1 Introduction

A class of nonlinear wave equation with a quintic term

includes many important models in applied mathematics and mechanics.

If γ = 0 and a₅ = 0,Eq. (1) becomes the Klein-Gordon equation^[1]

which describes relativistic electrons. Wazwaz^[2],Sirendaoreji^[3],Jang^[4],and Ye and Zhang^[5] have studied the compactons,solitons,and solitary wave solutions.

If a₅ = 0,Eq. (1) becomes nonlinear telegraph equation^[6]

Equation (3) also can describe the pressure produced by pulsation when the blood flows in arterial. Many results on it can be seen in Refs. ^{[7, 8, 9, 10]}. To speak of Ref. ^[10],the relation between the shape of solitary wave solution for Eq. (3) and the dissipation effect was analyzed by the theory and method of planar dynamic systems. In some ways,this paper can be regarded as an extent of Ref. ^[10]. The interaction of a higher-order term (quintic term) and a dissipation term will make the traveling wave solutions of Eq. (1) more complex. We have seen little literature on this point so far. This is also the gist of our paper.

If γ = 0,Eq. (1) becomes

Equation (4) can be regarded as the approximate equation of and because sin u and sinh u can be approached by the polynomials of odd number power about u. References ^{[11, 12, 13, 14]} investigated the solitary wave solutions of Eq. (4). In Ref. ^[15],Li and Zhang analyzed the bifurcations of traveling wave solutions for Eq. (4),and obtained all of its explicit solitary wave solutions and kink wave solutions by bifurcation theory of planar dynamical systems. Recently,the theory and method of planar dynamical systems have been developed and widely used to study the traveling wave solutions of integral systems with singular lines. Many good results have been obtained in Refs. ^{[16, 17]}. As far as we know,dissipation is inevitable in practical problem. Therefore,it is necessary to study the case of ≠ 0. If ≠ 0,Eq. (1) can be regarded as the approximate equation of Eqs. (5) and (6),which is accompanied by a damping term. Since Eqs. (5) and (6) are very important in mechanics and particle physics,it is meaningful to study Eq. (1).

Making a general survey to the previous references,we find that they all obtained some solutions purely,without giving us the global structure for the traveling wave solutions of Eq. (1). Inspired by Refs. ^{[15, 16, 17]},we employ the method of planar dynamical systems to study the dissipative system corresponding to Eq. (1),and give the global phase portraits. Then,we use these global phase portraits to study the shapes and existent number of all bounded traveling wave solutions. Next,we discuss the relation between the shapes of bounded traveling wave solutions and dissipation coefficient γ ,and obtain two critical values which can characterize the scale of the dissipation effect. It can be proved that when the dissipation effect is large,the bounded traveling wave solutions of Eq. (1) appear as the kink solitary wave solutions; when it is small,they appear as the damped oscillatory solutions. By using the undetermined coefficients method,we find out the bounded traveling wave solutions of Eq. (1),including exact bell and kink profile solitary wave solutions and approximate damped oscillatory solutions. Furthermore,by using the homogenization principle,we will give the error estimates for the approximate damped oscillatory solutions by establishing the integral equations which reflects the relation between exact solutions and approximate solutions.

The organization of this paper is as follows. In Section 2,we make a qualitative analysis of the dynamical system which the traveling wave solutions of Eq. (1) correspond to. In Section 3,we study the dissipation effect on the shape of bounded traveling wave solutions. In Section 4,we present the bounded traveling wave solutions of Eq. (1),including exact bell and kink profile solitary wave solutions and approximate damped oscillatory solutions. In Section 5,we give the error estimate for approximate damped oscillatory solutions obtained in previous sections.

2 Qualitative analysis to corresponding planar dynamical system

Assume that Eq. (1) has the traveling wave solution in the form of u(x,t) = u(ξ) = u(x−ct),where c is the wave speed. Thus,Eq. (1) satisfies

Not loss generality,we suppose c² − β > 0.

Let x = u(ξ),y = u_ξ(ξ). Then,Eq. (1) can be converted into the following planar dynamical system:

where r = −cγ . We assume r ≤ 0. For the case of r > 0,it can be discussed similarly.

Denote Thus,the number of singular points for system (8) can be determined by the number of real roots of f(x). Denote Δ = a²₃ − 4a₁a₅. Equation (1) has bounded traveling wave solutions only when f(x) at least has two real roots. Therefore,we suppose Δ > 0. Denote x₁ = ,x₂ = ,and x₃ = . The possible cases are listed as follows.

(i) If a₁ < 0,Δ > 0,and x₂ < 0 < x₁,f(x) has three different real roots,where x₀ = 0 and x_1± = ±.

(ii) If a₁ > 0 and a₃ < 0,(a) if Δ > 0,0 < x₂ < x₁,f(x) has five different real roots,where x₀ = 0,x_1± = ±,and x_2± = ±; (b) if Δ = 0,x₃ > 0,f(x) has three different real roots,where x₀ = 0 and x_3± = ±.

Denote the Jacobian matrices of system (8) at P₀(0,0) and P_i±(x_i±,0) as

For the sake of discussion,we let Δ₀ = r² − 4a₁(c² − β) and Δ_i = r² − 4(c² − β)(5a₅x²_i + 3a₃x_i + a₁),i = 1,2,3.

We apply the theory and method of planar dynamical systems^{[18, 19]} to make a qualitative analysis for system (8) from three aspects (finite singular points,infinite singular points,and existence of limit cycle).

2.1 Types of finite singular points for system (8)

(I) In the case of r = 0.

At this time,system (8) has the first integral

Through analysis,we know that P₀ is a saddle point if a₁ < 0 and a center if a₁ > 0. The types of P_i±,i = 1,2,3,are listed as follows.

(i) a₁ < 0. System (8) has three different finite singular points: P₀(0,0) and P_1±(x_1±,0). Since f′(x_1±) > 0 and Eq. (9) holds,P_1±(x_1±,0) are centers.

(ii) a₁ > 0 and a₃ < 0. (a) If Δ > 0,system (8) has five finite singular points: P₀(0,0),P_1±(x_1±,0),and P_2±(x_2±,0). Since f′(x_1±) > 0 and (9) holds,P_1±(x_1±,0) are centers. Since f′(x_2±) < 0,both P_2±(x_2±,0) are saddle points. (b) If Δ = 0,system (8) has three finite singular points: P₀(0,0) and P_3±(x_3±,0). Since f′(x_3±) = 0 and the Poincar´e index is 0,both P_3±(x_3±,0) are cusps.

(II) In the case of r < 0.

P₀ is a saddle point if a₁ < 0. If a₁ > 0 and Δ₀ > 0,P₀ is an unstable node point; if a₁ > 0 and Δ₀ < 0,P₀ is an unstable focus point. The types of other singular points are listed as follows.

(i) a₁ < 0. System (8) has three different finite singular points: P₀(0,0) and P_1±(x_1±,0). Since f′(x_1±) > 0 ,P_1±(x_1±,0) are unstable node points if Δ₁ > 0; while P_1±(x_1±,0) are unstable focus points if Δ₁ < 0.

(ii) a₁ > 0 and a₃ < 0. (a) If Δ > 0,system (8) has five finite singular points: P₀(0,0),P_1±(x_1±,0),and P_2±(x_2±,0). Since f′(x_1±) > 0,P_1±(x_1±,0) are unstable focus points if Δ₁ < 0 and unstable node points if Δ₁ > 0. Since f′(x_2±) < 0,both P_2±(x_2±,0) are saddle points; (b) If Δ = 0,system (8) has three finite singular points: P₀(0,0) and P_3±(x_3±,0). It is easy to prove that both P_3±(x_3±,0) are saddle-node points.

2.2 Types of infinite singular points for system (8)

We use Poincar´e transformation to analyze the singular points at infinity of system (8). It is easy to know that there exists a couple of singular points at infinity on the y-axis which are denoted by A₁ and A₂,respectively. There exists a hyperbolic type region around A_i,i = 1,2,respectively. Moreover,the circumference of Poincar´e disk is orbits.

2.3 Existence of limit cycle for system (8)

Since ,for system (8) and the Bendixson-Dulac criterion,we can have the following proposition.

Proposition 1 If r ≠ 0,system (8) does not have any closed orbit or singular closed orbit on (x,y) phase plane. This shows that Eq. (1) has neither periodic traveling wave solution nor bell profile solitary wave solution as r ≠ 0.

2.4 Global phase portraits of system (8)

Furthermore,based on the above qualitative analysis,we present all possible global phase portraits for system (8) in the case of r = 0 and r < 0 (see Figs. 1-6).

(i) The global phase portrait in the case of r = 0 (see Figs. 1-3).

(ii) The global phase portrait in the case of r < 0 (see Figs. 4-6).

We take the case in Fig. 4(a) for example to explain how to establish the global phase portraits. Based on the results in the previous qualitative analysis,we know that the key step to determine the global structure of the orbit is to determine the directions and relative position of the separatrix for saddle point P₀(0,0). Obviously,x = 0 and y = 0 are not the orbits of system (8). Therefore,there must exist an angle between the separatrix and coordinate axis. From,we know that x increases as ξ increases when y > 0 while decreases as ξ increases when y < 0. The directions of separatrix are shown in Fig. 4(a).

Secondly,we determine the relative position of the separatrix. We know from subsection 2.2 that there is a hyperbolic type region around infinite singular points A_i,i = 1,2. This means that A_i,i = 1,2,are neither source points nor sink points; moreover,there is no orbit entering or exiting from A_i,i = 1,2. Meanwhile,P_1±(0,0) are unstable node points. Therefore,the separatrix away from saddle point P₀(0,0) in the forward direction can neither approach to unstable mode points P_1±(0,0) nor infinite. And the separatrix entering saddle point P₀(0,0) in the forward direction can only come from P_1±(0,0). Thus,all directions of the orbits are determined (see Fig. 4(a)). Other global phase portraits can be established in the same way.

From Figs. 1-6,we have the following proposition.

Proposition 2 Except singular points P₀ and P_i,i = 1,2,3,and orbits L(P₀,P_i±),i = 1,2,3,and L(P_1±,P_2±),the non-periodic solutions of system (8) are unbounded. Moreover,the coordinate values of the points on these orbits tend to infinite.

2.5 Existence of bounded traveling wave solutions

It is well-known that a homoclinic orbit of planar dynamical system corresponds to a bell profile solitary wave solution of its corresponding nonlinear evolution equation; the heteroclinic orbit corresponds to a kink profile solitary or an oscillatory wave solution; the closed orbit corresponds to a periodic wave solution. Then,we can derive the following theorem from Figs. 1-6 and Proposition 2.

Theorem 1 (i) If r = 0,Eq. (1) at most has two bounded traveling wave solutions. Two bell profile solitary traveling wave solutions correspond to the homoclinic orbits L(P₀,P₀) in Fig. 1 or L(P_2±,P_2±) in Fig. 3; or two kink profile solitary traveling wave solutions correspond to the heteroclinic orbits L(P_2±,) in Fig. 2 or L(P_2±,) in Fig. 3.

(ii) If r < 0,Eq. (1) at most has four bounded traveling wave solutions. Four kink pro- file solitary wave solutions correspond to the heteroclinic orbits L(P_1±,P_2±) and L(P₀,P_2±) in Fig. 6(a); or four oscillatory traveling wave solutions correspond to the heteroclinic orbits L(P_1±,P_2±) and L(P₀,P_2±) in Fig. 6(d); or two kink profile solitary wave solutions correspond to the heteroclinic orbits L(P₀,P_2±) in Fig. 6(b) or L(P_1±,P_2±) in Fig. 6(c),and two oscilla- tory traveling wave solutions correspond to the heteroclinic orbits L(P_1±,P_2±) in Fig. 6(b) or L(P₀,P_2±) in Fig. 6(c).

3 Relation between shapes of bounded traveling wave solutions for Eq. (1) and dissipation coefficient γ

Theorem 2 Suppose a₁ < 0. Then,we have the following results:

(i) If |γ | > ,Eq. (1) has a monotone decreasing kink profile solitary wave solution u(ξ),satisfying u(−∞) = and u(+∞) = 0; and a monotone increasing kink profile solitary wave solution satisfying u(−∞) = − and u(+∞) = 0. Here,u(ξ) corresponds to the orbits L(P_1±,P₀) in Fig. 4(a).

(ii) If |γ | < ,Eq. (1) has two damped oscillatory traveling wave solutions u(ξ),satisfying u(−∞) = ,u(+∞) = 0,and u(−∞) = −,u(+∞) = 0,respectively. Here,u(ξ) corresponds to the orbits L(P_1±,P₀) in Fig. 4(b).

Proof (i) From the theory of planar dynamical systems,we know that P_1± are unstable node points if Δ₁ > 0,i.e.,Δ₁ = r² −4(c² −β)(5a₅x₂1+3a₃x₁ +a₁) > 0. Since x₁ also satisfies a₅x₂1 + a₃x₁ + a₁ = 0 and r = −c ,we have |γ | > . Meanwhile,P₀ is a saddle point. There exists an orbit L(P₁₊,P₀) in {(x,y)|x > 0,−∞ < y < +∞} and an orbit L(P_1-,P₀) in {(x,y)|x < 0,−∞ < y < +∞} for system (2).

Next,we prove that the kink profile solitary wave solutions are monotone if |γ | > . From the relation between the traveling wave solutions and the orbits,we only need to prove the exact position of the orbits when |γ | > .

Since system (2) is symmetrical about the origin,we just need to study the relation be- tween the shape of the bounded traveling wave solution corresponding to L(P₁₊,P₀) and the dissipation coefficient . Denote k_1,2 = . Since |γ | > ,there exists k₀ ∈ (k₁,k₂) such that 4(c² −β)k²₀ −cγk₀ +a₅(x₁ − x₃) < 0 holds. Further,we have > 2k₀. Denote the curve l as y = k₀(x₂ −x₁). Then,there is a closed region AP₀P₁₊ inside l,x-axis,and y-axis (see Fig. 7).

Substitute the coordinate values of the point on the arc ₁₊,segment P₀P₁₊,and P₀A into system (8),respectively. We have

Since = ∞ and > 0,the tangent slope on the arc₁₊,segment P₀P₁₊,and P₀A are shown in Fig. 7. Obviously,the region closed by₁₊,P₀P₁₊,and P₀A is generalized non-tangential. That is to say L(P₁₊,P₀) cannot pass through the closed region AP₀P₁₊. Actually,if L(P₁₊,P₀) passes through the closed region AP₀P₁₊,it cannot penetrate into the closed region AP₀P₁₊ again. Otherwise,there must exits a direction,which is tangent to₁₊,or P₀P₁₊,or P₀A,because of the direction changing continuously in vector field. This is contradictory with the generalized non-tangential. Therefore,L(P₁₊,P₀) cannot pass through the closed region AP₀P₁₊. From the symmetry of system (8) about the origin,the orbit L(P_1-,P₀) can satisfy the similarly condition. Therefore,the orbits L(P₁₊,P₀) are shown as Fig. 4(a).

The above proof also reflects the correctness of the global phase portraits in Fig. 4(a).

From the relation between the traveling wave solutions and the orbits,we have that the kink profile solitary wave solution u(ξ) corresponding to L(P₁₊,P₀) (L(P_1-,P₀)) satisfies y = u′(ξ) < 0 (y = u′(ξ) > 0). Thus,it is monotone decreasing (increasing).

(ii) From the theory of planar dynamical systems,we know that P_1± are unstable focus points if Δ₁ < 0,i.e.,|γ | < and P₀ is a saddle point. There exists an orbit L(P₁₊,P₀) in {(x,y)|x > 0,−∞ < y < +∞} and an orbit L(P_1-,P₀) in {(x,y)|x < 0,−∞ < y < +∞} for system (8). Hence,from the relation between Eq. (1) and system (8),we can derive that Eq. (1) has two damped oscillatory traveling wave solutions.

The shape of damped oscillatory traveling wave solutions for Eq. (1) described in Theorem2 can be expressed by Fig. 8.

Similarly,we can prove the following theorems.

Theorem 3 Suppose a₁ > 0,a₃ < 0,and a²₃ − 4a₁a₅ = 0,we have the following results:

(i) If |γ | > ,Eq. (1) has a monotone increasing kink profile solitary wave solution u(ξ) satisfying u(−∞) = 0 and u(+∞) = and a monotone decreasing kink profile solitary wave solution satisfying u(−∞) = 0 and u(+∞) = −. Here,u(ξ) corresponds to the orbits L(P₀,P_3±) in Fig. 5(a).

(ii) If |γ | < ,Eq. (1) has two damped oscillatory traveling wave solutions u(ξ) satisfying u(−∞) = 0,u(+∞) = and u(−∞) = 0,u(+∞) = −,respectively. Here,u(ξ) corresponds to the orbits L(P₀,P_3±) in Fig. 5(b).

Theorem 4 Suppose a₁ > 0,a₃ < 0,and a23 − 4a₁a₅ > 0. Then,we have the following results:

(i) If |γ | > ,Eq. (1) has a monotone decreasing kink profile solitary wave solution u(ξ) satisfying u(−∞) = and u(+∞) = and a monotone increasing kink profile solitary wave solution satisfying u(−∞) = − and u(+∞) = −. Here,u(ξ) corresponds to the orbits L(P_1±,P_2±) in Fig. 6(a) or Fig. 6(c).

(ii) If |γ | < ,Eq. (1) has two damped oscillatory traveling wave solu- tions u(ξ) satisfying u(−∞) = ,u(+∞) = and u(−∞) = −,u(+∞) = −,respectively. Here,u(ξ) corresponds to the orbits L(P_1±,P_2±) in Fig. 6(b) or Fig. 6(d).

(iii) If |γ | > ,Eq. (1) has a monotone increasing kink profile solitary wave solution u(ξ) satisfying u(−∞) = 0,u(+∞) = and a monotone decreasing kink profile solitary wave solution satisfying u(−∞) = 0,u(+∞) = −. Here,u(ξ) corresponds to the orbits L(P₀,P_2±) in Fig. 6(a) or Fig. 6(b).

(iv) If |γ | < ,Eq. (1) has two damped oscillatory traveling wave solutions u(ξ) satisfying u(−∞) = 0,u(+∞) = and u(−∞) = 0,u(+∞) = −,respectively. Here,u(ξ) corresponds to the orbits L(P₀,P_2±) in Fig. 6(c) or Fig. 6(d).

4 Solitary wave solutions and approximate damped oscillatory solutions of Eq. (1) 4.1 Bell profile solitary solutions of Eq. (1)

In light of Ref. ^[9],we suppose that Eq. (1) has the bell profile solitary solutions in the form

Substitution of Eq. (10) into Eq. (7) yields the following theorem.

Theorem 5 If γ = 0 and a₁ < 0,Eq. (1) has two bell profile solitary wave solutions u₁^± (ξ) = ± ,where

Here,u₁^± (ξ) = ± correspond to two symmetrical homoclinic orbits L(P₀,P₀) in Fig. 1.

Remark 1 We can verify that Eq. (11) is equivalent to Eq. (45) in Ref. ^[11] and Eq. (11) in Ref. ^[12].

Obviously,the asymptotic values of solutions u₁^± (ξ) are zero. While the solutions,whose asymptotic value are not zero,are listed as follows.

We suppose that Eq. (1) has the bell profile solitary solutions in the form

Substitution of Eq. (12) into Eq. (7) yields the following theorem.

Theorem 6 If γ = 0,Eq. (1) has two bell profile solitary wave solutions u₂^± (ξ) = ± ,where

Here,u₂^± (ξ) = ± correspond to two symmetrical homoclinic orbits L(P_2±,P_2±) in Fig. 3.

4.2 Kink profile solitary solutions of Eq. (1)

To obtain the kink profile solitary wave solutions of Eq. (1) when γ = 0,we suppose that they are in the form

Substituting Eq. (14) into Eq. (7) and computing it by MAPLE,we have the following theorem.

Theorem 7 If γ = 0,Eq. (1) has two kink profile solitary wave solutions,

Here,u₃^± (ξ) correspond to the heteroclinic orbits L(,P_3±) in Fig. 2 orL(,P_2±) in Fig. 3.

To obtain the kink profile solitary wave solutions of Eq. (1) when ≠ 0,we suppose that they are in the form

Substituting (16) into (7) and computing it by MAPLE,we have the following theorems.

Theorem 8 If a₁ < 0 and γ² = ,Eq. (1) has two kink profile solitary wave solutions,

where Here,u₄^± (ξ) correspond to the heteroclinic orbits L(P₀,) in Fig. 4(a).

Theorem 9 If a₁ > 0 and γ² = ,Eq. (1) has two kink profile solitary wave solutions,

where Here,u₅^± (ξ) correspond to the heteroclinic orbits L(P₀,P_3±) in Fig. 5 or L(P₀,P_2±) in Fig. 6(a) or Fig. 6(b).

4.3 Approximate damped oscillatory solutions of Eq. (1)

In this part,we give the approximate expressions of damped oscillatory solutions for Eq. (1) according to their structures (see Fig. 8). We take the damped oscillatory solution corresponding to the focus-saddle orbit L(P₁₊,P₀) in Fig. 4(b) for example,while other cases can be studied similarly. By the rotation vector field theory in planar dynamical systems,it is easy to see that the focus-saddle orbit L(P₁₊,P₀) in Fig. 4(b) comes from the break of the left homoclinic orbit L(P₀,P₀) in Fig. 2 under the effect of dissipation term γu_t (the dissipation coefficient γ satisfies |γ | < ). Hence,the non-oscillatory part of the damped oscillatory solution corresponding to L(P₁₊,P₀) can be expressed by the bell profile solitary wave solution in Theorem 5 as follows:

We use the form

to approximatively express the oscillatory part of this damped oscillatory solution,where A₁,A₂,B,C,and α are undetermined constants. The reason why we choose Eq. (20) is that it has both damped and oscillatory properties. e^{α(ξ−ξ₀)} has damped property while A₁ cos(B(ξ − ξ₀)) − A₂ sin(B(ξ − ξ₀)) has oscillatory property.

Substituting Eq. (20) into Eq. (7),and neglecting the terms including O(e^{α(ξ−ξ₀)}),we have

In order to derive the approximate damped oscillatory solution of Eq. (1),there still requires some conditions to connect Eqs. (19) and (20). Since a traveling wave solution u(ξ) keeps the shape and speed unchanged when parallel shifting on the ξ-axis,we let ξ₀ = 0. From Fig. 4(b),we know that the best selection of the connective point is the maximum point of u(ξ),which means u′(ξ) = 0. Therefore,we take ξ = 0 as the connective point,and choose

as the connective conditions. Then,we have

Since Eq. (20) tends to as ξ → −∞,C = . Then,we have

From the above analysis,we can derive the following theorems.

Theorem 10 If a₁ < 0 and |γ | < ,Eq. (1) has a damped oscillatory solution corresponding to the focus-saddle orbit L(P₁₊,P₀) in Fig. 4(b),whose approximate expression is

where Ψ₁(ξ),B,A₁,and A₂ are given by Eqs. (11) and (24).

Meanwhile,Eq. (1) has a damped oscillatory solution corresponding to the focus-saddle orbit L(P_1-,P₀) in Fig. 4(b),whose approximate expression is

where Ψ₁(ξ) and B are given by Eqs. (11) and (24),A₁ = − + ,and A₂ = .

Remark 2 The value of A₁ cos(Bξ) − A₂ sin(Bξ) is independent of the value of B. That is to say,whether B > 0 or B < 0,its value never changes.

Theorem 11 If a₁ > 0,a₃ < 0,and |γ | < ,Eq. (1) has a damped oscillatory solution corresponding to the focus-saddle orbits L(P₁₊,P₂₊) in Fig. 6(b) or Fig. 6(d),whose approximate expression is

where Ψ₂(ξ) and B are given by Eqs. (13) and (24),A₁ = − ,and A₂ = .

Meanwhile,Eq. (1) has a damped oscillatory solution corresponding to the focus-saddle orbits L(P_1-,P_2-) in Fig. 6(b) or Fig. 6(d),whose approximate expression is

where Ψ₂(ξ) and B are given by Eqs. (13) and (24),A₁ = − + ,and A₂ = .

Similarly,we can derive the approximate damped oscillatory solutions,corresponding to the heteroclinic orbits L(P₀,P_3±) in Fig. 5(b) and L(P₀,P_2±) in Fig. 6(c) or Fig. 6(d),which break from the heteroclinic orbits L(,P_3±) in Fig. 2 and L(,P_2±) in Fig. 3,respectively,under the effect of dissipation term ut. The only difference is the selection of the connective point. If we still select ξ = 0 as the connective point in this case,oscillatory curve will distort seriously. Thus,we let the connective point ξ′ in the hollow neighborhood at the left side of the inflection point ξ = 0,whose radius is very small. The connective conditions will not change. Then,we can imitate the reduction of Theorem 10 to obtain the following theorem.

Theorem 12 If a₁ > 0,a₃ < 0,and |γ | < ,Eq. (1) has two damped oscillatory solutions corresponding to the focus-saddle orbits L(P₀±,P_3±) in Fig. 5(b) and L(P₀,P_2±) in Fig. 6(c) or Fig. 6(d). Their approximate expressions are

where u₃^± (ξ) and B are given by Eqs. (16) and (23),

5 Error estimates of approximate damped oscillatory solutions

In this section,we investigate error estimates of the approximate damped oscillatory so- lutions given in the previous sections. To this end,we need to find the relation between the exact and approximate damped oscillatory solutions since it is too hard to find out the exact expressions of damped oscillatory solutions. We still take the approximate damped oscillatory solution (25) corresponding to the focus-saddle orbit L(P₀,P₁₊) in Fig. 4(b) for example. Then,other cases can be discussed similarly.

Substitute V (ξ) = and ξ = −η (η > 0) into Eq. (7). Consequently,the problem of finding the exact damped oscillatory solution for Eq. (7),which satisfies u(0) = and u′(0) = 0,is converted into solving the following initial value problem:

h(η) satisfies where

By the principle of homogenization,we can solve the initial value problem (30) and obtain the implicit expression of damped oscillatory solution

where

Substitute η = −ξ and V (ξ) = into Eq. (31),and make the transformation t = −τ. Then,we have

where h₁(t) = ,c1 = − ,and c² = . Obviously,β1,c1,and c² are equal to B,A₁,and −A₂ in Theorem 10,respectively.

Since damped oscillatory solution u(ξ) is bounded,there exists M > 0 satisfying |u(ξ)| < M. Moreover,from Eq. (32),we have

where By using the Gronwall inequality,the above formula becomes where

Equation (34) is the amplitude estimate of damped oscillatory solution for Eq. (1). From Fq. (34),it is easy to see that u(ξ) tends to rapidly as ξ → −∞.

From Eqs. (32)-(34),we have

Equation (35) shows that the error estimate between the approximate solution (26) and its exact damped oscillatory solution is less than ε(ξ) = . Due to ε1(ξ) = O,ξ → −∞,Eq. (25) is meaningful to be an approximate solution of Eq. (1) when the conditions in Theorem 10 hold.

By using the similar method,we can get error estimates between other approximate damped oscillatory solutions obtained above and their exact solutions. Their errors are all infinitesimal decreasing in the exponential form.

6 Conclusions

In this paper,we analyze the shapes of the bounded traveling wave solutions for the nonlin- ear wave equation with a quintic term,and obtain the expressions according to their shapes,including exact bell and kink profile solitary solutions and approximate damped oscillatory wave solutions. Two critical values which can characterize the scale of this effect are given,where γ₁ = and γ₂ = . From these two critical values,we can clearly see the dissipation influences on the shape evolution of bounded traveling wave solutions: the bounded traveling wave solutions of Eq. (1) appear as kink profile solitary waves if dissipation is large,i.e.,the dissipation coefficient γ is larger than a certain critical value; while they ap- pear as damped oscillatory waves if dissipation is small,i.e.,the dissipation coefficient γ is less than a certain critical value. We hope that the above point will help people control the system determined by Eq. (1). The approximate damped oscillatory wave solution has been proved to be reasonable. We still gave its error estimate through establishing the integral equation which reflects the relation between the exact and approximate damped oscillatory solutions by the idea of homogenization principle. The error is infinitesimal decreasing in exponential form,which also shows the advantage of this paper.