- Email: [email protected]
Propagating wave patterns and “peakons” of the Davey–Stewartson system
Chaos, Solitons and Fractals 27 (2006) 561–567 www.elsevier.com/locate/chaos
Propagating wave patterns and ‘‘peakons’’ of the Davey–Stewartson system K.W. Chow b
a,*
, S.Y. Lou
b
a Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China
Accepted 31 March 2005
Abstract Two exact, doubly periodic, propagating wave patterns of the Davey–Stewartson system are computed analytically by a special separation of variables procedure. For the first solution there is a cluster of smaller peaks within each period. The second one consists of a rectangular array of ÔplatesÕ joined together by sharp edges, and is thus a kind of ÔpeakonsÕ for this system of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations. A long wave limit will yield exponentially localized waves different from the conventional dromion. The stability properties and nonlinear dynamics must await further investigations. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The Davey–Stewartson model (DS) is an important system of evolution equations, both from the perspectives of theory and applications. DS can arise in hydrodynamics [1] and plasma physics [2]. Theoretically, many techniques of the modern theory of nonlinear waves are relevant, e.g., special Hirota bilinear forms [3], Darboux transformations [4], symmetries [5], rich soliton and related structures [6,7]. We shall take DS as oA 1 o2 A o2 A i þ þ þ vA2 A ¼ QA; ot 2 on2 og2 ð1:1Þ o2 Q o2 Q o2 ¼ 2v ðAA Þ. on2 on2 og2 In the hydrodynamic context, A is the envelope of the wave packet while Q is the induced mean flow. New coordinates X, Y are defined by X þY n ¼ pffiffiffi ; 2
*
X Y g ¼ pffiffiffi 2
Corresponding author. Tel.: +852 2859 2641; fax: +852 2858 5415. E-mail addresses: [email protected] (K.W. Chow), [email protected] (S.Y. Lou).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.036
ð1:2Þ
562
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
and the transformed DS considered in the present work will be oA 1 o2 A o2 A þ vA2 A ¼ QA; þ i þ ot 2 oX 2 oY 2 2 o2 Q o o 2 ðAA Þ. ¼v þ oX oY oX oY
ð1:3Þ
Recently a class of novel, exact solutions of DS and other (2 + 1) (2 spatial and 1 temporal) dimensional nonlinear evolution equations can be obtained by a special separation of variables approach [8]. More precisely, one exact solution of the system (1.3) is p ¼ pðxÞ;
q ¼ qðyÞ;
p0 ¼ p0 ðxÞ;
q0 ¼ q0 ðyÞ;
r ¼ rðxÞ;
s ¼ sðyÞ;
2
pxx þ qyy ðpx þ qy Þ þ ; a0 þ p þ q ða0 þ p þ qÞ2 rffiffiffi pffiffiffiffiffiffiffiffiffi px qy 2 f expðiðr þ sÞÞ; rx ¼ c1 þ ; A¼ v a0 þ p þ q px Q ¼ p0 þ q0
p0 ¼
pxxx p2xx d c21 f2 2 þ 2; 4px 8px 8 2 2px
x ¼ X c1 t;
q0 ¼
sy ¼ c 2
f ; qy
ð1:4Þ
qyyy q2yy d c22 f2 2þ þ 2. 4qy 8qy 8 2 2qy
y ¼ Y c2 t
ð1:5Þ
are the coordinates translating with the wave pattern. a0, c1, c2, f are constants, and the relevant functions depend on the indicated variables only. More complicated solutions with terms involving products of p and q in the denominator can be constructed, but details will be left for future studies. The choice of exponential functions as the basis functions in (1.4) leads to generalized solutions of localized solitons or dromions. The purpose of the present note is to demonstrate that the choice of the Jacobi elliptic functions as building blocks (or p 0 (x), q 0 (y) in (1.4)) is feasible too, and will result in doubly periodic, propagating wave patterns for DS. Two constraints will dictate the choice of elliptic functions. Firstly, the building block functions need to be nonnegative as a square root is taken in the process. Simple choices like the functions sn and cn, which oscillate in a sinusoidal manner, must be rejected. Secondly, for analytical convenience, we restrict the attention to simple cases where both A and Q can be evaluated in simple, closed forms which do not involve the elliptic integrals in this paper. The selections of the Jacobi elliptic function dn [9,10] and its reciprocal will satisfy these requirements, and will now lead to these two new wave patterns for DS (Section 2). Further properties like the long wave limit and the boundary conditions will also be investigated (Section 3). 2. Doubly periodic wave patterns 2.1. First solution By choosing both basis functions p 0 (x), q 0 (y) in (1.4) in the x, y directions as ÔdnÕ, we obtain rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dnðax; kÞdnðby; k 1 Þ A¼ expðiU 1 Þ; v R1 f cnðax; kÞ f cnðby; k 1 Þ þ c2 y þ qffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 ; U 1 ¼ c1 x pffiffiffiffiffiffiffiffiffiffiffiffi2ffi sin1 dnðax; kÞ dnðby; k 1 Þ a 1k b 1 k2
ð2:1Þ ð2:2Þ
1
1 1 R1 ¼ a0 þ sin1 ½snðax; kÞ þ sin1 ½snðby; k 1 Þ; a b Q¼
k 2 a2 4
ð2:3Þ !
k 2 sn2 ðax; kÞcn2 ðax; kÞ c21 f2 þ 2 2dn2 ðax; kÞ 2dn2 ðax; kÞ ! k 2 sn2 ðby; k 1 Þcn2 ðby; k 1 Þ c2 f2 þ 2 sn2 ðby; k 1 Þ cn2 ðby; k 1 Þ 1 2 2 2 2dn ðby; k 1 Þ 2dn ðby; k 1 Þ
sn2 ðax; kÞ cn2 ðax; kÞ
þ
k 21 b2 4
þ
½k 2 asnðax; kÞcnðax; kÞ þ k 21 bsnðby; k 1 Þcnðby; k 1 Þ ½dnðax; kÞ þ dnðby; k 1 Þ2 þ . R1 R21
ð2:4Þ
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
563
Fig. 1. Intensity jAj2 versus x, y for the first solution (2.1)–(2.4) of the Davey–Stewartson system, a = b = 1, a0 = 8, k2 = 0.5, k 21 ¼ 0.6, v = 2.
a, b are the wave numbers in the x and y directions respectively. k and k1 are the distinct, independent moduli of the Jacobi elliptic functions. The parameters c1, c2 measure the speeds of propagation in the x, y directions. The arbitrary constant a0 must be sufficiently large to avoid any singularity. To ensure the accuracy of the solutions, direct substitution in (1.3) and differentiation for solutions (2.1)–(2.4) by this separation of variables procedure are performed independently with the computer software Mathematica. As expected from the formulation (2.1)–(2.4), the solution is nonsingular, doubly periodic, and translating steadily in the x, y directions. There are clusters of smaller peaks within each period (Fig. 1). 2.2. Second solution The second solution is obtained by choosing the reciprocal of the Jacobi elliptic function dn as the building block for the formulation [8]. The profile for the intensity jAj2 resembles a sequence or a rectangular array of ÔplatesÕ, and is a kind of (2 + 1) (2 spatial and 1 temporal) dimensional ÔpeakonÕ solutions of nonlinear evolution equations. As the intensity profile of the wave pattern is continuous, but has a jump in the slope (or a sharp edge), we use the term ÔpeakonÕ [11–15] here as well. This usage is loose in the sense that no strict analytical comparison has been performed nor implied. A (1 + 1) dimensional ÔpeakonÕ will have a sharp corner. More precisely, the wave packet is now: rffiffiffi 2 1 expðiU 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A¼ ; v dnðax; kÞdnðby; k 1 Þ R2
ð2:5Þ
f f U 2 ¼ c1 x þ sin1 ½snðax; kÞ þ c2 y sin1 ½snðby; k 1 Þ; a b 1 cnðax; kÞ 1 cnðby; k 1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 R2 ¼ a0 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi sin1 dnðax; kÞ dnðby; k 1 Þ a 1k b 1 k2 1
a2 k 2 k 2 cn2 ðax; kÞsn2 ðax; kÞ c2 f2 dn2 ðax; kÞ 2 2 Q¼ 1 þ ðk 2Þsn ðax; kÞ þ 1 2 2 2 2 4dn ðax; kÞ 2 2 2 2 2 2 b k1 k cn ðby; k 1 Þsn ðby; k 1 Þ c f2 dn2 ðby; k 1 Þ 1 þ ðk 21 2Þsn2 ðby; k 1 Þ 1 þ 2 þ 2 2 2 2 4dn ðby; k 1 Þ
ð2:6Þ ð2:7Þ
564
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
ak 2 cnðax; kÞsnðax; kÞdn2 ðby; k 1 Þ þ bk 21 cnðby; k 1 Þsnðby; k 1 Þdn2 ðax; kÞ dn2 ðax; kÞdn2 ðby; k 1 ÞR2
þ
½dnðax; kÞ þ dnðby; k 1 Þ2 ; dn2 ðax; kÞdn2 ðby; k 1 ÞR22
ð2:8Þ
where x, y are defined by Eq. (1.5), c1, c2 are again the speeds of propagation. Fig. 2 shows that (2.5)–(2.8) represent a kind of ÔpeakonsÕ for DS. 3. Further investigations 3.1. Long wave limit To demonstrate that the present family of solutions is truly different from ones found earlier in the literature, an instructive perspective is to study the long wave limits (k, k1 going to 1) of (2.1)–(2.4). As sn, cn, dn tend to tanh, sech, sech respectively, one nonsingular limit is rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sech ax sech by 2 A¼ ð3:1Þ expðic1 x þ ic2 yÞ; v R10 R10 ¼ a0 þ Q¼
sin1 ðtanh axÞ sin1 ðtanh byÞ þ ; a b
a2 c2 b2 c2 ð1 3sech2 axÞ þ 1 þ ð1 3sech2 byÞ þ 2 8 2 8 2 ða tanh ax sech ax þ b tanh by sech byÞ ðsech ax þ sech byÞ2 þ . þ R10 R210
ð3:2Þ
ð3:3Þ
While jAj2 described by (3.1) is still exponentially localized, it differs from the known dromion of the Davey–Stewartson system [16,17] and deserves further studies.
Fig. 2. Intensity jAj2 versus x, y for the second solution (2.5)–(2.8) of the Davey–Stewartson system, a = b = 1, a0 = 8, k2 = 0.2, k 21 ¼ 0.3, v = 2.
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
565
3.2. Boundary conditions for the mean flow Q Another very important aspect to consider is the boundary condition for the mean flow Q, as the conventional localized dromion jAj2 is typically situated at the intersection(s) of the ÔtracksÕ provided by Q [16–19]. In deriving the localized solutions of the Davey–Stewartson equations in terms of the calculus of variations, a Hamiltonian integral will work smoothly for the wave packet A [16]. However, additional constraints must apply for the mean flow Q. More precisely, the cross sections of the mean flow Q in the far field positions must match. For (3.1)–(3.3), the asymptotic forms, Qjy!1 ¼
a2 c2 c2 b2 a tanh ax sech ax sech2 ax ð1 3sech2 axÞ þ 1 þ 2 þ þ þ ; R 8 2 2 8 R2
R ¼ a0 þ
sin1 ðtanh axÞ p ; a 2b
Qjy!þ1
a2 c2 c2 b2 a tanh ax sech ax sech2 ax ¼ ð1 3sech2 axÞ þ 1 þ 2 þ þ þ ; Rþ 8 2 2 8 R2þ
ð3:4Þ
sin1 ðtanh axÞ p þ a 2b are different (Figs. 3 and 4), and hence a Hamiltonian will not exist in this case. Stability of the localized solution (3.1) probably needs to be studied numerically. Rþ ¼ a0 þ
3.3. Semi-localized solutions An additional degree of freedom is to take the long wave limit in one of the moduli only (say k1 tends to one but still 0 < k < 1): rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dnðax; kÞsechby A¼ expðic1 x þ ic2 yÞ; R11 v 1 sin1 ðtanh byÞ R11 ¼ a0 þ sin1 ðsnðax; kÞÞ þ ; a b ! Q¼
k 2 a2 4
sn2 ðax; kÞ cn2 ðax; kÞ
k 2 sn2 ðax; kÞcn2 ðax; kÞ 2dn2 ðax; kÞ
þ
c21 c22 þ 2 2
b2 ½k 2 asnðax; kÞcnðax; kÞ þ b tanh by sech by ð1 3sech2 byÞ þ R11 8 ½dnðax; kÞ þ sech by2 þ . R211
þ
Fig. 3. The mean flow Q versus x for the limit y going to negative infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.
ð3:5Þ
566
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
Fig. 4. The mean flow Q versus x for the limit y going to positive infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.
Fig. 5. The intensity jAj2 versus x, y for a Ôsemi-localizedÕ solution (periodic in x, localized in y), Eq. (3.5), a0 = 4, a = 1, b = 2, k = 0.5, v = 2.
Physically this corresponds to wave patterns localized in one direction but periodic in the other direction (Fig. 5). The ÔpeakonÕ behavior is again evident as there are sharp edges where the derivatives will be discontinuous. Depending of the value of a0 the wave form generally is not symmetric for a cross section in a direction the pattern is localized (y in the present case).
4. Conclusions In conclusions two new exact solutions of the DS system are derived, and differ from some earlier works in the literature as these new solutions are doubly periodic (versus growing and decaying, or localized, or periodic in one direction [20,21]), steadily propagating (versus standing [22,23]) waves. Indeed the Jacobi elliptic functions have been employed earlier to obtain solutions for some (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations related to the DS system [24]. The difference in terms of the analytic structure is that rational functions are employed in the
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
567
present separation of variables approach, as opposed to a polynomial in Jacobi elliptic functions. Furthermore, inverse trigonometric functions of the elliptic functions are involved in the present formulation, and hence we believe that the present approach is novel. Solutions in the form of a ÔpeakonÕ, where derivatives might be discontinuous, are possible. In terms of applications, previously ignored physics, or higher order nonlinear effects, might be needed in the vicinity of the sharp edges. The long wave limits for these doubly periodic wave patterns yield an exponentially localized structure that differs from the conventional dromion. This aspect deserves further studies. As the mismatch in far field conditions shows that a Hamiltonian probably does not exist, the stability of these structures must be investigated numerically. Strictly speaking, the DS system considered in the present paper is the DS I system, as the mean flow Eq. (1.1) for Q is hyperbolic. We anticipate that similar solutions will also exist for DS II and other related evolution equations [24,25], but details will be left for future studies.
Acknowledgement Partial financial support has been provided by the Research Grants Council contracts HKU7184/04E and HKU7006/02E.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM; 1981. Nishinari K, Abe K, Satsuma J. J Phys Soc Japan 1993;62:2021. Hietarinta J. Phys Lett A 1991;149:133. Matveev VB, Salle MA. Darboux transformations and solitons. Berlin: Springer Verlag; 1991. Lou SY, Hu XB. J Phys A 1994;27:L207. Fokas AS, Santini PM. Phys Rev Lett 1989;63:1329. Boiti M, Leon J, Marina J, Penpinelli F. Phys Lett A 1988;132:432. Tang XY, Lou SY, Zhang Y. Phys Rev E 2002;66:046601. Abramowitz M, Stegun I. Handbook of mathematical functions. New York: Dover; 1964. Lawden DF. Elliptic functions and applications. Applied mathematical sciences, vol. 80. New York: Springer Verlag; 1989. Kraenkel RA, Senthilvelan M, Zenchuk AI. Phys Lett A 2000;273:183. Qian T, Tang M. Chaos, Solitons & Fractals 2001;12:1347. Johnson RS. Proc Royal Soc A 2003;459:1687. Vakhnenko VO, Parkes EJ. Chaos, Solitons & Fractals 2004;20:1059. Bai CL. J Phys Soc Japan 2004;73:37. Hietarinta J, Hirota R. Phys Lett A 1990;145:237. Nishinari K, Yajima T. Phys Rev E 1995;51:4986. Nishinari K, Yajima T, Nakao T. J Phys A 1996;29:4237. Yajima T, Nishinari K. J Phys Soc Japan 1998;67:1157. Tajiri M, Miura H, Arai T. Phys Rev E 2002;66:067601. Tajiri M, Takeuchi H, Arai T. Phys Rev E 2001;64:056622. Chow KW. J Math Phys 1995;36:4125. Chow KW. Wave Motion 2002;35:71. Yan Z. Chaos, Solitons & Fractals 2003;18:299. Zayed EME, Zedan HA, Gepreel KA. Chaos, Solitons & Fractals 2004;22:285.
Propagating wave patterns and ‘‘peakons’’ of the Davey–Stewartson system K.W. Chow b
a,*
, S.Y. Lou
b
a Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China
Accepted 31 March 2005
Abstract Two exact, doubly periodic, propagating wave patterns of the Davey–Stewartson system are computed analytically by a special separation of variables procedure. For the first solution there is a cluster of smaller peaks within each period. The second one consists of a rectangular array of ÔplatesÕ joined together by sharp edges, and is thus a kind of ÔpeakonsÕ for this system of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations. A long wave limit will yield exponentially localized waves different from the conventional dromion. The stability properties and nonlinear dynamics must await further investigations. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The Davey–Stewartson model (DS) is an important system of evolution equations, both from the perspectives of theory and applications. DS can arise in hydrodynamics [1] and plasma physics [2]. Theoretically, many techniques of the modern theory of nonlinear waves are relevant, e.g., special Hirota bilinear forms [3], Darboux transformations [4], symmetries [5], rich soliton and related structures [6,7]. We shall take DS as oA 1 o2 A o2 A i þ þ þ vA2 A ¼ QA; ot 2 on2 og2 ð1:1Þ o2 Q o2 Q o2 ¼ 2v ðAA Þ. on2 on2 og2 In the hydrodynamic context, A is the envelope of the wave packet while Q is the induced mean flow. New coordinates X, Y are defined by X þY n ¼ pffiffiffi ; 2
*
X Y g ¼ pffiffiffi 2
Corresponding author. Tel.: +852 2859 2641; fax: +852 2858 5415. E-mail addresses: [email protected] (K.W. Chow), [email protected] (S.Y. Lou).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.036
ð1:2Þ
562
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
and the transformed DS considered in the present work will be oA 1 o2 A o2 A þ vA2 A ¼ QA; þ i þ ot 2 oX 2 oY 2 2 o2 Q o o 2 ðAA Þ. ¼v þ oX oY oX oY
ð1:3Þ
Recently a class of novel, exact solutions of DS and other (2 + 1) (2 spatial and 1 temporal) dimensional nonlinear evolution equations can be obtained by a special separation of variables approach [8]. More precisely, one exact solution of the system (1.3) is p ¼ pðxÞ;
q ¼ qðyÞ;
p0 ¼ p0 ðxÞ;
q0 ¼ q0 ðyÞ;
r ¼ rðxÞ;
s ¼ sðyÞ;
2
pxx þ qyy ðpx þ qy Þ þ ; a0 þ p þ q ða0 þ p þ qÞ2 rffiffiffi pffiffiffiffiffiffiffiffiffi px qy 2 f expðiðr þ sÞÞ; rx ¼ c1 þ ; A¼ v a0 þ p þ q px Q ¼ p0 þ q0
p0 ¼
pxxx p2xx d c21 f2 2 þ 2; 4px 8px 8 2 2px
x ¼ X c1 t;
q0 ¼
sy ¼ c 2
f ; qy
ð1:4Þ
qyyy q2yy d c22 f2 2þ þ 2. 4qy 8qy 8 2 2qy
y ¼ Y c2 t
ð1:5Þ
are the coordinates translating with the wave pattern. a0, c1, c2, f are constants, and the relevant functions depend on the indicated variables only. More complicated solutions with terms involving products of p and q in the denominator can be constructed, but details will be left for future studies. The choice of exponential functions as the basis functions in (1.4) leads to generalized solutions of localized solitons or dromions. The purpose of the present note is to demonstrate that the choice of the Jacobi elliptic functions as building blocks (or p 0 (x), q 0 (y) in (1.4)) is feasible too, and will result in doubly periodic, propagating wave patterns for DS. Two constraints will dictate the choice of elliptic functions. Firstly, the building block functions need to be nonnegative as a square root is taken in the process. Simple choices like the functions sn and cn, which oscillate in a sinusoidal manner, must be rejected. Secondly, for analytical convenience, we restrict the attention to simple cases where both A and Q can be evaluated in simple, closed forms which do not involve the elliptic integrals in this paper. The selections of the Jacobi elliptic function dn [9,10] and its reciprocal will satisfy these requirements, and will now lead to these two new wave patterns for DS (Section 2). Further properties like the long wave limit and the boundary conditions will also be investigated (Section 3). 2. Doubly periodic wave patterns 2.1. First solution By choosing both basis functions p 0 (x), q 0 (y) in (1.4) in the x, y directions as ÔdnÕ, we obtain rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dnðax; kÞdnðby; k 1 Þ A¼ expðiU 1 Þ; v R1 f cnðax; kÞ f cnðby; k 1 Þ þ c2 y þ qffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 ; U 1 ¼ c1 x pffiffiffiffiffiffiffiffiffiffiffiffi2ffi sin1 dnðax; kÞ dnðby; k 1 Þ a 1k b 1 k2
ð2:1Þ ð2:2Þ
1
1 1 R1 ¼ a0 þ sin1 ½snðax; kÞ þ sin1 ½snðby; k 1 Þ; a b Q¼
k 2 a2 4
ð2:3Þ !
k 2 sn2 ðax; kÞcn2 ðax; kÞ c21 f2 þ 2 2dn2 ðax; kÞ 2dn2 ðax; kÞ ! k 2 sn2 ðby; k 1 Þcn2 ðby; k 1 Þ c2 f2 þ 2 sn2 ðby; k 1 Þ cn2 ðby; k 1 Þ 1 2 2 2 2dn ðby; k 1 Þ 2dn ðby; k 1 Þ
sn2 ðax; kÞ cn2 ðax; kÞ
þ
k 21 b2 4
þ
½k 2 asnðax; kÞcnðax; kÞ þ k 21 bsnðby; k 1 Þcnðby; k 1 Þ ½dnðax; kÞ þ dnðby; k 1 Þ2 þ . R1 R21
ð2:4Þ
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
563
Fig. 1. Intensity jAj2 versus x, y for the first solution (2.1)–(2.4) of the Davey–Stewartson system, a = b = 1, a0 = 8, k2 = 0.5, k 21 ¼ 0.6, v = 2.
a, b are the wave numbers in the x and y directions respectively. k and k1 are the distinct, independent moduli of the Jacobi elliptic functions. The parameters c1, c2 measure the speeds of propagation in the x, y directions. The arbitrary constant a0 must be sufficiently large to avoid any singularity. To ensure the accuracy of the solutions, direct substitution in (1.3) and differentiation for solutions (2.1)–(2.4) by this separation of variables procedure are performed independently with the computer software Mathematica. As expected from the formulation (2.1)–(2.4), the solution is nonsingular, doubly periodic, and translating steadily in the x, y directions. There are clusters of smaller peaks within each period (Fig. 1). 2.2. Second solution The second solution is obtained by choosing the reciprocal of the Jacobi elliptic function dn as the building block for the formulation [8]. The profile for the intensity jAj2 resembles a sequence or a rectangular array of ÔplatesÕ, and is a kind of (2 + 1) (2 spatial and 1 temporal) dimensional ÔpeakonÕ solutions of nonlinear evolution equations. As the intensity profile of the wave pattern is continuous, but has a jump in the slope (or a sharp edge), we use the term ÔpeakonÕ [11–15] here as well. This usage is loose in the sense that no strict analytical comparison has been performed nor implied. A (1 + 1) dimensional ÔpeakonÕ will have a sharp corner. More precisely, the wave packet is now: rffiffiffi 2 1 expðiU 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A¼ ; v dnðax; kÞdnðby; k 1 Þ R2
ð2:5Þ
f f U 2 ¼ c1 x þ sin1 ½snðax; kÞ þ c2 y sin1 ½snðby; k 1 Þ; a b 1 cnðax; kÞ 1 cnðby; k 1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffi sin1 R2 ¼ a0 pffiffiffiffiffiffiffiffiffiffiffiffi2ffi sin1 dnðax; kÞ dnðby; k 1 Þ a 1k b 1 k2 1
a2 k 2 k 2 cn2 ðax; kÞsn2 ðax; kÞ c2 f2 dn2 ðax; kÞ 2 2 Q¼ 1 þ ðk 2Þsn ðax; kÞ þ 1 2 2 2 2 4dn ðax; kÞ 2 2 2 2 2 2 b k1 k cn ðby; k 1 Þsn ðby; k 1 Þ c f2 dn2 ðby; k 1 Þ 1 þ ðk 21 2Þsn2 ðby; k 1 Þ 1 þ 2 þ 2 2 2 2 4dn ðby; k 1 Þ
ð2:6Þ ð2:7Þ
564
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
ak 2 cnðax; kÞsnðax; kÞdn2 ðby; k 1 Þ þ bk 21 cnðby; k 1 Þsnðby; k 1 Þdn2 ðax; kÞ dn2 ðax; kÞdn2 ðby; k 1 ÞR2
þ
½dnðax; kÞ þ dnðby; k 1 Þ2 ; dn2 ðax; kÞdn2 ðby; k 1 ÞR22
ð2:8Þ
where x, y are defined by Eq. (1.5), c1, c2 are again the speeds of propagation. Fig. 2 shows that (2.5)–(2.8) represent a kind of ÔpeakonsÕ for DS. 3. Further investigations 3.1. Long wave limit To demonstrate that the present family of solutions is truly different from ones found earlier in the literature, an instructive perspective is to study the long wave limits (k, k1 going to 1) of (2.1)–(2.4). As sn, cn, dn tend to tanh, sech, sech respectively, one nonsingular limit is rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sech ax sech by 2 A¼ ð3:1Þ expðic1 x þ ic2 yÞ; v R10 R10 ¼ a0 þ Q¼
sin1 ðtanh axÞ sin1 ðtanh byÞ þ ; a b
a2 c2 b2 c2 ð1 3sech2 axÞ þ 1 þ ð1 3sech2 byÞ þ 2 8 2 8 2 ða tanh ax sech ax þ b tanh by sech byÞ ðsech ax þ sech byÞ2 þ . þ R10 R210
ð3:2Þ
ð3:3Þ
While jAj2 described by (3.1) is still exponentially localized, it differs from the known dromion of the Davey–Stewartson system [16,17] and deserves further studies.
Fig. 2. Intensity jAj2 versus x, y for the second solution (2.5)–(2.8) of the Davey–Stewartson system, a = b = 1, a0 = 8, k2 = 0.2, k 21 ¼ 0.3, v = 2.
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
565
3.2. Boundary conditions for the mean flow Q Another very important aspect to consider is the boundary condition for the mean flow Q, as the conventional localized dromion jAj2 is typically situated at the intersection(s) of the ÔtracksÕ provided by Q [16–19]. In deriving the localized solutions of the Davey–Stewartson equations in terms of the calculus of variations, a Hamiltonian integral will work smoothly for the wave packet A [16]. However, additional constraints must apply for the mean flow Q. More precisely, the cross sections of the mean flow Q in the far field positions must match. For (3.1)–(3.3), the asymptotic forms, Qjy!1 ¼
a2 c2 c2 b2 a tanh ax sech ax sech2 ax ð1 3sech2 axÞ þ 1 þ 2 þ þ þ ; R 8 2 2 8 R2
R ¼ a0 þ
sin1 ðtanh axÞ p ; a 2b
Qjy!þ1
a2 c2 c2 b2 a tanh ax sech ax sech2 ax ¼ ð1 3sech2 axÞ þ 1 þ 2 þ þ þ ; Rþ 8 2 2 8 R2þ
ð3:4Þ
sin1 ðtanh axÞ p þ a 2b are different (Figs. 3 and 4), and hence a Hamiltonian will not exist in this case. Stability of the localized solution (3.1) probably needs to be studied numerically. Rþ ¼ a0 þ
3.3. Semi-localized solutions An additional degree of freedom is to take the long wave limit in one of the moduli only (say k1 tends to one but still 0 < k < 1): rffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dnðax; kÞsechby A¼ expðic1 x þ ic2 yÞ; R11 v 1 sin1 ðtanh byÞ R11 ¼ a0 þ sin1 ðsnðax; kÞÞ þ ; a b ! Q¼
k 2 a2 4
sn2 ðax; kÞ cn2 ðax; kÞ
k 2 sn2 ðax; kÞcn2 ðax; kÞ 2dn2 ðax; kÞ
þ
c21 c22 þ 2 2
b2 ½k 2 asnðax; kÞcnðax; kÞ þ b tanh by sech by ð1 3sech2 byÞ þ R11 8 ½dnðax; kÞ þ sech by2 þ . R211
þ
Fig. 3. The mean flow Q versus x for the limit y going to negative infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.
ð3:5Þ
566
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
Fig. 4. The mean flow Q versus x for the limit y going to positive infinity, Eq. (3.4), a0 = 4, a = 1, b = 2, v = 2.
Fig. 5. The intensity jAj2 versus x, y for a Ôsemi-localizedÕ solution (periodic in x, localized in y), Eq. (3.5), a0 = 4, a = 1, b = 2, k = 0.5, v = 2.
Physically this corresponds to wave patterns localized in one direction but periodic in the other direction (Fig. 5). The ÔpeakonÕ behavior is again evident as there are sharp edges where the derivatives will be discontinuous. Depending of the value of a0 the wave form generally is not symmetric for a cross section in a direction the pattern is localized (y in the present case).
4. Conclusions In conclusions two new exact solutions of the DS system are derived, and differ from some earlier works in the literature as these new solutions are doubly periodic (versus growing and decaying, or localized, or periodic in one direction [20,21]), steadily propagating (versus standing [22,23]) waves. Indeed the Jacobi elliptic functions have been employed earlier to obtain solutions for some (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations related to the DS system [24]. The difference in terms of the analytic structure is that rational functions are employed in the
K.W. Chow, S.Y. Lou / Chaos, Solitons and Fractals 27 (2006) 561–567
567
present separation of variables approach, as opposed to a polynomial in Jacobi elliptic functions. Furthermore, inverse trigonometric functions of the elliptic functions are involved in the present formulation, and hence we believe that the present approach is novel. Solutions in the form of a ÔpeakonÕ, where derivatives might be discontinuous, are possible. In terms of applications, previously ignored physics, or higher order nonlinear effects, might be needed in the vicinity of the sharp edges. The long wave limits for these doubly periodic wave patterns yield an exponentially localized structure that differs from the conventional dromion. This aspect deserves further studies. As the mismatch in far field conditions shows that a Hamiltonian probably does not exist, the stability of these structures must be investigated numerically. Strictly speaking, the DS system considered in the present paper is the DS I system, as the mean flow Eq. (1.1) for Q is hyperbolic. We anticipate that similar solutions will also exist for DS II and other related evolution equations [24,25], but details will be left for future studies.
Acknowledgement Partial financial support has been provided by the Research Grants Council contracts HKU7184/04E and HKU7006/02E.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. Philadelphia: SIAM; 1981. Nishinari K, Abe K, Satsuma J. J Phys Soc Japan 1993;62:2021. Hietarinta J. Phys Lett A 1991;149:133. Matveev VB, Salle MA. Darboux transformations and solitons. Berlin: Springer Verlag; 1991. Lou SY, Hu XB. J Phys A 1994;27:L207. Fokas AS, Santini PM. Phys Rev Lett 1989;63:1329. Boiti M, Leon J, Marina J, Penpinelli F. Phys Lett A 1988;132:432. Tang XY, Lou SY, Zhang Y. Phys Rev E 2002;66:046601. Abramowitz M, Stegun I. Handbook of mathematical functions. New York: Dover; 1964. Lawden DF. Elliptic functions and applications. Applied mathematical sciences, vol. 80. New York: Springer Verlag; 1989. Kraenkel RA, Senthilvelan M, Zenchuk AI. Phys Lett A 2000;273:183. Qian T, Tang M. Chaos, Solitons & Fractals 2001;12:1347. Johnson RS. Proc Royal Soc A 2003;459:1687. Vakhnenko VO, Parkes EJ. Chaos, Solitons & Fractals 2004;20:1059. Bai CL. J Phys Soc Japan 2004;73:37. Hietarinta J, Hirota R. Phys Lett A 1990;145:237. Nishinari K, Yajima T. Phys Rev E 1995;51:4986. Nishinari K, Yajima T, Nakao T. J Phys A 1996;29:4237. Yajima T, Nishinari K. J Phys Soc Japan 1998;67:1157. Tajiri M, Miura H, Arai T. Phys Rev E 2002;66:067601. Tajiri M, Takeuchi H, Arai T. Phys Rev E 2001;64:056622. Chow KW. J Math Phys 1995;36:4125. Chow KW. Wave Motion 2002;35:71. Yan Z. Chaos, Solitons & Fractals 2003;18:299. Zayed EME, Zedan HA, Gepreel KA. Chaos, Solitons & Fractals 2004;22:285.