Periodic Loop Solutions and Their Limit Forms for the Kudryashov-Sinelshchikov Equation

Abstract

The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. We show that the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. Also, some new exact travelling wave solutions are presented through some special phase orbits.

1. Introduction

A mixture of liquid and gas bubbles of the same size may be considered as an example of a classic nonlinear medium. In practice, analysis of propagation of the pressure waves in a liquid with gas bubbles is important problem. We know that there are solitary and periodic waves in a mixture of a liquid and gas bubbles and these waves can be described by nonlinear partial differential equations. As for examples of nonlinear differential equations to describe the pressure waves in bubbly liquids, we can point out the Burgers equation, the Korteweg-de Vries equation, the Burgers-Korteweg-de Vries equation, and so on [1].

In 2010, Kudryashov and Sinelshchikov [1] obtained a more common nonlinear partial differential equation for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, and the equation reads as follows: 𝑢_𝑡+𝛼𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥−𝛽𝑢_𝑥𝑢_𝑥𝑥=0, (1.1) where 𝑢 is a density and which model heat transfer and viscosity, 𝛼,𝛽 are real parameters. Equation (1.1) is called Kudryashov-Sinelshchikov equation, it is generalization of the KdV and the BKdV equation and similar but not identical to the Camassa-Holm equation. Undistorted waves are governed by a corresponding ordinary differential equation which, for special values of some integration constant, is solved analytically in [1]. Ryabov [2] obtained some exact solutions for 𝛽=−3 and 𝛽=−4 using a modification of the truncated expansion method. Solutions are derived in a more straightforward manner and cast into a simpler form, and some new types of solutions which contain solitary wave and periodic wave solutions are presented in [3].

In this paper, we focus on the case 𝛽=2 of (1.1) using the bifurcation theory and the method of phase portraits analysis [4–6], we will investigate periodic loop solutions and their limit forms and give some new exact travelling wave solutions.

2. Preliminary

In this paper, we always consider the case 𝛽=2, so from now on we assume 𝛽=2 in (1.1) without mentioning it further.

Substituting 𝑢(𝑥,𝑡)=1−𝜙(𝑥+𝛼𝑡)=1−𝜙(𝜉) into (1.1) and integrating the resulting equation once with respect to 𝜉, we obtain 𝛼𝑔−2𝛼𝜙+2𝜙²−𝜙𝜙^′′−𝜙^′2=0, (2.1) where 𝑔 is the integral constant.

Letting 𝑦=𝑑𝜙/𝑑𝜉, we get the following planar system:𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=()𝜙𝑑𝜉𝑔−2𝛼𝜙+𝛼/2²−𝑦²𝜙. (2.2)

Using the transformation 𝑑𝜉=𝜙𝑑𝜏, it carries (2.2) into the Hamiltonian system: 𝑑𝜙𝑑𝜏=𝜙𝑦,𝑑𝑦𝛼𝑑𝜏=𝑔−2𝛼𝜙+2𝜙²−𝑦². (2.3)

Since both system (2.2) and (2.3) have the same first integral: 𝜙²𝑦²4−𝑔+31𝛼𝜙−4𝛼𝜙²=ℎ, (2.4) then the two systems above have the same topological phase portraits except the line 𝜙=0. Therefore, we can obtain the bifurcation phase portraits of system (2.2) from that of system (2.3).

Write Δ=2𝛼(2𝛼−𝑔). Clearly, when Δ>0, system (2.3) has two equilibrium points at (𝜙_1,2,0) in 𝜙-axis, where 𝜙_1,2=(2𝛼±√Δ)/𝛼. When Δ=0, system (2.3) has only one equilibrium point at (2,0) in 𝜙-axis. When Δ<0, system (2.3) has no any equilibrium point in 𝜙-axis. When 𝑔>0, there exist two equilibrium points of system (2.3) in line 𝜙=0 at (0,±√𝑔).

Let 𝑀(𝜙_𝑒,𝑦_𝑒) be the coefficient matrix of the linearized system of (2.3) at equilibrium point (𝜙_𝑒,𝑦_𝑒),𝐽=det(𝑀(𝜙_𝑒,𝑦_𝑒)), and 𝑇=trace(𝑀(𝜙_𝑒,𝑦_𝑒)). By the theory of planar dynamical systems, we know that for an equilibrium point (𝜙_𝑒,𝑦_𝑒) of a planar integrable system, (𝜙_𝑒,𝑦_𝑒) is a saddle point if 𝐽<0, a center point if 𝐽>0 and 𝑇=0, a cusp if 𝐽=0 and the Poincaré index of (𝜙_𝑒,𝑦_𝑒) is zero. By using the properties of equilibrium points and bifurcation method of dynamical systems, we can show that the bifurcation phase portraits of systems (2.2) and (2.3) is as drawn in Figure 1.

Figure 1 

The bifurcation phase portraits of systems (2.2) and (2.3).

From Figures 1(b), 1(c), 1(d), 1(e), and 1(l), we have the following results.

3. Main Results

Proposition 3.1. (i) When 𝛼<0,𝑔=2𝛼,  for ℎ=(−4/3)𝛼 defined by (2.4), (1.1) has a loop-soliton solution.
(ii) When 𝛼<0,𝑔=2𝛼, for ℎ∈(0,(−4/3)𝛼), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the loop-soliton solution as ℎ approaches (−4/3)𝛼.
(iii) When 𝛼<0,𝑔=2𝛼,   for ℎ∈((−4/3)𝛼,+∞), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the loop-soliton solution as ℎ approaches (−4/3)𝛼.

Proposition 3.2. Denote that ℎ₁=𝐻(𝜙₁,0) and ℎ₂=𝐻(𝜙₂,0).
(i) When 𝛼<0,2𝛼<𝑔<0, for ℎ=ℎ₁ defined by (2.4), (1.1) has a loop-soliton solution and has a solitary wave solution.
(ii) When 𝛼<0,2𝛼<𝑔<0, for ℎ∈(ℎ₂,ℎ₁), there exist a family of uncountably infinite many periodic loop solutions and a family of uncountably infinite many smooth periodic wave solutions of (1.1). Moreover, the periodic loop solutions converge to the loop-soliton solution and the smooth periodic wave solutions converge to the solitary wave solution as ℎ approaches ℎ₁.
(iii) When 𝛼<0,2𝛼<𝑔<0, for ℎ∈(ℎ₁,+∞), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the loop-soliton solution as ℎ approaches ℎ₁.
(iv) When 𝛼<0,2𝛼<𝑔<0, for ℎ∈(0,ℎ₂], there exists a family of uncountably infinite many periodic loop solutions of (1.1).

Proposition 3.3. (i) When 𝛼<0,𝑔=0,  for ℎ=0 defined by (2.4), (1.1) has a smooth periodic wave solution.
(ii) When 𝛼<0,𝑔=0,  for ℎ∈(0,+∞), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the smooth periodic wave solution as ℎ approaches 0.

Proposition 3.4. (i) When 𝛼<0,𝑔>0,  for ℎ=0 defined by (2.4), (1.1) has two cusp periodic wave solutions.
(ii) When 𝛼<0,𝑔>0, for ℎ∈(0,+∞), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the cusp periodic wave solutions as ℎ approaches 0.

Proposition 3.5. Denote that ℎ₂=𝐻(𝜙₂,0).
(i) When 𝛼>0,𝑔<0, for ℎ=ℎ₂ defined by (2.4), (1.1) has a loop-soliton solution.
(ii) When 𝛼>0,𝑔<0, for ℎ∈(0,ℎ₂), there exists a family of uncountably infinite many periodic loop solutions of (1.1). Moreover, the periodic loop solutions converge to the loop-soliton solution as ℎ approaches ℎ₂.

4. Exact Traveling Wave Solutions of the Kudryashov-Sinelshchikov Equation

Corresponding to Figure 1(b), the graph defined by 𝐻(𝜙,𝑦)=(−4/3)𝛼 consist of two hyperbolic sectors of the cusp (2,0) and an open-end curve Γ₀ passing through the point (−2/3,0). It follows from (2.4) that𝑦=±2−𝜙−𝛼2−𝜙𝜙+2/322𝜙,−3≤𝜙<2,𝜙≠0. (4.1) Substituting (4.1) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₀ and noting that 𝑢(𝑥,𝑡)=1−𝜙(𝑥+𝛼𝑡)=1−𝜙(𝜉), we obtain the following representation of loop-soliton solution: 𝑢𝜒=1−2sin²𝜒+23cos²𝜒,𝜉𝜒=4√3−𝛼4tan𝜒−𝜒, (4.2) where 𝜒 is a new parametric variable.

Corresponding to Figure 1(c), the graph defined by 𝐻(𝜙,𝑦)=𝐻(𝜙₁,0) consists of an open-end curve Γ₁ passing through the point (𝜙_𝑚,0) and a homoclinic orbit connecting with saddle point (𝜙₁,0) and passing point (𝜙_𝑀,0), where 𝜙_𝑚=(3√Δ−2𝛼−22𝛼(2𝛼−3√Δ))/−3𝛼, 𝜙_𝑀=(3√Δ−2𝛼+22𝛼(2𝛼−3√Δ))/−3𝛼. It follows from (2.4) that𝜙𝑦=±¹−𝜙−𝛼𝜙_𝑀−𝜙𝜙−𝜙_𝑚2𝜙,𝜙_𝑚≤𝜙<𝜙₁,𝜙≠0,𝑦=±𝜙−𝜙₁−𝛼𝜙_𝑀−𝜙𝜙−𝜙_𝑚2𝜙,𝜙₁≤𝜙≤𝜙_𝑀. (4.3)(4.4) Substituting (4.3) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₁, we can obtain the following representation of loop-soliton solution: 𝑢𝜒𝜙=1−¹𝜙_𝑀sinh²𝜔𝜒−𝜙_𝑚cosh²𝜔𝜒+𝜙_𝑚𝜙_𝑀𝜙_𝑀cosh²𝜔𝜒−𝜙_𝑚sinh²𝜔𝜒−𝜙₁,𝜉𝜒=2√−𝛼𝜒−2arctan𝜙₁−𝜙_𝑚𝜙_𝑀−𝜙₁tanh𝜔𝜒, (4.5) where 𝜔=(𝜙₁−𝜙_𝑚)(𝜙_𝑀−𝜙₁)/2𝜙₁.

Substituting (4.4) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the homoclinic orbit, we can obtain the following representation of solitary wave solution: 𝑢𝜒𝜙=1−¹𝜙_𝑀cosh²𝜔𝜒−𝜙_𝑚sinh²𝜔𝜒−𝜙_𝑚𝜙_𝑀𝜙_𝑀sinh²𝜔𝜒−𝜙_𝑚cosh²𝜔𝜒+𝜙₁,𝜉𝜒=2√−𝛼𝜒−2arctan𝜙_𝑀−𝜙₁𝜙₁−𝜙_𝑚tanh𝜔𝜒, (4.6) where 𝜔=(𝜙₁−𝜙_𝑚)(𝜙_𝑀−𝜙₁)/2𝜙₁.

Moreover, the graph defined by 𝐻(𝜙,𝑦)=ℎ, ℎ∈(𝐻(𝜙₂,0), 𝐻(𝜙₁,0)), consists of two open-end curves Γ₂,Γ₃, and a periodic orbit, say Ψ, enclosing the center point (𝜙₂,0). The curve Ψ passes through the points (𝛾₁,0) and (𝛾₂,0), while Γ₂,Γ₃ pass through the points (𝛾₃,0) and (𝛾₄,0), respectively, where 𝛾₁,𝛾₂,𝛾₃,𝛾₄(𝛾₄<0<𝛾₃<𝛾₂<𝛾₁) are four real roots of 𝜓⁴−(16/3)𝜓³+(4𝑔/𝛼)𝜓²+(4ℎ/𝛼)=0. It follows from (2.4) that𝑦=±−𝛼𝛾₁−𝜙𝛾₂−𝜙𝛾₃−𝜙𝜙−𝛾₄2𝜙,𝛾₄≤𝜙≤𝛾₃,𝜙≠0,𝑦=±−𝛼𝛾₁−𝜙𝜙−𝛾₂𝜙−𝛾₃𝜙−𝛾₄2𝜙,𝛾₂≤𝜙≤𝛾₁. (4.7)(4.8)

Let us denote by F(⋅,𝑘) and Π(⋅,⋅,𝑘) the Legendre’s incomplete elliptic integrals of the first and third kinds, respectively, with the modulus 𝑘 (see [7]).

Substituting (4.7) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₂, we can obtain the implicit representation of periodic loop solution for 𝑢∈[1−𝛾₃,1−𝛾₄]:4𝛾₃𝛼²₁−𝛼𝛾₁−𝛾₃𝛾₂−𝛾₄×𝛼²₁−𝛼²₂Πarcsin𝛾₂−𝛾₄𝛾₃+𝑢−1𝛾₃−𝛾₄𝛾₂+𝑢−1,𝛼²₁,𝑘+𝛼²₂Farcsin𝛾₂−𝛾₄𝛾₃+𝑢−1𝛾₃−𝛾₄𝛾₂+𝑢−1,𝑘=±𝜉, (4.9) where 𝛼²₁=(𝛾₃−𝛾₄)/(𝛾₂−𝛾₄), 𝛼²₂=𝛾₂(𝛾₃−𝛾₄)/𝛾₃(𝛾₂−𝛾₄), 𝑘=(𝛾₁−𝛾₂)(𝛾₃−𝛾₄)/(𝛾₁−𝛾₃)(𝛾₂−𝛾₄).

Substituting (4.8) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the periodic orbit, we can obtain the implicit representation of smooth periodic wave solution for 𝑢∈[1−𝛾₁,1−𝛾₂]:4𝛾₂𝛼²₁−𝛼𝛾₁−𝛾₃𝛾₂−𝛾₄×𝛼²₁−𝛼²₂Πarcsin𝛾₁−𝛾₃1−𝑢−𝛾₂𝛾₁−𝛾₂1−𝑢−𝛾₃,𝛼²₁,𝑘+𝛼²₂Farcsin𝛾₁−𝛾₃1−𝑢−𝛾₂𝛾₁−𝛾₂1−𝑢−𝛾₃,𝑘=±𝜉, (4.10) where 𝛼²₁=(𝛾₁−𝛾₂)/(𝛾₁−𝛾₃), 𝛼²₂=𝛾₃(𝛾₁−𝛾₂)/𝛾₂(𝛾₁−𝛾₃), 𝑘=(𝛾₁−𝛾₂)(𝛾₃−𝛾₄)/(𝛾₁−𝛾₃)(𝛾₂−𝛾₄).

Corresponding to Figure 1(d), the graph defined by 𝐻(𝜙,𝑦)=0 is a periodic orbit enclosing the center point (2(𝛼−𝛼(𝛼−1))/𝛼,0) and passing through the points (0,0),(16/3,0). It follows from (2.4) that 1𝑦=±2−𝛼𝜙163−𝜙,0≤𝜙≤163. (4.11) Substituting (4.11) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the periodic orbit, we can obtain the following representation of smooth periodic wave solution: 𝑢(𝑥,𝑡)=1−163cos²(𝜔(𝑥+𝛼𝑡)), (4.12) where 𝜔=(1/4)√−𝛼.

Corresponding to Figure 1(e), the graph defined by 𝐻(𝜙,𝑦)=0 consists of four heteroclinic orbits: two of them connecting the saddle points (0,±√𝑔) with (𝜙_𝑚,0), and the others connecting saddle points (0,±√𝑔) with (𝜙_𝑀,0), where 𝜙_𝑚=2(4𝛼+𝛼(16𝛼−9𝑔))/3𝛼, 𝜙_𝑀=2(4𝛼−𝛼(16𝛼−9𝑔))/3𝛼. It follows from (2.4) that1𝑦=±2−𝛼𝜙−𝜙_𝑚𝜙_𝑀−𝜙,𝜙_𝑚1≤𝜙≤0,𝑦=±2−𝛼𝜙−𝜙_𝑚𝜙_𝑀−𝜙,0≤𝜙≤𝜙_𝑀. (4.13)(4.14) Substituting (4.13) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the heteroclinic orbit, we can obtain the following representation of cusp periodic wave solution:𝑢(𝑥,𝑡)=1−𝜙_𝑀sin²(Ω−𝜔|𝑥+𝛼𝑡−2𝑛𝑇|)−𝜙_𝑚cos²(Ω−𝜔|𝑥+𝛼𝑡−2𝑛𝑇|), (4.15) where 𝜔=(1/4)√−𝛼, Ω=arctan(−𝜙_𝑚/𝜙_𝑀), 𝑇=2|Ω|, 𝑛=0,±1,±2,…,(2𝑛−1)𝑇≤𝑥+𝛼𝑡≤(2𝑛+1)𝑇.

Substituting (4.14) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the heteroclinic orbit, we can obtain the following representation of cusp periodic wave solution: 𝑢(𝑥,𝑡)=1−𝜙_𝑀cos²(Ω−𝜔|𝑥+𝛼𝑡−2𝑛𝑇|)−𝜙_𝑚sin²(Ω−𝜔|𝑥+𝛼𝑡−2𝑛𝑇|), (4.16) where 𝜔=(1/4)√−𝛼, Ω=arctan(−𝜙_𝑀/𝜙_𝑚), 𝑇=2|Ω|, 𝑛=0,±1,±2,…, (2𝑛−1)𝑇≤𝑥+𝛼𝑡≤(2𝑛+1)𝑇.

Moreover, the graph defined by 𝐻(𝜙,𝑦)=ℎ,ℎ∈(0,+∞) consists of two open-end curves Γ₄,Γ₅ passing through the points (𝜙_𝑚,0) and (𝜙_𝑀,0), respectively, where (𝜙_𝑀−𝜙)(𝜙−𝜙_𝑚)[(𝜙−𝑏₁)²+𝑎²₁]=−𝜙⁴+(16/3)𝜙³−(4𝑔/𝛼)𝜙²−4ℎ/𝛼. It follows from (2.4) that 𝑦=±−𝛼𝜙_𝑀−𝜙𝜙−𝜙_𝑚𝜙−𝑏₁²+𝑎²₁2𝜙,𝜙_𝑚≤𝜙≤𝜙_𝑀,𝜙≠0. (4.17) Substituting (4.17) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₄, we can obtain the implicit representation of periodic loop solution for 𝑢∈[1−𝜙_𝑀,1−𝜙_𝑚]: 2𝜙_𝑚𝐴+𝜙_𝑀𝐵(𝐴−𝐵)√𝛼𝐴𝐵²F+𝛼𝜑,𝑘¹−𝛼₂1−𝛼²₁Π𝛼𝜑,²¹𝛼²₁−1,𝑘−𝛼₁𝑓₁−𝜂₀=±𝜉, (4.18) where 𝐴=(𝜙_𝑀−𝑏₁)²+𝑎²₁, 𝐵=(𝜙_𝑚−𝑏₁)²+𝑎²₁,𝛼₁=(𝐴−𝐵)/(𝐴+𝐵), 𝛼₂=(𝜙_𝑚𝐴−𝜙_𝑀𝐵)/(𝜙_𝑚𝐴+𝜙_𝑀𝐵), 𝑘=((𝜙_𝑀−𝜙_𝑚)²−(𝐴−𝐵)²)/4𝐴𝐵, 𝑘₁=√1−𝑘², 𝜂₀=[𝛼²₂F(𝜑,𝑘)+((𝛼₁−𝛼₂)/(1−𝛼²₁))(Π(𝜑,𝛼²₁/(𝛼²₁−1),𝑘)−𝛼₁𝑓₁)]|_{𝑢=1−𝜙𝑀}, 𝜑=arccos(((𝜙_𝑀+𝑢−1)𝐵+(𝜙_𝑚+𝑢−1)𝐴)/((𝜙_𝑀+𝑢−1)𝐵−(𝜙_𝑚+𝑢−1)𝐴)), 𝑓₁=(1−𝛼²₁)/(𝑘²+𝑘²₁𝛼²₁)arctan((sin²𝜑(𝑘²+𝑘²₁𝛼²₁))/(1−𝑘²sin²𝜑)(1−𝛼²₁)).

Corresponding to Figure 1(l), the graph defined by 𝐻(𝜙,𝑦)=𝐻(𝜙₂,0) consists of two hyperbolic sectors of the saddle point (𝜙₂,0) and two open-end curves Γ₆,Γ₇ passing through the points (𝜙_𝑚,0),(𝜙_𝑀,0), respectively, where 𝜙_𝑚=(2𝛼+3√Δ−22𝛼(2𝛼+3√Δ))/3𝛼, 𝜙_𝑀=(2𝛼+3√Δ+22𝛼(2𝛼+3√Δ))/3𝛼. It follows from (2.4) that 𝑦=±𝜙−𝜙₂𝛼𝜙_𝑚−𝜙𝜙_𝑀−𝜙2𝜙,𝜙₂<𝜙≤𝜙_𝑚,𝜙≠0. (4.19) Substituting (4.19) into the 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₆, we can obtain the following representation of loop-soliton solution: 𝑢𝜒𝜙=1−²𝜙_𝑀sinh²𝜔𝜒−𝜙_𝑚cosh²𝜔𝜒+𝜙_𝑚𝜙_𝑀𝜙_𝑀cosh²𝜔𝜒−𝜙_𝑚sinh²𝜔𝜒−𝜙₂,𝜉𝜒=2√𝛼𝜒−2tanh⁻¹𝜙_𝑚−𝜙₂𝜙_𝑀−𝜙₂tanh𝜔𝜒, (4.20) where 𝜔=(𝜙_𝑚−𝜙₂)(𝜙_𝑀−𝜙₂)/2𝜙₂,tanh⁻¹(⋅) is the inverse function of the hyperbolic function tanh(⋅), see [7].

Moreover, the graph defined by 𝐻(𝜙,𝑦)=ℎ,ℎ∈(0,𝐻(𝜙₂,0)) consist of four open-end curves Γ₈,Γ₉,Γ₁₀ and Γ₁₁ passing through the points (𝛾₄,0),(𝛾₃,0),(𝛾₂,0),(𝛾₁,0) respectively, where 𝛾₁,𝛾₂,𝛾₃,𝛾₄(𝛾₄<𝛾₃<0<𝛾₂<𝛾₁) are four real roots of 𝜓⁴−(16/3)𝜓³+(4𝑔/𝛼)𝜓²+(4ℎ/𝛼)=0. It follows from (2.4) that𝑦=±𝛼𝛾₁−𝜙𝛾₂−𝜙𝜙−𝛾₃𝜙−𝛾₄2𝜙,𝛾₃≤𝜙≤𝛾₂,𝜙≠0. (4.21) Substituting (4.21) into 𝑑𝜙/𝑑𝜉=𝑦 and integrating along the curve Γ₁₀, we can obtain the implicit representation of periodic loop solution for 𝑢∈[1−𝛾₂,1−𝛾₃]: 4𝛾₂𝛼²₁𝛼𝛾₁−𝛾₃𝛾₂−𝛾₄×𝛼²₁−𝛼²₂Πarcsin𝛾₁−𝛾₃𝛾₂+𝑢−1𝛾₂−𝛾₃𝛾₁+𝑢−1,𝛼²₁,𝑘+𝛼²₂Farcsin𝛾₁−𝛾₃𝛾₂+𝑢−1𝛾₂−𝛾₃𝛾₁+𝑢−1,𝑘=±𝜉, (4.22) where 𝛼²₁=(𝛾₂−𝛾₃)/(𝛾₁−𝛾₃), 𝛼²₂=𝛾₁(𝛾₂−𝛾₃)/𝛾₂(𝛾₁−𝛾₃), 𝑘=(𝛾₂−𝛾₃)(𝛾₁−𝛾₄)/(𝛾₁−𝛾₃)(𝛾₂−𝛾₄).

Remark 4.1. Denote that (i)  𝛼<0,𝑔=2𝛼, ℎ∈(0,(−4/3)𝛼),(ii) 𝛼<0,𝑔=2𝛼, ℎ∈((−4/3)𝛼,+∞), (iii) 𝛼<0, 2𝛼<𝑔<0, ℎ∈(0,𝐻(𝜙₂,0)],   (iv) 𝛼<0,2𝛼<𝑔<0,ℎ∈(𝐻(𝜙₁,0),+∞),  (v) 𝛼<0,𝑔=0,ℎ∈(0,+∞), we can obtain the implicit representation of periodic loop solution similar to (4.18) when 𝛽,𝛼,𝑔, and ℎ satisfy one and only one of above conditions, we omit it for brevity.

Example 4.2. Taking 𝛼=−1,𝑔=1 and ℎ=1, we get the approximations of 𝐴,𝐵,𝜙_𝑚,𝜙_𝑀,𝑎₁,𝑏₁,𝛼₁,𝛼₂,𝑘,𝑘₁ in the formula (4.18), where 𝐴≐5.846662930, 𝐵≐1.525667184, 𝜙_𝑚≐−1.117067993, 𝜙_𝑀≐6.016534182, 𝑎₁≐0.7403378831, 𝑏₁≐0.2169335722, 𝛼₁≐0.586109911, 𝛼₂≐−5.932667554, 𝑘≐0.9502338139, 𝑘₁≐0.3115376364.

5. Conclusion

In this paper, using the bifurcation theory and the method of phase portraits analysis, we investigated periodic loop solutions and their limit forms of the Kudryashov-Sinelshchikov equation and show that the limit forms contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. We also obtain the exact parametric representations above travelling wave solutions. The results of this paper have enriched results of [1–3]. We would like to study the Kudryashov-Sinelshchikov equation further.

Acknowledgments

The authors thank the referees for some perceptive comments and for some valuable suggestions. This work is supported by the National Natural Science Foundation of China (no. 11161020).