INTRODUCTION
In this study, we consider the following Generalized Special DoddBulloughMikhailov
(GSDBM) equation
where α, β are two nonzero real number, m, n≥1 are positive integer.
Specially, when m = 1, n = 2 and α = β = –1, (1) is called the
DoddBulloughMikhailov equation (DBM), when m = 1, n = 1 and α = β
= –1/2, (1) is called the sinhGordon equation. This equation appears in
problems varying from fluid flow to quantum field theory. Recently, by using
the tanh method (Wazwaz, 2005) considered some solitary wave and periodic wave
solutions for the special DBM equation. In this study, we investigate dynamical
behavior of solutions of Eq. (1). To answer this question,
we shall consider the bifurcations of travelling wave solutions of (1) in the
fiveparameter space (α, β, m, n, c).
Making the transformations u(x, t) = In v(x, t), (1) becomes
Let v(x –ct) = φ(x – ct) = φ(ξ). Substituting φ(x – ct) into (2), we obtain
where “/” is the derivative with respect to ξ. Clearly, (3) is equivalent to the following twodimensional system:
System (4) has the first integrals
System (4) is planar dynamical systems defined in the 5parameter space (α, β, m, n, c).
Because the phase orbits defined by the vector fields of (4) determine all travelling wave solutions, we will investigate bifurcations of phase portraits of this system as these parameters are varied.
Usually, a solitary wave solution of a nonlinear wave equation corresponds to a homoclinic orbit of its travelling wave equation; a kink (or antikink) wave solution corresponds to a heteroclinic orbit (or connecting orbit). Similarly, a periodic orbit of a travelling wave equation corresponds to a periodic travelling wave solution of the nonlinear wave equation. To find all possible bifurcations of solitary waves, periodic waves, kink and antikink wave of a nonlinear wave equation, we need to investigate the existence of all homoclinic, heteroclinic orbits and periodic orbits for its travelling wave equation in the parameter space. In doing so, the bifurcation theory of dynamical systems (Chow and Hale, 1981) is very important and useful.
We notice that by using transformation u(x, t) = In φ(x, t), we make (1)
and become the traveling Eq. (4). Therefore, we are only interesting
the positive boundary solutions of φ(ξ). In addition, if a solution
φ(ξ) of (4) can approach to φ(ξ) = 0, then In(φ(ξ))
approach to – ∞. In other words, this solution determines an unbounded
travelling wave solution of (1).
It is easy to see that the righthand side of the second equation in (4) is generally not continuous when φ(ξ) = 0. In other words, on such straight lines in the phase plane (φ, y), the function φ″_{ξ} is not welldefined. It implies that the smooth system (1) sometimes have nonsmooth travelling wave solutions. This phenomenon has been considered before by (Li and Chen, 2005; Li and Zhenrong, 2000, 2002) in which the authors had already pointed out that the existence of such a singular straight line for a travelling wave equation is the very reason why travelling waves can lose their smoothness.
BIFURCATIONS OF PHASE PORTRAITS OF SYSTEMS (4)
System (4) have the same phase orbits for the cases n = 1, or n≥2 as the following systems, respectively
and
except for the straight line φ = 0, where dξ = (c^{2} – 1)φdτ and dξ = (c^{2} – 1)φ^{n–1}dτ. By using (5) for φ ≠ 0, we define
Without loss of generality, we can assume that c^{2} – 1 > 0. We see from (6) and (7) that for the equilibrium points of these two systems, the following conclusions hold.
• 
When n = 1, m = 2k, k ε Z^{+}, system (6) has
two equilibrium points at A_{+}(φ_{+}, 0) and S(0,
0), where 
• 
When n = 1, m = 2k – 1, k ε Z^{+}, if αβ
< 0, (6) has three equilibrium points at A_{±}(φ_{±},
0) and S(0, 0), where 
if αβ > 0, (6) has one equilibrium point at S(0, 0).
• 
When n = 2, m = 2k – 1, k ε Z^{+}, if (c^{2}
– 1)β < 0, (6) has three equilibrium points at A_{+}(φ_{+},
0) and S_{±}(0, Y_{±}), where 
(7) has one equilibrium point at A_{+}(φ_{+}, 0).
• 
When n = 2, m = 2k, k ε Z^{+}, if αβ
< 0, (c^{2} – 1) β < 0, (7) has four equilibrium
points at A_{±}(φ_{±}, 0) and S_{±}(0,
Y_{±}), where 
if αβ > 0, (c^{2} – 1)β < 0, (6) has two equilibrium points at S_{±}(0, Y_{±}); if αβ > 0, (c^{2} – 1) β > 0, (7) has no equilibrium point.
• 
When n ≥ 3, m + n = 2k + 1 ≥ 5, k ε Z^{+},
(7) has one equilibrium point at A_{+}(φ_{+}, 0), where 
• 
When n ≥ 3, m + n = 2k ≥ 4, k ε Z^{+},
if αβ < 0, (7) has two equilibrium points at A_{±}(φ_{±},
0), where 
if αβ > 0, (7) has no equilibrium point.
Let M_{1}(φ_{i}, y_{i}) and M_{2}(φ_{i}, y_{i}) be the coefficient matrix of the linearized system of (6) and (7) at an equilibrium point (φ_{i}, y_{i}), respectively. Then we have Trace (M_{1}(φ_{±}, 0)) = 0 and J_{1}(M_{1}(φ_{±}, 0)) = det M_{1}(φ_{±}, 0)) = (c^{2} – 1) (m + 1) βφ_{+}, J_{1}(M_{1}(0, 0)) = 0. For n = 2, we have Trace M_{2}(φ_{±}, 0) = 0, Trace (M_{2}(0, ±Y_{+})) = 0 and J_{2}(M_{2}(φ_{±}, 0)) = det (M_{2}(φ_{±}, 0)) = (c^{2} – 1) (m + 2) β, J_{2}(M_{2}(0, ±Y_{+})) = 2(c^{2} – 1)^{2} Y^{2}_{+}. For n ≥ 3, we have, Trace (M_{2}(φ_{±}, 0)) and J_{2}(M_{2}(φ_{±}, 0)) = det (M_{2}(φ_{±}, 0)) = β(c^{2} – 1)^{2} (m + 2) (±φ_{+})^{n–2}.
By the theory of planar dynamical systems, we know that for an equilibrium
point (φ_{i}, y_{i}), of a planar integrable system, if
J < 0 then the equilibrium point is a saddle point; if J > 0 and Trace
(M(φ_{i}, y_{i}) = 0 then it is a center point; if J >
0 and (Trace (M(φ_{i}, y_{i})))^{2} – 4J(M(φ_{i},
y_{i})) > 0 then it is a node; if J = 0 and the index of the equilibrium
point is zero then it is a cusp; if J = 0 and the index of the equilibrium point
is not zero then it is a high order equilibrium point.
For H(φ, y) defined by (8), we have
For a fixed h the level curve H(φ, y) = h defined by (8) determines a set of invariant curves of (6) and (7), except for the straight line φ = 0, which contains different branches of curves. As h is varied, it defines different families of orbits of (6) and (7) with different dynamical behaviors.
From the above analysis we obtain the different phase portraits of (4) shown in Fig. 1 (k, l ≥ 1).
EXACT EXPLICIT TRAVELLING WAVE SOLUTIONS OF (1) FOR m = 1, n = 1 OR m = 1, n = 2 OR m = 2, n = 1 OR m = 2, n = 2
For m = n = 1, (1) becomes
When α = β = –1/2, (9) was called the sinhGordon equation by Wazwaz (2005). In this case, (4) have the following forms:
with the first integrals
For m = 1, n = 2, (1) becomes
When α = β = –1, (12) was called the special (DBM) equation by Wazwaz (2005). In this case, (4) have the following forms:
with the first integrals
For m = 2, n = 1, (1) becomes
In this case, (4) have the following forms:
with the first integrals
For m 2, n = 2, (1) becomes
In this case, (4) have the following forms:
with the first integrals
By using (11) and (14) and (17) and (20) and the first equations of (10) and (13) and (16) and (19) to do integrations, we can obtain some exact explicit parametric representations for the breaking wave solutions and periodic wave solutions of (9) and (12) and (15) and (18). Because the singular straight line φ = 0 intersects at two node points with other orbits of (11) and (14) and (17) and (20), so that, corresponding to these orbits, the travelling wave solutions of (9) and (12) and (15) and (18) is breaking waves.
Unbounded wave solutions of (9) and (12) and (15) and (18): For system (10), when αβ < 0, β(c^{2} – 1) < 0 or αβ < 0, β(c^{2} – 1) > 0 (Fig. 1 (1), (2)), we have
Corresponding to H_{1}(±φ, 0) = h_{1} defined by
(11), system (10) has two orbits connecting the saddle A_{+}(φ_{+},
0). Two orbits have the same algebraic equation for αβ < 0, β(c^{2}
– 1) < 0
and for αβ < 0, β(c^{2} – 1) > 0

Fig. 1: 
The phase portraits of (1) n = 1, m = 2k
– 1, αβ < 0, β(c^{2} – 1) > 0, (2)
n = 1, m = 2k + 1, αβ < 0, β(c^{2} – 1) <
0, (3) n = 1, m = 2k, αβ < 0, β(c^{2} – 1)
> 0, (4) n = 1, m = 2k, αβ < 0, β(c^{2} –
1) < 0, (5) n = 2, m = 2k, αβ > 0, β(c^{2} – 1) > 0, (6) n = 2, m = 2k, αβ > 0, β(c^{2} – 1) < 0, (7) n = 2, m = 2k  1, αβ < 0, β(c^{2} – 1) < 0, (8) n = 2, m = 2k – 1, αβ > 0, β(c^{2} – 1) < 0, (9) n = 2, m = 2k, αβ < 0, β(c^{2} – 1) < 0, (10) n = 2, m = 2k, αβ > 0, β(c^{2} – 1) < 0, (11) n = 2l + 1, m = 2k + 1, αβ <
0, β(c^{2} – 1) > 0, (12) n = 2l + 1, m = 2k
+ 1, αβ < 0, β(c^{2} – 1) < 0, (13) n
= 2l – 1, m = 2k, αβ < 0, β(c^{2} – 1) > 0, (14) n = 2l – 1, m = 2k, αβ <
0, β(c^{2} – 1) < 0, (15) n = 2l – 1, m
= 2k, αβ > 0, β(c^{2} – 1) > 0, (16) n
= 2l – 1, m = 2k, αβ > 0, β(c^{2} – 1) > 0, (17) n = 2l, m = 2k – 1, αβ <
0, β(c^{2} – 1) > 0, (18) n = 2(l + 1), m = 2k –
1, αβ < 0, β (c^{2} – 1) < 0, (19) n =
2l, m = 2k – 1, αβ > 0, β(c^{2} –
1) > 0, (20) n = 2(l + 1), m = 2k, αβ > 0, β(c^{2} – 1) < 0, (21) n = 2l, m = 2k, αβ < 0, β(c^{2} – 1) > 0, (22) n = 2(l + 1), m = 2k, αβ < 0, β(c^{2} – 1) < 0, (23) n = 2l, m = 2k, αβ > 0, β(c^{2} – 1) < 0, (24) n = 2l ≥ 2, m = 2k, αβ > 0,
β (c^{2} – 1) > 0 
Thus, from (21), we obtain the parametric representations of the arch orbit, for ξε(0, ∞) as follows:
where
clearly, we have that φ(0) = 0, φ(±∞) = φ_{+}.
It follows that Eq. (9) has one unbounded wave solution with
the parametric representations
From (22), we obtain the parametric representations of the arch orbit for ξε(0, T) as follows:
where
clearly, we have that φ(0) = 0, φ(T) = ∞.
It follows that Eq. (9) has one unbounded wave solution with
the parametric representations
For system (13), when α <, β > 0, c^{2} – 1 < 0 or α < 0, β < 0, c^{2} – 1 > 0 (Fig. 1 (7), (8)), we have
Corresponding to H_{1}(φ_{+}, 0) defined by (14), system (13) has two orbits connecting the node points S_{±} and the saddle A_{+}(φ_{+}, 0) and also has an arch orbit connecting two node S_{±} in the left (or right) side of the straight line φ = 0. Three orbits have the same algebraic equation for α < 0, β > 0, c^{2} – 1 < 0
or α < 0, β < 0, c^{2} – 1 > 0
Thus, from (27), we obtain the parametric representations of the arch orbit of (13), respectively, for ξε(–T_{1}, T_{1}), ξε(T_{1}, ∞) and ξε(–∞, –T_{1}) as follows:
where
Clearly, we have that φ(T_{1}) = 0, φ(±∞) = φ_{+}.
It follows that Eq. (12) has two unbounded wave solutions
with the parametric representations
for x – ctε(–T_{1}, T_{1}), x – ctε(T_{1}, ∞) and x – ctε(–∞, –T_{1}).
From (18), we obtain the parametric representations of the arch orbit of (13), for ξε(–T_{2}, T_{2}), ξε(T_{2}, ∞) and ξε(–∞, –T_{2}) as follows:
where
Clearly, we have that φ(T_{2}) = 0, φ(±∞) = φ_{+}.
It follows that Eq. (12). has two unbounded wave solutions
with the parametric representations for x – ctε(–T_{2},
T_{2}), x – ctε(T_{2}, ∞) and x – ctε(–∞,
–T_{2}).
Especially, when α = β = –1, the special DBM equation has the unbounded wave solution for x – ctε(–T_{2}, T_{2})
where
For system (16), when α < 0, β > 0, c^{2} – 1 < 0 (Fig. 1 (4)), we have
Corresponding to H_{1}(φ_{+}, 0) = h_{1} defined by (17), system (16) has two orbits connecting the saddle A_{+}(φ_{+}, 0). Two orbits have the same algebraic equation for α <, β > 0, c^{2} – 1 < 0
Thus, from (34), we obtain the parametric representations of the arch orbit for x – ctε(–T_{3}, T_{3}), x – ctε(T_{3}, ∞) and x – ctε(–∞, –T_{3}) as follows:
where
Clearly, we have that φ(T_{3}) = ∞, φ(±∞) = 0.
It follows that Eq. (15) has two unbounded wave solutions
with the parametric representations for x – ctε(T_{3}, ∞)
and x – ctε(–∞, –T_{3}) as follows:
For system (19), when αβ < 0, β(c^{2} – 1) < 0 (Fig. 1 (9)), we have
Corresponding to H_{1}(φ_{+}, 0) = h_{1} defined by (20), system (19) has two orbits connecting the node points S_{±} and the saddle A_{±}(φ_{±}, 0), respectively. Four orbits have the same algebraic equation for αβ < 0, β(c^{2} – 1) < 0
Thus, from (37), we obtain the parametric representations of the arch orbit for ξε(0, ∞) and ξε(–∞, 0) as follows:
where
Clearly, we have that φ(0) = 0, φ(±∞) = ±φ_{+}.
It follows that Eq. (18) has two unbounded wave solutions
with the parametric representations for x – ctε(0, ∞) and x
– ctε(–∞, 0)
Uncountable infinite many exact explicit unbounded wave solutions: For system (13), when α < 0, β > 0, c^{2} – 1 < 0 or α < 0, β > 0, c^{2} – 1 > 0 (Fig. 1 (7), (8)), we have
Corresponding to H_{1}(φ_{+}, 0) = h_{1}, hε(–∞, 0) defined by (14), system (13) has two families of arch orbits connecting two node points S_{±} which lie in the left (or right) side of the straight line φ = 0, respectively. These orbits have the algebraic equation for α < 0, β > 0, c^{2} – 1 < 0
and for α < 0, β > 0, c^{2} – 1 < 0
By using (40), we obtain the parametric representations for the right arch orbits of (13), for α < 0, β > 0, c^{2} – 1 < 0 as follows:
where sn (x, k) is the Jacobin elliptic functions with the modulo k and
By using (41), we obtain the parametric representations for the right arch orbits of (13) for α < 0, β < 0, c^{2} – 1 > 0
where
Thus, (42) and (43) give rise to the following uncountable infinite many exact explicit unbounded wave solutions of (12) for α < 0, β > 0, c^{2} – 1 > 0
and for α < 0, β < 0, c^{2} – 1 > 0
For system (19), when αβ < 0, β(c^{2} – 1) < 0 (Fig. 1 (9)), we have
Corresponding to H_{1}(φ, 0) = h, hε(–4, 0) defined by (20), system (19) has two families of arch orbits connecting two node points S_{±} which lie in the left (or right) side of the straight line φ = 0, respectively. These orbits have the algebraic equation for
By using (46), we obtain the parametric representations for the right arch orbits of (19) for αβ < 0, β(c^{2} – 1) < 0
where
Thus, (47) give rise to the following uncountable infinite many exact explicit unbounded wave solutions of (18) for αβ < 0, β(c^{2} – 1) < 0
Uncountable infinite many exact explicit periodic wave solutions: For n = 1, m = 1, αβ < 0, β(c^{2} – 1) > 0 corresponding to H_{1}(φ, y) = h, hε(h_{1}, ∞),
defined by (11), system (10) have a family of periodic solutions enclosing the center (φ_{+}, 0) which lie in the right side of the straight line φ = 0, these orbits determine uncountable infinite many periodic wave solutions of (9) (Fig. 1 (1)).
Thus, from (49), we obtain the parametric representations of the arch orbit for ξε(0, P_{4}) as follows:
where
Clearly, we have that φ(0) = φ_{M}, φ(P_{4}) = 0.
Thus, Eq. (12) has uncountable infinite many periodic wave
solutions,
For n = 2, m = 1, αβ < 0, β(c^{2} – 1) > 0 corresponding to H(φ, y) = h, hε(h_{1}, ∞),
defined by (14), system (13) have a family of periodic solutions enclosing the center (φ_{+}, 0) which lie in the right side of the straight line φ = 0, these orbits determine uncountable infinite many periodic wave solutions of (12) (Fig. 1 (17)).
Thus, from (52), we obtain the parametric representations of the arch orbit for ξε(–P_{5}, P_{5}) as follows:
where
Clearly, we have that φ(P_{5}) = 0
Thus, Eq. (12) has uncountable infinite many periodic wave
solutions,
For n = 1, m = 2, αβ < 0, β(c^{2} – 1) > 0 corresponding to H(φ, y) = h, hε(h_{1}, ∞),
defined by (17), system (16) have a family of periodic solutions enclosing the center (φ_{+}, 0) which lie in the right side of the straight line φ = 0, these orbits determine uncountable infinite many periodic wave solutions of (15) (Fig. 1 (3)).
Thus, we obtain the parametric representations of the arch orbit for ξε(–P_{6}, P_{6}) as follows:
where
Clearly, we have that φ(P_{6}) = ∞
Thus, Eq. (15) have uncountable infinite many periodic wave
solutions,
EXISTENCE OF UNBOUNDED WAVE SOLUTIONS AND PERIODIC WAVE SOLUTIONS OF (1)
Here, by using the phase portraits we show the existence of unbounded wave
and periodic wave solutions of (1) for any integer m = 2k +1, m = 2k. We have
mentioned that we are only interesting the positive solutions of φ(ξ),
because of u(x, t) = In φ(x – ct).
We see from Fig. 1(23) or Fig. 1(16) that corresponding to a branch of the curves H(φ, y) = h, or hε(–∞, ∞_{1}) or given by (8), in the right side of the (φ, y)phase plane, there exist uncountable infinity many bounded solutions of φ(ξ) (but, φ΄(ξ) are unbounded). These solutions approach to φ = 0 as ξ → ±∞. These φ(ξ) are breaking solutions of (4) near. φ = 0 Similarly, some solution families in Fig. 1 (14), (18), (20), (22) or (12) have the same dynamical behavior. We use Fig. 2 (21)(27) to show these wave profiles (k, l ε Z^{+}).
From the above discussion, we have the following conclusions.
Theorem 1
(i) 
Suppose that m = 2k, n = 2l + 1, αβ >
0, β (c^{2} – 1) > 0, hε(h_{1}, ∞).
Then, Eq. (1) has a family of uncountable infinity many
periodic wave solutions which correspond to a branch of the curves H(φ,
y) = h given by (8) in the right side of the (φ, y)phase plane (Fig.
1 (13)). 
(ii) 
Suppose that m = 2k – 1, n = 2l, αβ < 0, β
(c^{2} – 1) > 0, hε(h_{1}, ∞). Then,
Eq. (1) has a family of uncountable infinity many periodic
wave solutions which corresponds to a branch of the curves H(φ, y)
= h given by (8) in the right side of the (φ, y)phase plane (Fig.
1 (17) ). 
(iii) 
Suppose that Then, Eq. (1) has a family of uncountable
infinity many periodic wave solutions which corresponds to a branch of the
curves H(φ, y) = h given by (8) in the right side of the (φ, y)plane
(Fig. 1 (21)). 
Theorem 2
(i) 
Suppose that m = 2k – 1, n = 2l – 1, αβ
< 0, β(c^{2} – 1) < 0, hε(–∞, h_{1}).
Then, Eq. (1) has a family of uncountable infinity many
unbounded wave solutions which corresponds to a branch of the curves H(φ,
y) = h given by (8) in the right side of the (φ, y)phase plane (Fig.
1 (12)). 
(ii) 
Suppose that m = 2k, n = 2l + 1, αβ < 0, β(c^{2}
– 1) < 0, hε(–∞, h_{1}). Then, Eq.
(1) has a family of uncountable infinity many unbounded wave solutions
which corresponds to a branch of the curves H(φ, y) = h given by (8)
in the right side of the phase (φ, y)plane (Fig. 1
(14)). 
(iii) 
Suppose that m = 2k – 1, n = 2l, αβ
< 0, β(c^{2} – 1) < 0, hε(–∞, h_{1}).
Then, Eq. (1) has a family of uncountable infinity many
unbounded wave solutions which corresponds to a branch of the curves H(φ,
y) = h given by (8) in the right side of the (φ, y)phase plane (Fig.
1 (18)). 
(iv) 
Suppose that m = 2k, n = 2l, αβ < 0, β (c^{2}
– 1) > 0, hε(–∞, h_{1}). Then, Eq.
(1) has a family of uncountable infinity many unbounded wave solutions
which corresponds to a branch of the curves H(φ, y) = h given by (8)
in the right side of the (φ, y)phase plane (Fig. 1
(22)). 
(v) 
Suppose that m = 2k – 1, n = 2l + 1, αβ > 0,
β(c^{2} – 1) < 0, hε(–∞, +∞).
Then, Eq. (1) has a family of uncountable infinity many
unbounded wave solutions which corresponds to a branch of the curves H(φ,
y) = h given by (8) in the right side of the (φ, y)plane (Fig.
1 (16)). 
(vi) 
Suppose that m = 2k – 1, n = 2l, αβ > 0, β(c^{2}
– 1) < 0, hε(–∞, +∞) Then, Eq.
(1) has a family of uncountable infinity many unbounded wave solutions
which corresponds to a branch of the curves H(φ, y) = h given by (8)
in the right side of the (φ, y)phase plane (Fig. 1
(20) ). 
(vii) 
Suppose that m = 2k, n = 2l, αβ > 0, β(c^{2}
– 1) < 0, hε(–∞, +∞). Then, Eq.
(1) has a family of uncountable infinity many unbounded wave solutions
whichcorresponds to a branch of the curves H(φ, y) = h given by (8)
in the right side of the (φ, y)phase plane (Fig. 1
(23)). 

Fig. 2: 
The wave profiles of bounded solutions of
(21) m = 2k – 1, n = 2l + 1, αβ < 0, β(c^{2} – 1) < 0, (22) m = 2k, n = 2l + 1, αβ < 0,
β(c^{2} – 1) < 0, (23) m = 2k, n = 2l + 1,
αβ > 0, β(c^{2} – 1) < 0, (24) m = 2k
– 1, n = 2l + 1, αβ < 0, β(c^{2} –
1) < 0, (25) m = 2k – 1, n = 2l, αβ > 0, β
(c^{2} – 1) < 0, (26) m = 2k, n = 2(l + 1), αβ
< 0, β(c^{2} – 1) < 0, (27) m = 2k, n = 2(l + 1),
αβ > 0, β(c^{2} – 1) > 0 
ACKNOWLEDGMENT
This research was supported by Science Foundation of Guangxi Province, China (0575092).