton solution as follows: √𝑢=𝜆+2𝐴sech𝐴𝜏𝜖√Δ−𝐵sech√,()2𝐴𝜏𝜉=1−𝜆𝜏+√𝐶tanh⁻¹𝐵+𝜖√Δ2√√𝐴𝐶tanh𝐴2𝜏. (26)(vi)When 𝐴>0,Δ>0, and 𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: √𝑢=𝜆+2𝐴sech𝐴𝜏𝜖√Δ−𝐵sech√,()2𝐴𝜏𝜉=1−𝜆𝜏−√√−𝐶arctan𝐵+𝜖Δ2√√−𝐴𝐶tanh𝐴2𝜏. (27)(vii)When 𝐴<0,Δ>0, and 𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: √𝑢=𝜆+2𝐴sec−𝐴𝜏𝜖√Δ−𝐵sec√,()2−𝐴𝜏𝜉=1−𝜆𝜏−√√−𝐶arctan𝐵+𝜖Δ2√√𝐴𝐶tan−𝐴2𝜏. (28) The solution (28) contains four transformable wave-forms as the parameter 𝑐 varies; these four wave-forms are periodic compacton, smooth periodic wave, periodic overlap wave, and double periodic overlap wave, which are shown in Figures 6(a), 6(b), 6(c), and 6(d). (viii)When 𝐴>0,Δ<0, and 𝐶>0, by using (21) and (9), we obtain a soliton solution as follows: √𝑢=𝜆+2𝐴csch𝐴𝜏𝜖√−Δ−𝐵csch√,()2𝐴𝜏𝜉=1−𝜆𝜏+√𝐶×tanh⁻¹𝐵tanh√𝐴/2𝜏+𝜖√−Δ2√.𝐴𝐶 (29)(ix)When 𝐴<0,Δ>0, and 𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: √𝑢=𝜆+2𝐴csc−𝐴𝜏𝜖√Δ−𝐵csc√,()2−𝐴𝜏𝜉=1−𝜆𝜏+√√−𝐶×arctan−𝐵tan−𝐴/2𝜏+𝜖√Δ2√.𝐴𝐶 (30) The solution (30) contains three transformable wave-forms as the parameter 𝑐 varies; these three wave-forms are periodic loop wave, smooth periodic wave, and multiperiodic overlap wave, which are shown in Figures 7(a), 7(b), and 7(c). (x)When 𝐴<0,𝐶>0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆−𝐴csc²√−𝐴/2𝜏𝐵+2𝜖√−𝐴𝐶cot√𝜏()1−𝐴/2,𝜉=1−𝜆𝜏−𝜖√𝐶ln𝐵+2𝜖√√−𝐴𝐶cot−𝐴2𝜏. (31)(xi)When 𝐴>0,Δ=0, by using (21) and (9), we obtain a soliton solution as follows: 𝐴𝑢=𝜆−𝐵√1+𝜖tanh𝐴2𝜏,()𝐴𝜉=1−𝜆𝜏+𝐵√𝜏+2𝜖𝐴𝐵√lncosh𝐴2𝜏. (32) The solution (32) contains two generalized kink waves when 𝜖=±1, which are shown in Figures 8(a) and 8(b). (xii)When 𝐴>0,𝐵=0, and 𝐶<0, by using (21) and (9), we obtain a soliton solution as follows: 𝑢=𝜆+−𝐴𝐶sech𝜖√,()1𝐴𝜏𝜉=1−𝜆𝜏−𝜖√𝜖√−𝐶arctansinhA𝜏. (33) If 𝐴=0,𝐶<0𝑢=𝜆+4𝐵(𝐵𝜏)²,()2−4𝐶𝜉=1−𝜆𝜏−√−𝐶arctan𝐵𝜏2√.−𝐶 (34)

(a) Dark loop soliton

(b) Dark peakon

(c) Smooth-dark soliton

Figure 3 

The profile of soliton solutions (22) for (a) 𝛼=0.5,𝜆=2, and 𝑐=0.3, (b) 𝛼=0.5,𝜆=2,𝑐=0.875, and (c) 𝛼=0.5,𝜆=2, and 𝑐=0.9375.

(a) Dark loop soliton

(b) Smooth-bright soliton

Figure 4 

The profile of soliton solutions (23) for (a) 𝛼=−0.5,𝜆=2, and 𝑐=−5 and (b) 𝛼=−0.5,𝜆=2, and 𝑐=−1.15.

(a) Broken wave

(b) Broken loop soliton

(c) Dark loop soliton

(d) Smooth-bright soliton

Figure 5 

The profile of soliton solutions (25) for (a) 𝛼=−0.5,𝜆=2, and 𝑐=−1.0001, (b) 𝛼=−0.5, 𝜆=2, and 𝑐=−1.2, and (c) 𝛼=−0.5,𝜆=2, and 𝑐=−1.25, and (d) 𝛼=−0.5,𝜆=2, and 𝑐=−1.25000001.

(a) Periodic compacton

(b) Smooth periodic wave

(c) Periodic overlap wave

(d) Double periodic overlap wave

Figure 6 

The profile of soliton solutions (28) for (a) 𝛼=−0.5,𝜆=2, and 𝑐=−0.94, (b) 𝛼=−0.5,𝜆=2, and 𝑐=−0.499999, (c) 𝛼=−0.5,𝜆=2, and 𝑐=−0.2, and (d) 𝛼=−0.5,𝜆=2, and 𝑐=0.875.

(a) Periodic loop wave

(b) Smooth periodic wave

(c) Multiperiodic overlap wave

Figure 7 

The profile of soliton solutions (30) for (a) 𝛼=−0.5,𝜆=2, and 𝑐=−0.96, (b) 𝛼=−0.5,𝜆=2, and 𝑐=0.4999999, and (c) 𝛼=−0.5,𝜆=2, and 𝑐=2.

(a) Generalized kink wave

(b) Generalized kink wave

Figure 8 

The profiles of soliton solutions (32) for (a) 𝛼=0.5,𝜆=2, and 𝑐=0.875, and 𝜖=−1 and (b) 𝛼=0.5,𝜆=2,𝑐=0.875, and 𝜖=1.

4. Conclusion

In this work, by using the integral bifurcation method, we studied Kudryashov-Sinelshchikov equation in the special case 𝛽=2. Some singular and non-singular traveling wave solutions, such as periodic compacton solutions, non-continuous periodic loop-wave solutions, peakon solutions broken wave solutions, periodic cusp-wave solutions, smooth soliton solutions, continuous periodic loop-wave solutions, and smooth periodic wave solutions are obtained. Some solutions such as (19), (22), (23), (25), (28), and (30) contain multiwave forms; their wave-forms vary accordingly as the parameter 𝑐 varies. For example, the wave-forms of solution (19) show that the smooth periodic wave becomes the non-smooth periodic cusp wave as parameter 𝑐 varies. The waveforms of solution (22) show that the solitary wave becomes solitary cusp wave (peakon) when the value of 𝑐 arrives at certain limit value; then solitary cusp wave becomes the loop wave when the value of 𝑐 exceeds this limit value. Indeed, the non-smooth periodic cusp wave can be considered as a kind of long-wave limit of smooth periodic solutions; the loop wave also can be considered as limit wave of smooth solitary wave; these phenomena are very similar to the one in [23]. In [23], according to Braun and Randrüüt’s explaination, solitary wave can be considered as the long-wave limit of periodic solutions in KdV case. By the way, these results which obtained in this paper are different from those in [8–12].

Acknowledgments

The authors thank the reviewers very much for their useful comments and helpful suggestions. This research is supported by the National Natural Science Foundation of China (no. 11161020) and the Natural Science Foundations of Yunnan Province (nos. 2011FZ193 and 2013FZ117).