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Dynamical behavior of loop solutions for the K(2,2) equation
Physics Letters A 375 (2011) 2965–2968
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Dynamical behavior of loop solutions for the K (2, 2) equation Lina Zhang a,∗ , Jibin Li a,b a b
Center of Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan, 650093, PR China College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang, 321004, PR China
a r t i c l e
i n f o
Article history: Received 5 January 2011 Received in revised form 1 June 2011 Accepted 19 June 2011 Available online 23 June 2011 Communicated by A.R. Bishop
a b s t r a c t The asymptotic behavior of loop-soliton solution and periodic loop solutions is studied for the K (2, 2) equation. The results show that both of them consist of two or more branches of non-smooth solutions. © 2011 Elsevier B.V. All rights reserved.
Keywords: K (2, 2) equation Loop-soliton solution Periodic loop solution
1. Introduction Loop-soliton solutions and periodic loop solutions have attracted a great deal of interest [1–15] since Konno et al. [16] first reported the loop-soliton solution in a nonlinear oscillation model of an elastic beam with tension. The K (2, 2) equation
ut + u 2
x
+ u 2 xxx = 0
(1.1)
was introduced by Rosenau and Hyman in 1993 as a model to understand the role of nonlinear dispersion in the formation of patterns in liquid drops [17]. From then on, the K (2, 2) equation (1.1) and its generalizations have attracted many research attentions over years and have been studied successfully by many authors [18–30]. In [27], Yin and Tian obtained an abundance of traveling wave solutions including compacton solutions, cuspon solutions, loop-soliton solutions, periodic cusp (peak) wave solutions, stumpon solutions and fractal-like wave solutions of the K (2, 2) equation (1.1) by applying the qualitative analysis method given by Lenells [31]. In fact, it is very important to understand the dynamical behavior of solutions for traveling wave equations. The present Letter aims at studying the traveling wave solutions of the K (2, 2) equation (1.1) in the case of degenerate singular points. A loopsoliton solution and two families of periodic loop solutions are found by employing the method of the phase plane. In addition, the relationship between the loop-soliton solution and the periodic loop solutions is as well investigated. This work can be considered
*
Corresponding author. Tel.: +86 13605799536. E-mail address: [email protected] (L. Zhang).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.040
also as complementary to paper [27] as the analysis given may be helpful in understanding the significance of dynamical behavior of Eq. (1.1). 2. Main results By substituting u (x, t ) = u (x + ct ) = u (ξ ) into Eq. (1.1) and integrating the resulting equation once, we obtain
2
cu + u 2 + 2 u
+ 2uu = g ,
(2.1)
where g is an integration constant. Clearly, (2.1) is equivalent to the following planar dynamical system
du dξ
= y,
dy dξ
= −u 2 − 2 y 2 − cu + g (2u )
(2.2)
with the first integral
H (u , y ) = u 2 y 2 + u 2 /4 + cu /3 − g /2 = h.
(2.3)
Let dξ = 2u dτ , then system (2.2) becomes
du dτ
= 2u y ,
dy dτ
= −u 2 − 2 y 2 − cu + g .
(2.4)
System (2.4) has the same first integral H (u , y ) and the same topological phase portraits as system (2.2) except for the straight line u = 0. Since system (2.4) is invariant under the transformation u → −u , c → −c, we only consider the case c > 0 in the rest of this Letter for the sake of brevity. Let M (φe , y e ) be the coefficient matrix of the linearized system of (2.4) at the singular point (φe , y e ), J be its Jacobin determinant and T be its trace. By the theory of planar dynamical systems,
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L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
Fig. 2. Loop-soliton solution when h = h0 (c = 4, g = −4).
Fig. 1. Phase portraits of system (2–7) for g = −c 2 /4.
we know that for a singular point (φe , y e ) of a planar integrable system, (φe , y e ) is said to be non-degenerate if J = 0. Moreover, (φe , y e ) is a saddle if J < 0, a node if T 2 > 4 J > 0 (stable if T < 0, unstable if T > 0), a center if T = 0 < J . (φe , y e ) is a nilpotent singular point if T = J = 0 but M (φe , y e ) is not identically zero. Traveling wave solutions have been extensively discussed in the neighborhood of non-degenerate singular points, for instance, a solitary wave solution of Eq. (1.1) corresponds to a homoclinic orbit of system (2.4); a periodic wave solution of Eq. (1.1) corresponds to a periodic orbit of system (2.4). However, seldom have the traveling wave solutions been considered in the case of degenerate singular points. Here we concentrate on this special case. After simple analysis, we know that system (2.4) has a nilpotent singular point, namely a cusp (−c /2, 0) on the u-axis when g = −c 2 /4. We show the phase portrait of system (2.4) in Fig. 1. We have the following main result.
following representation of loop-soliton solution
±ξ = π − 2 arcsin
±ξ =
−
2c
3(c − 6u )
(3.1)
c + 2u
α1 F (ϕ , k) √ ( A − B) A B
α − α1 α2 + Π − − ϕ , , k α f η 1 0 , 1 − α2 α2 − 1 2(r2 A + r1 B )
where
A=
(r1 − b1 )2 + a21 ,
α= k2 =
A−B A+B
and
α1 =
,
B=
(3.2)
(r2 − b1 )2 + a21 ,
r2 A − r1 B r2 A + r1 B
,
(r1 − r2 )2 − ( A − B )2
ϕ = arccos
3. Exact traveling wave solutions of the K (2, 2) equation (1.1) In this section, we will give the exact representations of the loop-soliton solution and periodic loop solutions of the K (2, 2) equation (1.1). Note that F (·, k) and Π(·, ·, k) are the elliptic integrals of the first and third kind respectively with the modulus k. 1. Corresponding to Fig. 2, the graph defined by H (u , y ) = h0 consist of two hyperbolic sectors of the cusp (−c /2, 0) and an open-end curve Γ0 passing through the point (c /6, 0). It follows from (2.3) that y 2 = (c + 2u )3 (c − 6u )/(192u 2 ). Hence, integrating the first equation of system (2.2) along the curve Γ0 , we obtain the
with u ∈ (−c /2, c /6]. 2. Corresponding to Fig. 3, the graph defined by H (u , y ) = h, h ∈ (0, h0 ) consist of two open-end curves Γ1 and Γ2 passing through the points (r1 , 0) and (r2 , 0), respectively, where −c /2 < r2 < 0 < r1 < c /6. It follows from (2.3) that y 2 = (r1 − u )(u − r2 )[(u − b1 )2 + a21 ]/(4u 2 ). By using the first equation of system (2.2) to do integration along the curve Γ1 , we obtain the implicit representation of periodic loop solution for u ∈ [r2 , r1 ] (see [32])
Proposition 2.1. Denote that h0 = H (−c /2, 0). (i) For h = h0 defined by (2.3), Eq. (1.1) has a unique loop-soliton solution shown in Fig. 2. (ii) For h ∈ (0, h0 ), there exists a family of uncountably infinite many periodic loop solutions of Eq. (1.1) shown in Fig. 3. Moreover, the periodic loop solutions converge to the loop-soliton solution as h approaches h0 . (iii) For h ∈ (h0 , +∞), there exists a family of uncountably infinite many periodic loop solutions of Eq. (1.1) shown in Fig. 4. Moreover, the periodic loop solutions converge to the loop-soliton solution as h approaches h0 .
c + 6u
f1 =
, k21 = 1 − k2 , 4AB (r1 − u ) B + (r2 − u ) A (r1 − u ) B − (r2 − u ) A
1 − α2 k2 + k21 α 2
arctan
,
k2 + k21 α 2
sin2 ϕ
1 − α2
1 − k2 sin2 ϕ
α − α1 α2 Π η0 = α1 F (ϕ , k) + ϕ , , k − α f1 1 − α2 α2 − 1
,
(3.3)
. u =r1
(3.4) 3. Corresponding to Fig. 4, the graph defined by H (u , y ) = h, h ∈ (h0 , +∞) consists of two open-end curves Γ3 and Γ4 passing through the points (r1 , 0) and (r2 , 0) respectively, where
L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
2967
(a)
(a)
(b)
(b)
(c)
(c)
Fig. 3. Periodic loop solutions when h ∈ (0, h0 ) (c = 4, g = −4).
Fig. 4. Periodic loop solutions when h ∈ (h0 , +∞) (c = 4, g = −4).
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L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
r2 < −c /2 < 0 < c /6 < r1 . We have the representation of periodic loop solution as in (3.2) by doing similar procedure which is omitted here for simplicity. Example 3.1. Taking c = 4, g = −4 and h = 0.4, we get the approximations of A, B, r1 , r2 , a1 , b1 , α , α1 , k, k1 in the for. . . mula (3.2), where A = 3.072400, B = 2.170250, r1 = 0.394864, . . . . r2 = −0.544768, a1 = 0.721106, b1 = −2.591710, α = 0.172079, . . . α1 = 3.098340, k = 0.0508747, k1 = 0.998705. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 10671179 and 11071222). References [1] [2] [3] [4] [5] [6]
V.A. Vakhnenko, J. Phys. A: Math. Gen. 25 (1992) 4181. E.J. Parkes, J. Phys. A: Math. Gen. 26 (1993) 6469. V.O. Vakhnenko, E.J. Parkes, Nonlinearity 11 (1998) 1457. V.O. Vakhnenko, J. Math. Phys. 40 (1999) 2011. A.J. Morrison, E.J. Parkes, V.O. Vakhnenko, Nonlinearity 12 (1999) 1427. V.O. Vakhnenko, E.J. Parkes, Chaos Solitons Fractals 13 (2002) 1819.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Y.A. Stepanyants, Chaos Solitons Fractals 28 (2006) 193. E.J. Parkes, Chaos Solitons Fractals 31 (2007) 602. E.J. Parkes, Chaos Solitons Fractals 36 (2008) 602. E.J. Parkes, SIGMA 4 (2008) 1. J. Li, Sci. China Ser. A 50 (2007) 773. J. Li, Int. J. Bifur. Chaos 7 (2009) 2249. J. Li, Y. Zhang, Nonlinear Anal.: Real World Appl. 10 (2009) 2502. J. Li, Appl. Math. Mech. 30 (2009) 537. A. Chen, J. Li, Commun. Theor. Phys. 54 (2010) 297. K. Konno, Y.H. Ichikawa, M. Wadati, J. Phys. Soc. Jpn. 50 (1981) 1025. P. Rosenau, J.M. Hyman, Phys. Rev. Lett. 70 (1993) 564. A.M. Wazwaz, Appl. Math. Comput. 133 (2002) 213. A.M. Wazwaz, Chaos Solitons Fractals 13 (2002) 321. A.M. Wazwaz, Appl. Math. Comput. 133 (2002) 229. A.M. Wazwaz, Chaos Solitons Fractals 13 (2002) 161. A.M. Wazwaz, Appl. Math. Comput. 173 (2006) 213. Y. Shang, Appl. Math. Comput. 173 (2006) 1124. A. Biswas, Phys. Lett. A 372 (2008) 4601. C. Xu, L. Tian, Chaos Solitons Fractals 40 (2009) 893. J. Zhou, L. Tian, Phys. Lett. A 372 (2008) 6232. J. Yin, L. Tian, Nonlinear Anal. 73 (2010) 465. X. Deng, E.J. Parkes, J. Cao, Appl. Math. Comput. 217 (2010) 1566. A. Chen, J. Li, J. Math. Anal. Appl. 369 (2010) 758. J. Yin, L. Tian, X. Fan, Comput. Math. Appl. 59 (2010) 2756. J. Lenells, J. Differential Equations 217 (2005) 393. P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Dynamical behavior of loop solutions for the K (2, 2) equation Lina Zhang a,∗ , Jibin Li a,b a b
Center of Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan, 650093, PR China College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang, 321004, PR China
a r t i c l e
i n f o
Article history: Received 5 January 2011 Received in revised form 1 June 2011 Accepted 19 June 2011 Available online 23 June 2011 Communicated by A.R. Bishop
a b s t r a c t The asymptotic behavior of loop-soliton solution and periodic loop solutions is studied for the K (2, 2) equation. The results show that both of them consist of two or more branches of non-smooth solutions. © 2011 Elsevier B.V. All rights reserved.
Keywords: K (2, 2) equation Loop-soliton solution Periodic loop solution
1. Introduction Loop-soliton solutions and periodic loop solutions have attracted a great deal of interest [1–15] since Konno et al. [16] first reported the loop-soliton solution in a nonlinear oscillation model of an elastic beam with tension. The K (2, 2) equation
ut + u 2
x
+ u 2 xxx = 0
(1.1)
was introduced by Rosenau and Hyman in 1993 as a model to understand the role of nonlinear dispersion in the formation of patterns in liquid drops [17]. From then on, the K (2, 2) equation (1.1) and its generalizations have attracted many research attentions over years and have been studied successfully by many authors [18–30]. In [27], Yin and Tian obtained an abundance of traveling wave solutions including compacton solutions, cuspon solutions, loop-soliton solutions, periodic cusp (peak) wave solutions, stumpon solutions and fractal-like wave solutions of the K (2, 2) equation (1.1) by applying the qualitative analysis method given by Lenells [31]. In fact, it is very important to understand the dynamical behavior of solutions for traveling wave equations. The present Letter aims at studying the traveling wave solutions of the K (2, 2) equation (1.1) in the case of degenerate singular points. A loopsoliton solution and two families of periodic loop solutions are found by employing the method of the phase plane. In addition, the relationship between the loop-soliton solution and the periodic loop solutions is as well investigated. This work can be considered
*
Corresponding author. Tel.: +86 13605799536. E-mail address: [email protected] (L. Zhang).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.06.040
also as complementary to paper [27] as the analysis given may be helpful in understanding the significance of dynamical behavior of Eq. (1.1). 2. Main results By substituting u (x, t ) = u (x + ct ) = u (ξ ) into Eq. (1.1) and integrating the resulting equation once, we obtain
2
cu + u 2 + 2 u
+ 2uu = g ,
(2.1)
where g is an integration constant. Clearly, (2.1) is equivalent to the following planar dynamical system
du dξ
= y,
dy dξ
= −u 2 − 2 y 2 − cu + g (2u )
(2.2)
with the first integral
H (u , y ) = u 2 y 2 + u 2 /4 + cu /3 − g /2 = h.
(2.3)
Let dξ = 2u dτ , then system (2.2) becomes
du dτ
= 2u y ,
dy dτ
= −u 2 − 2 y 2 − cu + g .
(2.4)
System (2.4) has the same first integral H (u , y ) and the same topological phase portraits as system (2.2) except for the straight line u = 0. Since system (2.4) is invariant under the transformation u → −u , c → −c, we only consider the case c > 0 in the rest of this Letter for the sake of brevity. Let M (φe , y e ) be the coefficient matrix of the linearized system of (2.4) at the singular point (φe , y e ), J be its Jacobin determinant and T be its trace. By the theory of planar dynamical systems,
2966
L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
Fig. 2. Loop-soliton solution when h = h0 (c = 4, g = −4).
Fig. 1. Phase portraits of system (2–7) for g = −c 2 /4.
we know that for a singular point (φe , y e ) of a planar integrable system, (φe , y e ) is said to be non-degenerate if J = 0. Moreover, (φe , y e ) is a saddle if J < 0, a node if T 2 > 4 J > 0 (stable if T < 0, unstable if T > 0), a center if T = 0 < J . (φe , y e ) is a nilpotent singular point if T = J = 0 but M (φe , y e ) is not identically zero. Traveling wave solutions have been extensively discussed in the neighborhood of non-degenerate singular points, for instance, a solitary wave solution of Eq. (1.1) corresponds to a homoclinic orbit of system (2.4); a periodic wave solution of Eq. (1.1) corresponds to a periodic orbit of system (2.4). However, seldom have the traveling wave solutions been considered in the case of degenerate singular points. Here we concentrate on this special case. After simple analysis, we know that system (2.4) has a nilpotent singular point, namely a cusp (−c /2, 0) on the u-axis when g = −c 2 /4. We show the phase portrait of system (2.4) in Fig. 1. We have the following main result.
following representation of loop-soliton solution
±ξ = π − 2 arcsin
±ξ =
−
2c
3(c − 6u )
(3.1)
c + 2u
α1 F (ϕ , k) √ ( A − B) A B
α − α1 α2 + Π − − ϕ , , k α f η 1 0 , 1 − α2 α2 − 1 2(r2 A + r1 B )
where
A=
(r1 − b1 )2 + a21 ,
α= k2 =
A−B A+B
and
α1 =
,
B=
(3.2)
(r2 − b1 )2 + a21 ,
r2 A − r1 B r2 A + r1 B
,
(r1 − r2 )2 − ( A − B )2
ϕ = arccos
3. Exact traveling wave solutions of the K (2, 2) equation (1.1) In this section, we will give the exact representations of the loop-soliton solution and periodic loop solutions of the K (2, 2) equation (1.1). Note that F (·, k) and Π(·, ·, k) are the elliptic integrals of the first and third kind respectively with the modulus k. 1. Corresponding to Fig. 2, the graph defined by H (u , y ) = h0 consist of two hyperbolic sectors of the cusp (−c /2, 0) and an open-end curve Γ0 passing through the point (c /6, 0). It follows from (2.3) that y 2 = (c + 2u )3 (c − 6u )/(192u 2 ). Hence, integrating the first equation of system (2.2) along the curve Γ0 , we obtain the
with u ∈ (−c /2, c /6]. 2. Corresponding to Fig. 3, the graph defined by H (u , y ) = h, h ∈ (0, h0 ) consist of two open-end curves Γ1 and Γ2 passing through the points (r1 , 0) and (r2 , 0), respectively, where −c /2 < r2 < 0 < r1 < c /6. It follows from (2.3) that y 2 = (r1 − u )(u − r2 )[(u − b1 )2 + a21 ]/(4u 2 ). By using the first equation of system (2.2) to do integration along the curve Γ1 , we obtain the implicit representation of periodic loop solution for u ∈ [r2 , r1 ] (see [32])
Proposition 2.1. Denote that h0 = H (−c /2, 0). (i) For h = h0 defined by (2.3), Eq. (1.1) has a unique loop-soliton solution shown in Fig. 2. (ii) For h ∈ (0, h0 ), there exists a family of uncountably infinite many periodic loop solutions of Eq. (1.1) shown in Fig. 3. Moreover, the periodic loop solutions converge to the loop-soliton solution as h approaches h0 . (iii) For h ∈ (h0 , +∞), there exists a family of uncountably infinite many periodic loop solutions of Eq. (1.1) shown in Fig. 4. Moreover, the periodic loop solutions converge to the loop-soliton solution as h approaches h0 .
c + 6u
f1 =
, k21 = 1 − k2 , 4AB (r1 − u ) B + (r2 − u ) A (r1 − u ) B − (r2 − u ) A
1 − α2 k2 + k21 α 2
arctan
,
k2 + k21 α 2
sin2 ϕ
1 − α2
1 − k2 sin2 ϕ
α − α1 α2 Π η0 = α1 F (ϕ , k) + ϕ , , k − α f1 1 − α2 α2 − 1
,
(3.3)
. u =r1
(3.4) 3. Corresponding to Fig. 4, the graph defined by H (u , y ) = h, h ∈ (h0 , +∞) consists of two open-end curves Γ3 and Γ4 passing through the points (r1 , 0) and (r2 , 0) respectively, where
L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
2967
(a)
(a)
(b)
(b)
(c)
(c)
Fig. 3. Periodic loop solutions when h ∈ (0, h0 ) (c = 4, g = −4).
Fig. 4. Periodic loop solutions when h ∈ (h0 , +∞) (c = 4, g = −4).
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L. Zhang, J. Li / Physics Letters A 375 (2011) 2965–2968
r2 < −c /2 < 0 < c /6 < r1 . We have the representation of periodic loop solution as in (3.2) by doing similar procedure which is omitted here for simplicity. Example 3.1. Taking c = 4, g = −4 and h = 0.4, we get the approximations of A, B, r1 , r2 , a1 , b1 , α , α1 , k, k1 in the for. . . mula (3.2), where A = 3.072400, B = 2.170250, r1 = 0.394864, . . . . r2 = −0.544768, a1 = 0.721106, b1 = −2.591710, α = 0.172079, . . . α1 = 3.098340, k = 0.0508747, k1 = 0.998705. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 10671179 and 11071222). References [1] [2] [3] [4] [5] [6]
V.A. Vakhnenko, J. Phys. A: Math. Gen. 25 (1992) 4181. E.J. Parkes, J. Phys. A: Math. Gen. 26 (1993) 6469. V.O. Vakhnenko, E.J. Parkes, Nonlinearity 11 (1998) 1457. V.O. Vakhnenko, J. Math. Phys. 40 (1999) 2011. A.J. Morrison, E.J. Parkes, V.O. Vakhnenko, Nonlinearity 12 (1999) 1427. V.O. Vakhnenko, E.J. Parkes, Chaos Solitons Fractals 13 (2002) 1819.
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
Y.A. Stepanyants, Chaos Solitons Fractals 28 (2006) 193. E.J. Parkes, Chaos Solitons Fractals 31 (2007) 602. E.J. Parkes, Chaos Solitons Fractals 36 (2008) 602. E.J. Parkes, SIGMA 4 (2008) 1. J. Li, Sci. China Ser. A 50 (2007) 773. J. Li, Int. J. Bifur. Chaos 7 (2009) 2249. J. Li, Y. Zhang, Nonlinear Anal.: Real World Appl. 10 (2009) 2502. J. Li, Appl. Math. Mech. 30 (2009) 537. A. Chen, J. Li, Commun. Theor. Phys. 54 (2010) 297. K. Konno, Y.H. Ichikawa, M. Wadati, J. Phys. Soc. Jpn. 50 (1981) 1025. P. Rosenau, J.M. Hyman, Phys. Rev. Lett. 70 (1993) 564. A.M. Wazwaz, Appl. Math. Comput. 133 (2002) 213. A.M. Wazwaz, Chaos Solitons Fractals 13 (2002) 321. A.M. Wazwaz, Appl. Math. Comput. 133 (2002) 229. A.M. Wazwaz, Chaos Solitons Fractals 13 (2002) 161. A.M. Wazwaz, Appl. Math. Comput. 173 (2006) 213. Y. Shang, Appl. Math. Comput. 173 (2006) 1124. A. Biswas, Phys. Lett. A 372 (2008) 4601. C. Xu, L. Tian, Chaos Solitons Fractals 40 (2009) 893. J. Zhou, L. Tian, Phys. Lett. A 372 (2008) 6232. J. Yin, L. Tian, Nonlinear Anal. 73 (2010) 465. X. Deng, E.J. Parkes, J. Cao, Appl. Math. Comput. 217 (2010) 1566. A. Chen, J. Li, J. Math. Anal. Appl. 369 (2010) 758. J. Yin, L. Tian, X. Fan, Comput. Math. Appl. 59 (2010) 2756. J. Lenells, J. Differential Equations 217 (2005) 393. P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 1971.