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W-shape solitary wave solutions determined by a singular traveling wave equation
Nonlinear Analysis: Real World Applications 10 (2009) 1797–1802 www.elsevier.com/locate/nonrwa
Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation Jibin Li a,b,∗ , Yi Zhang a a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China b Kunming University of Science and Technology, Kunming 200062, PR China
Received 1 July 2007; accepted 13 February 2008
Abstract It had been found that some nonlinear wave equations have the so-called “W/M”-shape-peaks solitons. What is the dynamical behavior of these solutions? To answer this question, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Discontinuous planar dynamical system; “W/M”-shape-peaks soliton; Periodic wave solution; Bifurcation; Nonlinear wave equation
1. Introduction Very recently, Qiao [4] proposed the following new completely integrable wave equation u t − u x xt + 3u 2 u x − u 3x = (4u − 2u x x )u x u x x + (u 2 − u 2x )u x x x ,
(1)
namely, m t + m x (u 2 − u 2x ) + 2m 2 u x = 0,
m = u − ux ,
where u is the fluid velocity and subscripts denote the partial derivatives. This equation can be also derived from the two-dimensional Euler equation by using the approximation procedure. The author has proved that (1) has Lax pair and bi-Hamiltonian structures, which implies the integrability of the equation. In addition, the author found the so-called “W/M”-shape-peaks solitons and claimed that there exist no smooth solitons for this integrable water wave equation. In this paper, we would like to use the method of dynamical systems to the traveling wave system of (1), and show that there exists a smooth solitary solution of (1) when some parameter conditions are satisfied. This result also ∗ Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China. Tel.: +86 0579 82282836; fax: +86 0579 82298188. E-mail addresses: [email protected] (J. Li), [email protected] (Y. Zhang).
c 2008 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2008.02.016
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corrects the erroneous assertion that there exist no smooth solitons for Eq. (1). In the mean time, we shall explain why the so-called “W/M”-shape-peaks solitons can be created and give the determined parameter conditions and exact explicit parametric representations for all solitary wave solutions of (1). Setting u(x, t) = u(x − ct) ≡ φ(ξ ), where ξ = x − ct and c is the wave speed. Substituting it into (1), we have (φ 2 − φξ2 − c)(φ − φξ ξ )ξ + 2φξ (φ − φξ ξ )2 = 0. Integrating the above equation once, we obtain (φ − φξ ξ )(c − φ 2 − φξ2 ) = g,
(2)
where g is an integral constant. We suppose that g 6= 0. Otherwise, (2) becomes a linear equation. Eq. (2) is equivalent to the system dφ = y, dξ
dy g −g + φ(c − φ 2 + y 2 ) =φ− = , 2 2 dξ c−φ +y (c − φ 2 + y 2 )
(3)
which has the first integral H (φ, y) = (c − φ 2 + y 2 )2 + 4gφ = h.
(4)
Clearly, on the hyperbola φ 2 − y 2 = c, system (3) is discontinuous. Such a system is called a singular traveling wave system by one of the authors [2,3]. 2. Bifurcation set and phase portraits of (3) Let us consider all possible phase portraits of (3). We can also assume c > 0 in what follows, without loss of generality. It is known that system (3) has the same phase portraits as the system dφ = y(c − φ 2 + y 2 ), dζ
dy = −g + φ(c − φ 2 + y 2 ), dζ
(5)
where dξ = (c − φ 2 + y 2 )dζ , for (c − φ 2 + y 2 ) 6= 0. Denote that f (φ) = φ 3 − cφ + g, f 0 (φ) = 3φ 2 − c. It is easy to show for a fixed c > 0, the following facts hold. q q q c c 2c c (1) For g > 0, when g > 2c 3 3 , f (φ) only has a negative zero z 3 < − 3 ; when g = 3 3 , f (φ) has one simple q q q c 2c c zero z 3 and a double zero z 21 = 3 ; when g < 3 3 , f (φ) has three simple zeroes: z 3 < 0 < z 2 < 3c < z 1 < √ c. q q q c c 2c c (2) For g < 0, when |g| > 2c , f (φ) only has a positive zero z > ; when |g| = 1 3 3 3 3 , f (φ) has q q3 c one simple zero z 1 and a double zero z 23 = − 3c ; when |g| < 2c 3 3 , f (φ) has three simple zeroes: q q √ − c < z 3 < − 3c < z 2 < 0 < 3c < z 1 . Let M(z i , 0) be the coefficient matrix of the linearized system of (5) at an equilibrium point E i (z i , 0). We have J (z i , 0) = det M(z i , 0) = (c − z i2 )(3z i2 − c).
(6)
By the theory of planar dynamical systems (see [3,4]), for an equilibrium point of a planar integral system, if J < 0, then the equilibrium point is a saddle point; if J > 0, then it is a center point; if J = 0 and the Poincar´e index of the equilibrium point is 0, then this equilibrium point is a cusp. In the following we write that h i = H (z i , 0), q i = 1, 2, 3 whereqH is given by (4). Obviously, we see from (5) that when g >
g < 0), the unique equilibrium point E 3 (z 3 , 0) q c (or E 1 (z 1 , 0)) of (5) is a saddle point. We do not give the phase portrait in this case. When g ∗ < g < 2c 3 3 , the point E 1 (z 1 , 0) is a center; E 2 (z 2 , 0) is a saddle point, where for g = g ∗ , the homoclinic orbit defined by H (φ, y) = h 2 2c 3
c 3
(or |g| >
2c 3
c 3,
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J. Li, Y. Zhang / Nonlinear Analysis: Real World Applications 10 (2009) 1797–1802
Fig. 1. The change of phase portraits of (5) for a fixed c > 0.
Fig. 2. The change of phase portraits of (5) for a fixed c > 0.
√ √ to the saddle point E 2 (z 2 , 0) passes through the point E s ( c, 0) and we have h 2 = 4g ∗ c = (c − φ22 )2 + 4g ∗ φ2 . Similarly, we can discuss the case of g < 0. By using the above result to do qualitative analysis, we have the bifurcation set and the change of phase portraits of (5) shown in Figs. 1 and 2, respectively, for g > 0 and g < 0. We also draw the graph of the hyperbola φ 2 − y 2 = c in every phase plane in order to show the position of the singular curve. Clearly, Fig. 2(2-1)–(2-4) are just the reflections of Fig. 1(1-1)–(1-4) with respect to the y-axis. Therefore, in the next section, we only need to discuss the case g > 0. 3. The traveling wave solutions determined by phase portraits of (3) and their parametric representations It is well know that a homoclinic orbit of the traveling system gives rise to a solitary solution of the corresponding nonlinear wave equation. Weqalways assume that c > 0 and is fixed, without loss of generality. c 1. The case g ∗ < g < 2c 3 3. In this case, the homoclinic orbit defined by H (φ, y) = h 2 has no intersection point with the hyperbola φ 2 −y 2 = c. Thus, by the results in Section 2, we immediately obtain the following conclusions. q c Proposition 1. (1) For any fixed c > 0, when g ∗ < g < 2c 3 3 , Eq. (1) has a smooth solitary wave solution of peak type defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of smooth periodic wave solutions defined by H (φ, y) = h, h ∈ (h 1 , h 2 ) (see Fig. q 1(1-2)). 2c c (2) For any fixed c > 0, when − 3 3 < g < g∗ , Eq. (1) has a smooth solitary wave solution of valley type defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of smooth periodic wave solutions defined by H (φ, y) = h, h ∈ (h 3 , h 2 ) (see Fig. 2(2-2)). To find the exact explicit parametric representations of solitary wave solutions, we have from (4) that p y 2 = φ 2 ± h − 4gφ − c. √ The signs ± before the term h − 4gφ are dependent on the interval of φ. Let ψ 2 = h − 4gφ i.e., φ = Then, we obtain y2 =
1 [ψ 4 − 2hψ 2 ± 16g 2 ψ + (h 2 − 16g 2 c)]. 16g 2
(7) 1 4g (h
− ψ 2 ).
(8)
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q √ c 2 Under the condition g ∗ < g < 2c 3 3 , for φ ∈ (φ2 , c), we need to take + before the term 16g ψ. By the first equation of (3), corresponding to the homoclinic orbit defined by H (φ, y) = h 2 , we have ψdψ
q
1 = − dξ. 2 ψ 4 − 2h 2 ψ 2 + 16g 2 ψ + (h 22 − 16g 2 c)
(9)
Denote that (c − φ 2 )2 + 4gφ − h 2 = (φ − φ2 )2 (φ − φ M )(φ − φ3 ), where φ3 < φ2 < φ M . The point (φ M , 0) is the intersection point of the homoclinic orbit to (φ2 , 0) of (3) with the positive φ-axis. Thus, we have √ 4 − 2h ψ 2 + 16g 2 ψ + (h 2 − 16g 2 c) = (ψ − ψ )2 (ψ − ψ )(ψ − ψ ), where ψ = ψ h − 4gφ2 , ψm = m 1 3 1 2 2 2 √ √ h 2 − 4gφ M , ψ3 = − h 2 − 4gφ3 . By introducing a parametric variable χ and integrating (9) [1], we obtain 2(ψ1 − ψm )(ψ1 − ψ3 ) , √ (ψm − ψ3 ) cosh( (ψ1 − ψm )(ψ1 − ψ3 )χ ) − (2ψ1 − ψm − ψ3 ) √ 2 (ψ − ψm )(ψ − ψ3 ) + 2ψ − (ψm − ψ3 ) ξ(χ ) = x − ct = −2 ψ1 χ + ln . ψm − ψ3 ψ(χ ) = ψ1 −
(10)
Thus, we have the following exact explicit parametric representations of smooth solitary solution of (1): 1 (1 − ψ 2 (χ )), 4g √ 2 (ψ − ψm )(ψ − ψ3 ) + 2ψ − (ψm − ψ3 ) . ξ(χ ) = x − ct = −2 ψ1 χ + ln ψm − ψ3 φ(χ ) =
(11)
2. The case g = g ∗ . √ √ In this case, we have h 2 = 4g c and φ M = c, ψm = 0. By the same computation as the case 1, we obtain 2ψ1 (ψ1 − ψ3 ) , √ (−ψ3 ) cosh( ψ1 (ψ1 − ψ3 )χ ) − (2ψ1 − ψ3 ) √ 2 ψ(ψ − ψ3 ) + 2ψ + ψ3 ξ(χ ) = x − ct = −2 ψ1 χ + ln . (−ψ3 ) ψ(χ ) = ψ1 −
(12)
Hence, we have the following exact explicit parametric representations of solitary solution of (1): 1 (1 − ψ 2 (χ )), 4g √ 2 ψ(ψ − ψ3 ) + 2ψ + ψ3 . ξ(χ ) = x − ct = −2 ψ1 χ + ln (−ψ3 ) φ(χ ) =
(13)
3. The case 0 < g < g ∗ . We notice that unlike the case where there exists a singular straight line in Ref. [2], for system (5), the hyperbola φ 2 − y 2 = c is not a solution of (5). For every fixed c > 0, when 0 < g < g ∗ (or g∗ < g < 0), there exist uncountable infinite many periodic orbits and a homoclinic orbit of (5), which are transversely intersecting with the hyperbola φ 2 − y 2 = c (see Figs. 1(1-4), 2(2-4) and 3). We now consider the dynamical behavior of the homoclinic orbit given by Fig. 3(3-1). h2 The hyperbola φ 2 − y 2 = c intersects the homoclinic orbit of (5) at two points P(φs , ±ys ), where φs = 4g , ys = q h2 2 2 2 c − ( 4g ) . We see from (7) that in the left-hand side of the positive half branch of the hyperbola φ − y = c, we √ √ have y 2 = φ 2 + h − 4gφ − c; while in the right-hand side, we have y 2 = φ 2 − h − 4gφ − c. Therefore, we can respectively write that ψ 4 − 2h 2 ψ 2 + 16g 2 ψ + (h 22 − 16g 2 c) = (ψ − ψ1 )2 (ψ − ψm )(ψ − ψ3 ), ψ 4 − 2h 2 ψ 2 − 16g 2 ψ + (h 22 − 16g 2 c) = (ψ − ψb )2 (ψ − ψ M )(ψ − ψa ), √ where ψb = − h 2 − 4gφ2 .
(14)
J. Li, Y. Zhang / Nonlinear Analysis: Real World Applications 10 (2009) 1797–1802
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Fig. 3. The graphs of hyperbola and two homoclinic orbits (5).
Fig. 4. M-shape solitary wave solution of (1) in the (ξ, φ)-plane when 0 < g < g ∗ .
We next define a value χb of χ by satisfying φ(χb ) =
1 h2 (1 − ψ 2 (χb )) = ≡ φs , 4g 4g
(15)
where ψ(χ ) is given by (10). It is easy to see that for χ ∈ (−∞, −χb ) and χ ∈ (χb , ∞), we have the parametric representations of solitary solution of (1) as (11). For χ ∈ (−χb , χb ), we have from (14) that 2(ψb − ψ M )(ψb − ψa ) , √ (ψ M − ψa ) cosh( (ψb − ψ M )(ψb − ψa )χ ) − (2ψb − ψ M − ψa ) √ 2 (ψ − ψ M )(ψ − ψa ) + 2ψ − (ψ M − ψa ) ξ(χ ) = x − ct = −2 ψb χ + ln ψ M − ψa
(16)
1 (1 − ψ 2 (χ )), χ ∈ (−χb , χb ), 4g √ 2 (ψ − ψ M )(ψ − ψa ) + 2ψ − (ψ M − ψa ) ξ(χ ) = x − ct = −2 ψb χ + ln . ψ M − ψa
(17)
ψ(χ ) = ψb +
and φ(χ ) =
For the case g∗ < g < 0, the homoclinic orbit given by Fig. 3(3-2) gives rise to a W-shape solitary wave solution whose profile is the reflection of Fig. 4 with respect to the ξ -axis.
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To sum up, we have the following conclusions. Proposition 2. (1) For any fixed c > 0, when 0 < g < g ∗ , Eq. (1) has a M-shape solitary wave solution defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of periodic wave solutions defined by H (φ, y) = h, h ∈ (h 1 , h 2 ) (see Figs. 1(1-4) and 4). (2) For any fixed c > 0, when g∗ < g < 0, Eq. (1) has a W-shape solitary wave solution defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of periodic wave solutions defined by H (φ, y) = h, h ∈ (h 3 , h 2 ) (see Fig. 2(2-4)). In summary, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given. It is important to understand the dynamical behavior for the traveling wave solutions governed by singular traveling wave equations. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 10671179 and No. 10771196), the Natural Science Foundation of Zhejiang Province (No. Y605044). References [1] P.F. Byrd, M.D. Fridman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971. [2] J.B. Li, H.H. Dai, On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach, Science Press, Beijing, 2007. [3] J.B. Li, J.H. Wu, H.P. Zhu, Travelling waves for an integrable higher order KdV type wave equations, Int. J. Bifurcation Chaos 16 (8) (2006) 2235–2260. [4] Z.J. Qiao, A new integrable equation with cuspons and W/M-shape peaks solitons, J. Math. Phys. 47 (2006) 112701–112709.
Exact M/W-shape solitary wave solutions determined by a singular traveling wave equation Jibin Li a,b,∗ , Yi Zhang a a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China b Kunming University of Science and Technology, Kunming 200062, PR China
Received 1 July 2007; accepted 13 February 2008
Abstract It had been found that some nonlinear wave equations have the so-called “W/M”-shape-peaks solitons. What is the dynamical behavior of these solutions? To answer this question, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Discontinuous planar dynamical system; “W/M”-shape-peaks soliton; Periodic wave solution; Bifurcation; Nonlinear wave equation
1. Introduction Very recently, Qiao [4] proposed the following new completely integrable wave equation u t − u x xt + 3u 2 u x − u 3x = (4u − 2u x x )u x u x x + (u 2 − u 2x )u x x x ,
(1)
namely, m t + m x (u 2 − u 2x ) + 2m 2 u x = 0,
m = u − ux ,
where u is the fluid velocity and subscripts denote the partial derivatives. This equation can be also derived from the two-dimensional Euler equation by using the approximation procedure. The author has proved that (1) has Lax pair and bi-Hamiltonian structures, which implies the integrability of the equation. In addition, the author found the so-called “W/M”-shape-peaks solitons and claimed that there exist no smooth solitons for this integrable water wave equation. In this paper, we would like to use the method of dynamical systems to the traveling wave system of (1), and show that there exists a smooth solitary solution of (1) when some parameter conditions are satisfied. This result also ∗ Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China. Tel.: +86 0579 82282836; fax: +86 0579 82298188. E-mail addresses: [email protected] (J. Li), [email protected] (Y. Zhang).
c 2008 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2008.02.016
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J. Li, Y. Zhang / Nonlinear Analysis: Real World Applications 10 (2009) 1797–1802
corrects the erroneous assertion that there exist no smooth solitons for Eq. (1). In the mean time, we shall explain why the so-called “W/M”-shape-peaks solitons can be created and give the determined parameter conditions and exact explicit parametric representations for all solitary wave solutions of (1). Setting u(x, t) = u(x − ct) ≡ φ(ξ ), where ξ = x − ct and c is the wave speed. Substituting it into (1), we have (φ 2 − φξ2 − c)(φ − φξ ξ )ξ + 2φξ (φ − φξ ξ )2 = 0. Integrating the above equation once, we obtain (φ − φξ ξ )(c − φ 2 − φξ2 ) = g,
(2)
where g is an integral constant. We suppose that g 6= 0. Otherwise, (2) becomes a linear equation. Eq. (2) is equivalent to the system dφ = y, dξ
dy g −g + φ(c − φ 2 + y 2 ) =φ− = , 2 2 dξ c−φ +y (c − φ 2 + y 2 )
(3)
which has the first integral H (φ, y) = (c − φ 2 + y 2 )2 + 4gφ = h.
(4)
Clearly, on the hyperbola φ 2 − y 2 = c, system (3) is discontinuous. Such a system is called a singular traveling wave system by one of the authors [2,3]. 2. Bifurcation set and phase portraits of (3) Let us consider all possible phase portraits of (3). We can also assume c > 0 in what follows, without loss of generality. It is known that system (3) has the same phase portraits as the system dφ = y(c − φ 2 + y 2 ), dζ
dy = −g + φ(c − φ 2 + y 2 ), dζ
(5)
where dξ = (c − φ 2 + y 2 )dζ , for (c − φ 2 + y 2 ) 6= 0. Denote that f (φ) = φ 3 − cφ + g, f 0 (φ) = 3φ 2 − c. It is easy to show for a fixed c > 0, the following facts hold. q q q c c 2c c (1) For g > 0, when g > 2c 3 3 , f (φ) only has a negative zero z 3 < − 3 ; when g = 3 3 , f (φ) has one simple q q q c 2c c zero z 3 and a double zero z 21 = 3 ; when g < 3 3 , f (φ) has three simple zeroes: z 3 < 0 < z 2 < 3c < z 1 < √ c. q q q c c 2c c (2) For g < 0, when |g| > 2c , f (φ) only has a positive zero z > ; when |g| = 1 3 3 3 3 , f (φ) has q q3 c one simple zero z 1 and a double zero z 23 = − 3c ; when |g| < 2c 3 3 , f (φ) has three simple zeroes: q q √ − c < z 3 < − 3c < z 2 < 0 < 3c < z 1 . Let M(z i , 0) be the coefficient matrix of the linearized system of (5) at an equilibrium point E i (z i , 0). We have J (z i , 0) = det M(z i , 0) = (c − z i2 )(3z i2 − c).
(6)
By the theory of planar dynamical systems (see [3,4]), for an equilibrium point of a planar integral system, if J < 0, then the equilibrium point is a saddle point; if J > 0, then it is a center point; if J = 0 and the Poincar´e index of the equilibrium point is 0, then this equilibrium point is a cusp. In the following we write that h i = H (z i , 0), q i = 1, 2, 3 whereqH is given by (4). Obviously, we see from (5) that when g >
g < 0), the unique equilibrium point E 3 (z 3 , 0) q c (or E 1 (z 1 , 0)) of (5) is a saddle point. We do not give the phase portrait in this case. When g ∗ < g < 2c 3 3 , the point E 1 (z 1 , 0) is a center; E 2 (z 2 , 0) is a saddle point, where for g = g ∗ , the homoclinic orbit defined by H (φ, y) = h 2 2c 3
c 3
(or |g| >
2c 3
c 3,
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Fig. 1. The change of phase portraits of (5) for a fixed c > 0.
Fig. 2. The change of phase portraits of (5) for a fixed c > 0.
√ √ to the saddle point E 2 (z 2 , 0) passes through the point E s ( c, 0) and we have h 2 = 4g ∗ c = (c − φ22 )2 + 4g ∗ φ2 . Similarly, we can discuss the case of g < 0. By using the above result to do qualitative analysis, we have the bifurcation set and the change of phase portraits of (5) shown in Figs. 1 and 2, respectively, for g > 0 and g < 0. We also draw the graph of the hyperbola φ 2 − y 2 = c in every phase plane in order to show the position of the singular curve. Clearly, Fig. 2(2-1)–(2-4) are just the reflections of Fig. 1(1-1)–(1-4) with respect to the y-axis. Therefore, in the next section, we only need to discuss the case g > 0. 3. The traveling wave solutions determined by phase portraits of (3) and their parametric representations It is well know that a homoclinic orbit of the traveling system gives rise to a solitary solution of the corresponding nonlinear wave equation. Weqalways assume that c > 0 and is fixed, without loss of generality. c 1. The case g ∗ < g < 2c 3 3. In this case, the homoclinic orbit defined by H (φ, y) = h 2 has no intersection point with the hyperbola φ 2 −y 2 = c. Thus, by the results in Section 2, we immediately obtain the following conclusions. q c Proposition 1. (1) For any fixed c > 0, when g ∗ < g < 2c 3 3 , Eq. (1) has a smooth solitary wave solution of peak type defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of smooth periodic wave solutions defined by H (φ, y) = h, h ∈ (h 1 , h 2 ) (see Fig. q 1(1-2)). 2c c (2) For any fixed c > 0, when − 3 3 < g < g∗ , Eq. (1) has a smooth solitary wave solution of valley type defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of smooth periodic wave solutions defined by H (φ, y) = h, h ∈ (h 3 , h 2 ) (see Fig. 2(2-2)). To find the exact explicit parametric representations of solitary wave solutions, we have from (4) that p y 2 = φ 2 ± h − 4gφ − c. √ The signs ± before the term h − 4gφ are dependent on the interval of φ. Let ψ 2 = h − 4gφ i.e., φ = Then, we obtain y2 =
1 [ψ 4 − 2hψ 2 ± 16g 2 ψ + (h 2 − 16g 2 c)]. 16g 2
(7) 1 4g (h
− ψ 2 ).
(8)
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q √ c 2 Under the condition g ∗ < g < 2c 3 3 , for φ ∈ (φ2 , c), we need to take + before the term 16g ψ. By the first equation of (3), corresponding to the homoclinic orbit defined by H (φ, y) = h 2 , we have ψdψ
q
1 = − dξ. 2 ψ 4 − 2h 2 ψ 2 + 16g 2 ψ + (h 22 − 16g 2 c)
(9)
Denote that (c − φ 2 )2 + 4gφ − h 2 = (φ − φ2 )2 (φ − φ M )(φ − φ3 ), where φ3 < φ2 < φ M . The point (φ M , 0) is the intersection point of the homoclinic orbit to (φ2 , 0) of (3) with the positive φ-axis. Thus, we have √ 4 − 2h ψ 2 + 16g 2 ψ + (h 2 − 16g 2 c) = (ψ − ψ )2 (ψ − ψ )(ψ − ψ ), where ψ = ψ h − 4gφ2 , ψm = m 1 3 1 2 2 2 √ √ h 2 − 4gφ M , ψ3 = − h 2 − 4gφ3 . By introducing a parametric variable χ and integrating (9) [1], we obtain 2(ψ1 − ψm )(ψ1 − ψ3 ) , √ (ψm − ψ3 ) cosh( (ψ1 − ψm )(ψ1 − ψ3 )χ ) − (2ψ1 − ψm − ψ3 ) √ 2 (ψ − ψm )(ψ − ψ3 ) + 2ψ − (ψm − ψ3 ) ξ(χ ) = x − ct = −2 ψ1 χ + ln . ψm − ψ3 ψ(χ ) = ψ1 −
(10)
Thus, we have the following exact explicit parametric representations of smooth solitary solution of (1): 1 (1 − ψ 2 (χ )), 4g √ 2 (ψ − ψm )(ψ − ψ3 ) + 2ψ − (ψm − ψ3 ) . ξ(χ ) = x − ct = −2 ψ1 χ + ln ψm − ψ3 φ(χ ) =
(11)
2. The case g = g ∗ . √ √ In this case, we have h 2 = 4g c and φ M = c, ψm = 0. By the same computation as the case 1, we obtain 2ψ1 (ψ1 − ψ3 ) , √ (−ψ3 ) cosh( ψ1 (ψ1 − ψ3 )χ ) − (2ψ1 − ψ3 ) √ 2 ψ(ψ − ψ3 ) + 2ψ + ψ3 ξ(χ ) = x − ct = −2 ψ1 χ + ln . (−ψ3 ) ψ(χ ) = ψ1 −
(12)
Hence, we have the following exact explicit parametric representations of solitary solution of (1): 1 (1 − ψ 2 (χ )), 4g √ 2 ψ(ψ − ψ3 ) + 2ψ + ψ3 . ξ(χ ) = x − ct = −2 ψ1 χ + ln (−ψ3 ) φ(χ ) =
(13)
3. The case 0 < g < g ∗ . We notice that unlike the case where there exists a singular straight line in Ref. [2], for system (5), the hyperbola φ 2 − y 2 = c is not a solution of (5). For every fixed c > 0, when 0 < g < g ∗ (or g∗ < g < 0), there exist uncountable infinite many periodic orbits and a homoclinic orbit of (5), which are transversely intersecting with the hyperbola φ 2 − y 2 = c (see Figs. 1(1-4), 2(2-4) and 3). We now consider the dynamical behavior of the homoclinic orbit given by Fig. 3(3-1). h2 The hyperbola φ 2 − y 2 = c intersects the homoclinic orbit of (5) at two points P(φs , ±ys ), where φs = 4g , ys = q h2 2 2 2 c − ( 4g ) . We see from (7) that in the left-hand side of the positive half branch of the hyperbola φ − y = c, we √ √ have y 2 = φ 2 + h − 4gφ − c; while in the right-hand side, we have y 2 = φ 2 − h − 4gφ − c. Therefore, we can respectively write that ψ 4 − 2h 2 ψ 2 + 16g 2 ψ + (h 22 − 16g 2 c) = (ψ − ψ1 )2 (ψ − ψm )(ψ − ψ3 ), ψ 4 − 2h 2 ψ 2 − 16g 2 ψ + (h 22 − 16g 2 c) = (ψ − ψb )2 (ψ − ψ M )(ψ − ψa ), √ where ψb = − h 2 − 4gφ2 .
(14)
J. Li, Y. Zhang / Nonlinear Analysis: Real World Applications 10 (2009) 1797–1802
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Fig. 3. The graphs of hyperbola and two homoclinic orbits (5).
Fig. 4. M-shape solitary wave solution of (1) in the (ξ, φ)-plane when 0 < g < g ∗ .
We next define a value χb of χ by satisfying φ(χb ) =
1 h2 (1 − ψ 2 (χb )) = ≡ φs , 4g 4g
(15)
where ψ(χ ) is given by (10). It is easy to see that for χ ∈ (−∞, −χb ) and χ ∈ (χb , ∞), we have the parametric representations of solitary solution of (1) as (11). For χ ∈ (−χb , χb ), we have from (14) that 2(ψb − ψ M )(ψb − ψa ) , √ (ψ M − ψa ) cosh( (ψb − ψ M )(ψb − ψa )χ ) − (2ψb − ψ M − ψa ) √ 2 (ψ − ψ M )(ψ − ψa ) + 2ψ − (ψ M − ψa ) ξ(χ ) = x − ct = −2 ψb χ + ln ψ M − ψa
(16)
1 (1 − ψ 2 (χ )), χ ∈ (−χb , χb ), 4g √ 2 (ψ − ψ M )(ψ − ψa ) + 2ψ − (ψ M − ψa ) ξ(χ ) = x − ct = −2 ψb χ + ln . ψ M − ψa
(17)
ψ(χ ) = ψb +
and φ(χ ) =
For the case g∗ < g < 0, the homoclinic orbit given by Fig. 3(3-2) gives rise to a W-shape solitary wave solution whose profile is the reflection of Fig. 4 with respect to the ξ -axis.
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To sum up, we have the following conclusions. Proposition 2. (1) For any fixed c > 0, when 0 < g < g ∗ , Eq. (1) has a M-shape solitary wave solution defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of periodic wave solutions defined by H (φ, y) = h, h ∈ (h 1 , h 2 ) (see Figs. 1(1-4) and 4). (2) For any fixed c > 0, when g∗ < g < 0, Eq. (1) has a W-shape solitary wave solution defined by H (φ, y) = h 2 . In addition, Eq. (1) has a family of periodic wave solutions defined by H (φ, y) = h, h ∈ (h 3 , h 2 ) (see Fig. 2(2-4)). In summary, all traveling wave solutions in the parameter space are investigated for a integrable water wave equation from a dynamical systems theoretical point of view. Exact explicit parametric representations of all solitary wave solutions are given. It is important to understand the dynamical behavior for the traveling wave solutions governed by singular traveling wave equations. Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 10671179 and No. 10771196), the Natural Science Foundation of Zhejiang Province (No. Y605044). References [1] P.F. Byrd, M.D. Fridman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971. [2] J.B. Li, H.H. Dai, On the Study of Singular Nonlinear Traveling Wave Equations: Dynamical System Approach, Science Press, Beijing, 2007. [3] J.B. Li, J.H. Wu, H.P. Zhu, Travelling waves for an integrable higher order KdV type wave equations, Int. J. Bifurcation Chaos 16 (8) (2006) 2235–2260. [4] Z.J. Qiao, A new integrable equation with cuspons and W/M-shape peaks solitons, J. Math. Phys. 47 (2006) 112701–112709.