Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations where α, β, δ1, δ2, δ3, and cs are real parameters, E = E(x, t) is a complex function, and u = u(x, t) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.
1. Introduction
Since the exact solutions to nonlinear wave equations help to understand the characteristics of nonlinear equations, seeking exact solutions of nonlinear equations is an important subject. For this purpose, there have been many methods such as the Jacobi elliptic function method [1, 2], F-expansion method [3, 4], and (G′/G)-expansion method [5, 6].
Recently, the bifurcation method of dynamical systems [7–9] has been introduced to study the nonlinear partial differential equations. Up to now, the method is widely used in literatures such as [10–16].
In this paper, we consider the generalized Zakharov equations [17], which read as
()
where α, β, δ1, δ2, δ3, and cs are real parameters. E = E(x, t) is a complex function which represents the envelop of the electric field, and u = u(x, t) is a real function which represents the plasma density measured from its equilibrium value. Huang and Zhang [17] used Fan′s direct algebraic method to obtain some exact travelling wave solutions of (1) as follows:
()
where γandw are two constants and
()
()
()
()
()
()
()
()
()
()
When α = 1, β = −1, δ1 = −2, δ2 = 2λ, δ3 = 0, andcs = 1, (1) reduce to the equations
()
El-Wakil et al. [18] used the extended Jacobi elliptic function expansion method to obtain some Jacobi elliptic function expression solutions of (13).
By multisymplectic numerical method, Wang [19] proved the preservation of discrete normal conservation law of (14) theoretically and investigated the propagation and collision behaviors of the solitary waves numerically. There are also many other researchers studying (1) or its special case; for more information, one can see [20–24].
In this paper, we investigate the nonlinear waves and the bifurcation phenomena of (1). Coincidentally, under some transformations, (1) reduce to a planar system (54) which is similar to the planar system obtained by Feng and Li [16]. Many exact explicit parametric representations of solitary waves, kink and antikink waves, and periodic waves were obtained in [16], and their work is very important for the ϕ6 model. In order to find the travelling wave solutions of (1), here we consider (1) by using the bifurcation method mentioned above; firstly, we obtain three types of explicit nonlinear wave solutions, this is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. Secondly, we point out that these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves under different parameter conditions. Thirdly, we reveal five kinds of interesting bifurcation phenomena mentioned in the abstract above.
The remainder of this paper is organized as follows. In Section 2, we give some notations and state our main results. In Section 3, we give derivations for our results. A brief conclusion is given in Section 4.
2. Main Results
In this section, we state our main results. To relate conveniently, let us give some notations which will be used in the latter statement and the derivations.
Let li(i = 1,2, …, 7) represent the following seven curves:
()
()
()
()
()
()
()
Let Ai(i = 1,2, …, 12) represent the regions surrounded by the curves li(i = 1,2, …, 7) and the coordinate axes (see Figure 1).
The locations of the regions Ai(i = 1,2, …, 12) and curves li(i = 1,2, …, 7).
Let
()
()
()
()
()
()
where H(φ, y) is the first integral which will be given later and h is the integral constant.
In order to search for the solutions of (1) and studing the bifurcation phenomena, we only need to get the solution φ(ξ) according to (22) and (23). For convenience, throughout the following work we only discuss the solution φ(ξ). Now let us state our main results in the following Propositions 1, 2, and 3.
2.1. When the Orbit Γ Is Defined by H(φ, y) = H(0,0)
Proposition 1.
(i) For p = 0, (1) have two fractional nonlinear wave solutions
()
where
()
If (q, p) ∈ l4, then are 1-blow-up waves (refer to Figure 2(d)). If (q, p) ∈ l3, then are 2-blow-up waves (refer to Figure 3(d)). If (q, p) ∈ l5, then are symmetric solitary waves (refer to Figure 4(d)).
(ii) For p < 0, (1) have two nonlinear wave solutions
()
where
()
These solutions have the following properties and wave shapes.
(1) When (q, p) ∈ A1 or A2, then and are periodic-blow-up waves (refer to Figure 2(a) or Figure 3(a)). Specially, in region A1 when p → 0 − 0, the periodic-blow-up waves become 1-blow-up waves , and for the varying process, see Figure 2, while the periodic-blow-up waves become a trivial wave φ = 0. In region A2 when p → 0 − 0, the periodic-blow-up waves become 2-blow-up waves , and for the varying process, see Figure 3.
(2) When (q, p) ∈ A12, then and are symmetric periodic waves (refer to Figure 5(a)). If p → 0 − 0, the symmetric periodic waves become symmetric solitary waves , and for the varying process, see Figure 4. If p → 3q2/16r + 0, the symmetric periodic waves and become two trivial waves , and for the varying process, see Figure 5.
(iii) For p > 0, (1) have four nonlinear wave solutions
()
where
()
λ is a nonzero arbitrary real constant and λ0 = 3q2 − 16pr. Corresponding to λ > 0 or λ < 0, these solutions have the following properties and wave shapes.
(1) For the case of λ > 0, there are four properties as follows.
which represent four low-kink waves (refer to Figure 6 or Figure 7). Specially, let λ = −q > 0, thenandbecomeand.
(1) bIf (q, p) belongs to one ofA3, A7, A8, and, then , and they represent four symmetric solitary waves (refer to Figure 6). Specially, when (q, p) ∈ A3andp → 3q2/16r − 0, then the four symmetric solitary wavesandbecome the four low-kink wavesand , and for the varying process, see Figure 6.
(1) cIf (q, p) belongs to one ofA4, A5, A6,l1, and, then , and they represent four 1-blow-up waves (refer to Figure 7). Specially, when (q, p) ∈ A4andp → 3q2/16r + 0, then the four 1-blow-up wavesandbecome the four low-kink wavesand , and for the varying process, see Figure 7.
(1) dIf (q, p) belongs to one ofA3, A7, A8, and, then. Specially, when (q, p) ∈ A3andp → 3q2/16r − 0, thentend to two trivial solutions.
(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A1 and p → 0 − 0, where r = 1, q = 4 and (a) p = 0 − 10−1, (b) p = 0 − 10−2, (c) p = 0 − 10−3, and (d) p = 0 − 10−4.
(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A1 and p → 0 − 0, where r = 1, q = 4 and (a) p = 0 − 10−1, (b) p = 0 − 10−2, (c) p = 0 − 10−3, and (d) p = 0 − 10−4.
(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A1 and p → 0 − 0, where r = 1, q = 4 and (a) p = 0 − 10−1, (b) p = 0 − 10−2, (c) p = 0 − 10−3, and (d) p = 0 − 10−4.
(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A1 and p → 0 − 0, where r = 1, q = 4 and (a) p = 0 − 10−1, (b) p = 0 − 10−2, (c) p = 0 − 10−3, and (d) p = 0 − 10−4.
(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A2 and p → 0 − 0, where r = 1, q = −4 and (a) p = 0 − 1, (b) p = 0 − 0.5, (c) p = 0 − 0.1, and (d) p = 0 − 0.01.
(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A2 and p → 0 − 0, where r = 1, q = −4 and (a) p = 0 − 1, (b) p = 0 − 0.5, (c) p = 0 − 0.1, and (d) p = 0 − 0.01.
(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A2 and p → 0 − 0, where r = 1, q = −4 and (a) p = 0 − 1, (b) p = 0 − 0.5, (c) p = 0 − 0.1, and (d) p = 0 − 0.01.
(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of when (q, p) ∈ A2 and p → 0 − 0, where r = 1, q = −4 and (a) p = 0 − 1, (b) p = 0 − 0.5, (c) p = 0 − 0.1, and (d) p = 0 − 0.01.
(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of when (q, p) ∈ A12 and p → 0 − 0, where r = −1, q = 4 and (a) p = 0 − 0.5, (b) p = 0 − 0.1, (c) p = 0 − 0.01, and (d) p = 0 − 0.001.
(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of when (q, p) ∈ A12 and p → 0 − 0, where r = −1, q = 4 and (a) p = 0 − 0.5, (b) p = 0 − 0.1, (c) p = 0 − 0.01, and (d) p = 0 − 0.001.
(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of when (q, p) ∈ A12 and p → 0 − 0, where r = −1, q = 4 and (a) p = 0 − 0.5, (b) p = 0 − 0.1, (c) p = 0 − 0.01, and (d) p = 0 − 0.001.
(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of when (q, p) ∈ A12 and p → 0 − 0, where r = −1, q = 4 and (a) p = 0 − 0.5, (b) p = 0 − 0.1, (c) p = 0 − 0.01, and (d) p = 0 − 0.001.
(The symmetric periodic waves become two trivial waves.) The varying process for the figures of and when (q, p) ∈ A12 and p → 3q2/16r + 0, where r = −1, q = 4, and l6 : p = 3q2/16r = −3 and (a) p = −3 + 10−1, (b) p = −3 + 10−2, (c) p = −3 + 10−3, and (d) p = −3 + 10−6.
(The symmetric periodic waves become two trivial waves.) The varying process for the figures of and when (q, p) ∈ A12 and p → 3q2/16r + 0, where r = −1, q = 4, and l6 : p = 3q2/16r = −3 and (a) p = −3 + 10−1, (b) p = −3 + 10−2, (c) p = −3 + 10−3, and (d) p = −3 + 10−6.
(The symmetric periodic waves become two trivial waves.) The varying process for the figures of and when (q, p) ∈ A12 and p → 3q2/16r + 0, where r = −1, q = 4, and l6 : p = 3q2/16r = −3 and (a) p = −3 + 10−1, (b) p = −3 + 10−2, (c) p = −3 + 10−3, and (d) p = −3 + 10−6.
(The symmetric periodic waves become two trivial waves.) The varying process for the figures of and when (q, p) ∈ A12 and p → 3q2/16r + 0, where r = −1, q = 4, and l6 : p = 3q2/16r = −3 and (a) p = −3 + 10−1, (b) p = −3 + 10−2, (c) p = −3 + 10−3, and (d) p = −3 + 10−6.
(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of and when and p → 3q2/16r − 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−7, and (d) p = 3 − 10−9.
(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of and when and p → 3q2/16r − 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−7, and (d) p = 3 − 10−9.
(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of and when and p → 3q2/16r − 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−7, and (d) p = 3 − 10−9.
(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of and when and p → 3q2/16r − 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−7, and (d) p = 3 − 10−9.
(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r + 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r + 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r + 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r + 0, where r = 1, q = −4, λ = 4, and l2 : p = 3q2/16r = 3 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(2) For the case ofλ < 0, there are three properties as follows.
(2) aIf (q, p) ∈ l2, that is,r > 0, q < 0, andr = 3q2/16p, thenandbecomeand which represent four 1-blow-up waves (refer to Figure 8). Specially, let λ = q < 0, thenandbecome the hyperbolic 1-blow-up wave solutionsand
()
(2) bIf (q, p) ∈ A3and, then , and they represent four 2-blow-up waves (refer to Figure 8). Specially, when p → 3q2/16r − 0, then the four 2-blow-up waves become four 1-blow-up wavesand , and for the varying process, see Figure 8.
(2) cIf (q, p) ∈ A3and, thenof forms
()
which represent hyperbolic blow-up waves. Whenp → 3q2/16r − 0, thentend to two trivial solutions.
(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r − 0, where r = 1, q = −4, λ = −4, andl2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−6, and (d) p = 3 − 10−9.
(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r − 0, where r = 1, q = −4, λ = −4, andl2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−6, and (d) p = 3 − 10−9.
(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r − 0, where r = 1, q = −4, λ = −4, andl2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−6, and (d) p = 3 − 10−9.
(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of and when , and p → 3q2/16r − 0, where r = 1, q = −4, λ = −4, andl2 : p = 3q2/16r = 3 and (a) p = 3 − 10−1, (b) p = 3 − 10−3, (c) p = 3 − 10−6, and (d) p = 3 − 10−9.
2.2. When the Orbit Γ Is Defined by H(φ, y) = H(φ1, 0)
Proposition 2. If (q, p) belongs to one of the regions A1, A2, A3, A4, A11, andA12 or curves l1, l2, l3, l6, and l7, then (1) have four nonlinear wave solutions
()
where
()
μ is a nonzero arbitrary real constant, Δ is given in (24), and
()
Let
()
About μ0, one has the following fact:
()
Corresponding to μ > 0 and μ < 0, these solutions have the following properties and wave shapes.
(1o) For the case of μ > 0, there are six properties as follows.
(1o) a If (q, p) ∈ l2, that is, r > 0, q < 0, and r = 3q2/16p, then and become
()
()
which represent four low-kink waves (refer to Figure 9(d)). Specially, let μ = −(16p/q) > 0, then and .
(1o) b If (q, p) ∈ A1, A2, A3, and μ≠|μ0|, then , and they represent four tall-kink waves (refer to Figure 9(a)). Specially, when (q, p) ∈ A3 and p → 3q2/16r − 0, then and become and , and for the varying process, see Figure 9.
(1o) c If (q, p) ∈ A4, A11, A12, l6, and μ≠|μ0|, then . When (q, p) ∈ A4, they represent four antisymmetric solitary waves (refer to Figure 10(a)). When (q, p) ∈ A11, A12, andl6, they represent four symmetric solitary waves (refer to Figure 11(a)). Specially, when (q, p) ∈ A4, if p → 3q2/16r + 0, then and become and , and for the varying process, see Figure 10. When (q, p) ∈ A11 and p → q2/4r + 0, then and tend to two trivial solutions , and for the varying process, see Figure 11.
(1o) d If (q, p) ∈ A3 and μ = |μ0|, then of forms
()
which represent two tall-kink waves and tend to a trivial wave φ = 0 when p → 3q2/16r − 0.
(1o) e If (q, p) ∈ A4 and μ = |μ0|, then of forms
()
which represent two antisymmetric solitary waves, tend to a trivial wave φ = 0 when p → 3q2/16r + 0, and tend to two trivial waves when p → q2/4r − 0.
(1o) f If (q, p) ∈ A11, A12, l6 and μ = |μ0|, then which represent two symmetric solitary waves. Specially, when (q, p) ∈ A11 and p → q2/4r − 0, then tend to two trivial waves .
(2o) For the case of μ < 0, there are five properties as follows.
(2o) a If (q, p) ∈ l2, then and represent four 1-blow-up waves (refer to Figure 12(d)). Specially, let μ = 16p/q < 0, then and .
(2o) b If (q, p) ∈ A1, A2, A3, A4 and μ ≠ −|μ0|, then , and they represent four pairs of 1-blow-up waves (refer to Figure 12(a)). In particular, when (q, p) ∈ A4, if p → 3q2/16r + 0, then the four pairs of 1-blow-up waves become two pairs of 1-blow-up waves, and the varying process is displayed in Figure 12. If p → q2/4r − 0, then the four pairs of 1-blow-up waves become two trivial waves .
(2o) c If (q, p) ∈ A11, A12, l6 and μ ≠ −|μ0|, then , and they represent four tall-kink waves.
(2o) d If μ = −|μ0|, when (q, p) ∈ A3, then which represent two pairs of 1-blow-up waves. When (q, p) ∈ A4, then which represent two pairs of 1-blow-up waves.
(2o) e If (q, p) ∈ A11, A12, l6 and μ = −|μ0|, then which represent two tall-kink waves.
(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A3, and p → 3q2/16r − 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 − 0.5, (b) p = 3 − 10−2, (c) p = 3 − 10−4, and (d) p = 3 − 10−6.
(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A3, and p → 3q2/16r − 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 − 0.5, (b) p = 3 − 10−2, (c) p = 3 − 10−4, and (d) p = 3 − 10−6.
(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A3, and p → 3q2/16r − 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 − 0.5, (b) p = 3 − 10−2, (c) p = 3 − 10−4, and (d) p = 3 − 10−6.
(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A3, and p → 3q2/16r − 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 − 0.5, (b) p = 3 − 10−2, (c) p = 3 − 10−4, and (d) p = 3 − 10−6.
(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−5, and (d) p = 3 + 10−7.
(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−5, and (d) p = 3 + 10−7.
(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−5, and (d) p = 3 + 10−7.
(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, andμ = 4 and (a) p = 3 + 10−1, (b) p = 3 + 10−3, (c) p = 3 + 10−5, and (d) p = 3 + 10−7.
(The four symmetric solitary waves become two trivial waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A11, and p → q2/4r + 0, where r = −1, q = 4, l7 : p = q2/4r = −4, and μ = 1 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−2, and (d) p = −4 + 10−5.
(The four symmetric solitary waves become two trivial waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A11, and p → q2/4r + 0, where r = −1, q = 4, l7 : p = q2/4r = −4, and μ = 1 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−2, and (d) p = −4 + 10−5.
(The four symmetric solitary waves become two trivial waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A11, and p → q2/4r + 0, where r = −1, q = 4, l7 : p = q2/4r = −4, and μ = 1 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−2, and (d) p = −4 + 10−5.
(The four symmetric solitary waves become two trivial waves.) The varying process for the figures of and when μ > 0, μ≠|μ0 | , (q, p) ∈ A11, and p → q2/4r + 0, where r = −1, q = 4, l7 : p = q2/4r = −4, and μ = 1 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−2, and (d) p = −4 + 10−5.
(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of and when μ < 0, μ ≠ −|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, and μ = −4 and (a) p = 3 + 10−2, (b) p = 3 + 10−4, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of and when μ < 0, μ ≠ −|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, and μ = −4 and (a) p = 3 + 10−2, (b) p = 3 + 10−4, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of and when μ < 0, μ ≠ −|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, and μ = −4 and (a) p = 3 + 10−2, (b) p = 3 + 10−4, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of and when μ < 0, μ ≠ −|μ0 | , (q, p) ∈ A4, and p → 3q2/16r + 0, where r = 1, q = −4, l2 : p = 3q2/16r = 3, and μ = −4 and (a) p = 3 + 10−2, (b) p = 3 + 10−4, (c) p = 3 + 10−6, and (d) p = 3 + 10−9.
2.3. When the Orbit Γ Is Defined by H(φ, y) = H(φ2, 0)
Proposition 3.
(i) If (q, p) belongs to one of A3, A4, A11, and l2, then (1) have four nonlinear wave solutions
()
where
()
()
()
When (q, p) ∈ A3, A4, or l2, then and represent periodic blow-up wave solutions (refer to Figure 13(a)). When (q, p) ∈ A11, then and represent periodic wave solutions (refer to Figure 14(a)).
(ii) If (q, p) ∈ l1 or l7, then (1) have two fractional wave solutions
()
where
()
When (q, p) ∈ l1, represent two 1-blow-up waves (refer to Figure 13(d)). When (q, p) ∈ l7, represent two tall-kink waves (refer to Figure 14(d)).
(iii) If (q, p) ∈ A4 and p → q2/4r − 0, then the periodic blow-up waves tend to two trivial solutions , while tend to the 1-blow-up waves , and the varying process is displayed in Figure 13. If (q, p) ∈ A11 and p → q2/4r + 0, then the periodic waves tend to two trivial solutions , while tend to the tall-kink waves , and for the varying process, see Figure 14.
(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of when (q, p) ∈ A4 and p → q2/4r − 0, where r = 1, q = −4, and l1 : p = q2/4r = 4 and (a) p = 4 − 10−1, (b) p = 4 − 10−2, (c) p = 4 − 10−4, and (d) p = 4 − 10−6.
(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of when (q, p) ∈ A4 and p → q2/4r − 0, where r = 1, q = −4, and l1 : p = q2/4r = 4 and (a) p = 4 − 10−1, (b) p = 4 − 10−2, (c) p = 4 − 10−4, and (d) p = 4 − 10−6.
(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of when (q, p) ∈ A4 and p → q2/4r − 0, where r = 1, q = −4, and l1 : p = q2/4r = 4 and (a) p = 4 − 10−1, (b) p = 4 − 10−2, (c) p = 4 − 10−4, and (d) p = 4 − 10−6.
(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of when (q, p) ∈ A4 and p → q2/4r − 0, where r = 1, q = −4, and l1 : p = q2/4r = 4 and (a) p = 4 − 10−1, (b) p = 4 − 10−2, (c) p = 4 − 10−4, and (d) p = 4 − 10−6.
(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of when (q, p) ∈ A11 and p → q2/4r + 0, where r = −1, q = 4, and l7 : p = q2/4r = −4 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−3, and (d) p = −4 + 10−5.
(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of when (q, p) ∈ A11 and p → q2/4r + 0, where r = −1, q = 4, and l7 : p = q2/4r = −4 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−3, and (d) p = −4 + 10−5.
(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of when (q, p) ∈ A11 and p → q2/4r + 0, where r = −1, q = 4, and l7 : p = q2/4r = −4 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−3, and (d) p = −4 + 10−5.
(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of when (q, p) ∈ A11 and p → q2/4r + 0, where r = −1, q = 4, and l7 : p = q2/4r = −4 and (a) p = −3.5, (b) p = −4 + 10−1, (c) p = −4 + 10−3, and (d) p = −4 + 10−5.
3. The Derivations of Main Results
To derive our results, substituting (23) and u(x, t) = u(ξ) with ξ = x − ct into (1), it follows that
()
Integrating the first equation of (52) twice with respect to ξ and taking the integral constants to zero, we get (22). Substituting (22) into the second equation of (52) and letting c = 2αγ, it follows that
According to the qualitative theory, we obtain the bifurcation phase portraits of system (54) as Figures 15 and 16. Using the information given by Figures 15 and 16, we give derivations to Propositions 1, 2, and 3, respectively.
For p < 0, completing (57) and solving for φ, it follows that
()
where λ = λ(l) is an arbitrary real constant. Let λ = ±(π/2), respectively, and we obtain the solutions and as (3) and (31).
When (q, p) ∈ A1, that is, r > 0, q > 0, andp < 0, let
()
Thus, we have
()
When (q, p) ∈ A2, that is, r > 0, q < 0, andp < 0, similarly we get
()
When (q, p) ∈ A12, that is, r < 0, q > 0, andp < 0, if p → 3q2/16r + 0, then f(p) and g(p) → 0.
Thus,
()
For p > 0, completing (57) and solving for φ, it follows that
()
where λ = λ(l) is an arbitrary real constant.
Note that if φ(ξ) is a solution of (53), so is φ(−ξ). Thus, from (64) we obtain the solutions and as in (33).
For the case of λ > 0. When (q, p) ∈ l2, that is, r > 0, q < 0 and r = 3q2/16p. In (33) letting r = 3q2/16p, we get (34). Furthermore, in (34) letting λ = −q > 0, it follows that
For the case of λ < 0, similarly we can obtain (7), (35), and (36), and here we omit the process. Hereto, we have completed the derivations for Proposition 1.
When the orbit Γ is defined by H(φ, y) = H(φ1, 0), from (55) we obtain
()
where φ1 and μ0 are given in (25) and (40). Substituting (67) into the first equation of (54) and integrating it, we have
()
where m is an arbitrary constant or ±∞.
Completing the above integral and solving for φ, it follows that
()
where θ is given in (39), μ = μ(m) is an arbitrary constant, and
()
Similar to the derivations for and , we get and (see (38)) from (69).
For the case of μ > 0, when (q, p) ∈ l2, that is, r > 0, q < 0, andr = 3q2/16p, then and q + 2Δ = 0. From (38) and (41), it is easy to check that and become and (see (42)), respectively. Furthermore, in (42) letting μ = −(16p/q), it follows that
()
If (q, p) ∈ A3 and μ = |μ0|, that is, r > 0, μ = μ0 > 0, and 2Δ + q > 0, we have
()
If (q, p) ∈ A4 and μ = |μ0|, that is, r > 0, μ = −μ0 > 0, and 2Δ + q < 0, we have
()
If (q, p) ∈ A11, A12, l6 and μ = |μ0|, that is, r < 0, μ = −μ0 > 0, and 2Δ + q > 0, we have
()
For the case of μ < 0, similarly we can obtain the relations of the solutions , and , and here we omit the process. Hereto, we have completed the derivations for Proposition 2.
Hereto, we have completed the derivations for our main results.
4. Conclusions
In this paper, we have investigated the explicit expressions of the nonlinear waves and bifurcation phenomena in (1). Firstly, we obtained three types of explicit nonlinear wave solutions. The first type is the fractional expressions and (see (29) and (51)). The second type is the trigonometric expressions and (see (47) and (48)). The third type is the exp-function expressions , , , and (see (33) and (38)).
Secondly, we revealed five kinds of interesting bifurcation phenomena in (1). The first phenomena is that the 1-blow-up waves can be bifurcated from the periodic-blow-up (see Figure 2 or Figure 13) and 2-blow-up waves (see Figure 8). The second phenomena is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves (see Figure 3). The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves (see Figure 4). The fourth kind is that the low-kink waves can be bifurcated from the symmetric solitary waves (see Figure 6), the 1-blow-up waves (see Figure 7), the tall-kink waves (see Figure 9), and the antisymmetric solitary waves (see Figure 10). The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves (see Figure 14).
Thirdly, we showed that some previous results are our special cases. For instants, , , and are included in , , , and (see (4), (5), (7), (33), and (38)). are included in and (see (6) and (33)).
Finally, we employed the software Mathematica to check the correctness of these solutions. For example, the commands for ua and Ea are as follows:
w = αγ2,
p = (αγ2 − w)/α,
,
r = −δ3/α,
c = 2αγ,
ξ = x − ct,
,
,
Ea = φaExp[i(γx − wt)],
0
0
Acknowledgment
This paper is supported by the National Natural Science Foundation (no. 11171115), Natural Science Foundation of Guangdong Province (no. S2012040007959) and the Fundamental Research Funds for the Central Universities (no. 2012ZM0057).
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