homoclinic solutions for davey-stewartson equation
TRANSCRIPT
Chaos, Solitons and Fractals 35 (2008) 996–1002
www.elsevier.com/locate/chaos
Homoclinic solutions for Davey-Stewartson equation
Jian Huang a,*, Zhengde Dai b
a School of Management Science and Engineering, Nanjing University, Nanjing 210093, PR Chinab School of Mathematics and Physics, Yunnan University, Kunming 650091, PR China
Accepted 6 June 2006
Abstract
In this paper, we firstly prove the existence of homoclinic solutions for Davey-Stewartson I equation (DSI) with theperiodic boundary condition. Then we obtain a set of exact homoclinic solutions by the novel method-Hirota’s method.Moreover, the structure of homoclinic solutions has been investigated. At the same time, we give some numerical sim-ulations which validate these theoretical results.� 2006 Elsevier Ltd. All rights reserved.
1. Introduction
It is well known that the existence of homoclinic solution is very important to the study of complex dynamics inpartial differential equations. Recently, a set of methods were developed to prove the existence of homoclinic solutionsin some nonlinear evolutionary equations (NEE) such as nonlinear Schrodinger equation [1], Sine-Gordon equation [2],long-short wave equation [3], DS II equation [4] and Boussinesq equations [5]. Especially, in [4] and [5], we have estab-lished a novel method by which we found exact homoclinic solutions for DS II and Boussinesq equation. At the sametime, we have found periodic and soliton solutions for Boussinesq equations [5]. The outline of this method used in [4]and [5] is as following: Firstly, the fixed points or cycles of NEE are proved to be hyperbolic, which showes that thefixed points or cycles are saddle points or cycles. Secondly, by linearized stability analysis, the fixed points or cyclesare proved to be linear unstable. Therefore, these two steps prove whether homoclinic solutions exist in NEE fromthe theoretic aspect. Finally, by using the Hirota’s bilinear method, the exact homoclinic solution is obtained. Today,finding the exact homoclinic solution is an important way to explore the complex dynamics for NEE. Among NEE,Davey-Stewartson equation I (DSI) is an important equation to both physical and mathematical researches. In thispaper, we will focus only on DSI in this paper. By our new method, we firstly prove the existence of homoclinic solu-tions. Moreover, we obtain the exact expressions of homoclinic solutions and investigate the properties of homoclinicsolutions. In the coming researches, we will investigate other types of DSII.
DSI equation is as following
0960-0doi:10
* CoE-m
iqt þ qxx þ qyy ¼ �jqj2qþ qu
uxx � uyy ¼ 2ðjqj2Þxx
ð1:1Þ
779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved..1016/j.chaos.2006.06.022
rresponding author.ail address: [email protected] (J. Huang).
J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002 997
where q:Rx · Ry · Rt! C and u:Rx · Ry · Rt! R. Systems (1.1) and (1.2) were derived by Davey et al. [6] to model theevolution of a three-dimensional disturbance in the nonlinear regime of plane Poiseuille flow (fully developed steadyflow under a constant pressure gradient between fixed parallel planes). q(t,x,y) stands for the complex amplitudeand u(t,x,y) describes the perturbation of the real velocity.
Supplement with boundary conditions
uðx; y; tÞ ¼ uðxþ l1; y þ l2; tÞ ð1:2Þ
and initial conditions
uðx; y; 0Þ ¼ u0ðx; yÞ; u0ð�x;�yÞ ¼ u0ðx; yÞ x; y 2 X ð1:3Þ
The rest of this paper is organized as following: Section 2 gives the hyperbolic and linear stability analysis of fixedcycles for DSI. In Section 3, we obtain the exact homoclinic solutions by Hirota’s method and explore the structurecharacteristics of these homoclinic solutions. At the same time, we give some numerical simulations which validate thesetheoretical results. A conclusion is given in the end.
2. Homoclinic solutions for DSI
2.1. Hyperbolic analysis of fixed cycles
Fixed cycle is the extended form of fixed point in higher dimensional spaces. In this subsection, DSI will be proved tohave hyperbolic fixed cycles [7,8].
Denoting q = q1 + iq2, Eq. (1.1) can be transformed into the following equations
q1t þ q2xx þ q2yy þ q21q2 þ q3
2 � q2u ¼ 0
q2t � q1xx � q1yy � q1q22 � q3
1 þ q1u ¼ 0
uxx � uyy ¼ 4q21x þ 4q1q1xx þ 4q2
2x þ 4q2q2xx
ð2:1Þ
Obviously, ðaeia2 t; 0Þ is a fixed cycle. We linearize Eq. (2.1) on cycle ðaeia2t; 0Þ
Q1t � k21Q2 � k2
2Q2 þ 2q1Q1q2 þ q21Q2 þ 3q2
2Q2 � Q2u� q2/ ¼ 0
Q2t þ k21Q1 þ k2
2Q1 � 3q21Q1 � Q1q2
2 � 2q2q1Q2 þ Q1uþ q1/ ¼ 0
k22/� k2
1/ ¼ 8q1xQ1x þ 4Q1q1xx þ 4q1Q1xx þ 8q2xQ2x þ 4Q2q2xx þ 4q2Q2xx
ð2:2Þ
where we consider a simple case, i.e., only one wave number k1 (k2) in the x direction (y direction), the qi, u are theeigenfunctions of a spatial operator (oxx,oyy) around the cycle which has the form
� oxxqi ¼ k21qi; �oxxu ¼ k2
1u
� oyyqi ¼ k22qi; �oyyu ¼ k2
2u ði ¼ 1; 2Þ
Substituting / ¼ ðk22 � k2
1Þ�1ð�4k2
1Q1q1 � 4q2k21Q2Þ into (2.2) leads to
Q1t ¼ ðk21 þ k2
2 � a2 � 2a2 sin2 a2tÞQ2 þ4k2
1a2 sin2 a2t
k21 � k2
2
Q2 � 2a2 sin a2t cos a2tQ1 þ4k2
1a2 sin a2t cos a2t
k21 � k2
2
Q1
Q2t ¼ �ðk21 þ k2
2 � a2 � 2a2 cos2 a2tÞQ1 �4k2
1a2 cos2 a2t
k21 � k2
2
Q1 þ 2a2 sin a2t cos a2tQ2 �4k2
1a2 sin a2t cos a2t
k21 � k2
2
Q2
ð2:3Þ
Then, the eigenvalue matrix of (2.3) is
�2a2 sin a2t cos a2t þ 4k21a2 sin a2t cos a2t
k21 � k2
2
k21 þ k2
2 � a2 � 2a2 sin2 a2t þ 4k21a2 sin2 a2t
k21 � k2
2
�k21 þ k2
2 � a2 � 2a2 cos2 a2tÞ � 4k21a2 cos2 a2t
k21 � k2
2
2a2 sin a2t cos a2t � 4k21a2 sin a2t cos a2t
k21 � k2
2
0BBB@
1CCCA ð2:4Þ
Matrix (2.4) has the following eigenvalues
k2 ¼ k21 þ k2
2 � a2� �
k42 � k4
1 � 3a2k22 � a2k2
1
� �ð2:5Þ
998 J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002
Assuming k1 = kk2 (0 < k < 1), (2.5) becomes
k2 ¼ ðk2 þ 1Þk22 � a2
� �ð1� k4Þk2
2 � 3a2 � a2k2� �
ð2:6Þ
In order to obtain k2 > 0, the following inequalities must be satisfieda2 � ðk2 þ 1Þk22 > 0
a2k2 þ 3a2 þ ðk4 � 1Þk22 > 0
ð2:7Þ
From (2.7), we obtain
k22 <
a2
k2 þ 1
k22 <
a2ðk2 þ 3Þ1� k4
ð2:8Þ
Since 0 < k < 1, we have
k22 <
a2
k2 þ 1
Assume k2 = p2n (n is an positive integer), we obtain
n2 <a2
ðk2 þ 1Þp22
Especially, when p2 = 1, denoting N as the maximal module of n, the above inequality can be transformed into thefollowing
0 < N 2 <a2
k2 þ 1
i.e.,
0 < N <jajffiffiffiffiffiffiffiffiffiffiffiffiffi
k2 þ 1p
Therefore, the fixed cycle ðaeia2 t; 0Þ is hyperbolic when 0 < N < jajffiffiffiffiffiffiffik2þ1p .
2.2. Linearized stability analysis
In this subsection, we investigate the linear stability of fixed cycles ðaeia2t; 0Þ by considering a small perturbation ofthe form
q ¼ aeia2tð1þ Qeðx; y; tÞÞu ¼ ueðx; y; tÞ
ð2:9Þ
where jQe(x,y, t)j � 1, jue(x,y, t)j � 1. Substituting (2.9) into (1.1), we get linearized equations
iQet þ Qexx þ Qeyy ¼ �a2Q� a2Q� þ u
uexx � ueyy ¼ 2a2Qexx þ 2a2Q�exx
ð2:10Þ
where * denote the conjugation. We assume that Qe and ue have the following forms
Qe ¼ Aeiðlnxþ�lnyÞþrnt þ Be�iðlnxþ�lnyÞþrnt
ue ¼ Cðeiðlnxþ�lnyÞþrnt þ e�iðlnxþ�lnyÞþrntÞð2:11Þ
where A, B are complex constants and C is real, ln = p1n, �ln ¼ p2n and rn is the growth rate of the nth mode.Substitution (2.11) into (2.10) leads to
irn � l2n � �l2
n þ a2� �
A ¼ �a2B� þ C
irn � l2n � �l2
n þ a2� �
B ¼ �a2A� þ C
� Cl2n þ C�l2
n ¼ 2a2Al2n þ 2a2B�l2
n
Cl2n þ C�l2
n ¼ 2a2Bl2n þ 2a2A�l2
n
ð2:12Þ
J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002 999
Solving the equations of (2.12), we obtain
r2n ¼ a4 � a2 � l2
n � �l2n
� �2 ¼ l2n þ �l2
n
� �2a2 � l2
n � �l2n
� �ð2:13Þ
Since r2n > 0, we obtain
2a2 > l2n þ �l2
n
Assume ln ¼ k3�ln, k3 = p1/ p2
�l2n <
2a2
1þ k23
i.e.,
n2 <2a2
ð1þ k23Þp2
2
Thus the number of unstable modes which determines the complexity of the homoclinic structure is given by thefollowing largest integer N
0 < N <
ffiffiffi2pjaj
jp2jffiffiffiffiffiffiffiffiffiffiffiffiffi1þ k2
3
q ð2:14Þ
Especially, when p2 = 1
0 < N <
ffiffiffi2pjajffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ k23
q ð2:15Þ
3. Homoclinic solutions
In Sections 1, we have proved the existence of homoclinic solutions in DSI. In this section, we will obtain the exacthomoclinic solutions by Hirota’s method. Using this method, we have successfully obtained the periodic, homoclinicand soliton solutions for some equations such as Boussinesq equation, Zakharov system, DSII, coupled nonlinearSchrodinger system and long-short wave equation [3–5,10].
We form the main results into Theorem 1 and Theorem 2.
Theorem 1. There exists homoclinic solutions for Eq. (1.1) like
qj ¼ aeia2 t 1þ b1ðeip1xþip2y þ e�ip1x�ip2yÞeXj tþc þ b3e2Xj tþ2c
1þ b4ðeip1xþip2y þ e�ip1x�ip2yÞeXj tþc þ b5e2Xj tþ2c
uj ¼ �8p2 þ 2b2
4p2ðeip1xþip2y þ e�ip1x�ip2yÞðe�Xjt�c þ b5eXj tþcÞ½e�Xjt�c þ b4ðeip1xþip2y þ e�ip1x�ip2yÞ þ b5eXj tþc�2
j ¼ 1; 2
ð3:1Þ
where a, b1, b3, b4, b5, p1, p2, X1, X2 satisfy
k ¼ �a2; b1 ¼ b2 ¼iXþ p2
1 þ p22 þ p2
2
iX� p21 � p2
2
b4; b3 ¼iXþ p2
1 þ p22 þ p2
2
iX� p21 � p2
2
� �2
b5; b5 ¼X2 þ ðp2
1 þ p22Þ
2
X2b2
4;
X1 ¼ðp2
1 þ p22Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a2 � ðp2
2 � p21Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
2 � p21
p ;X2 ¼ �X1; p1 ¼ k4 sin w0; p2 ¼ sin w0; p2 <
ffiffiffi2pjajffiffiffiffiffiffiffiffiffiffiffiffiffi
1� k24
qk4 ¼ ln=�ln:
ð3:2Þ
Proof. Substituting q ¼ aeia2 tQðx; y; tÞ into (1.1) leads to
iQt þ ðQxx þ QyyÞ ¼ �a2jQj2Qþ a2Qþ Qu
uxx � uyy ¼ 2jaj2ðjQj2Þxx
ð3:3Þ
1000 J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002
By the dependent variable transformation
Q ¼ G=F ; u ¼ �4ðln F Þxx ð3:4Þ
where G is complex and F is real. Inserting (3.4) into (1.1) leads to the following bilinear formiDtG � F þ D2xG � F þ D2
y G � F � a2G � F ¼ kG � FD2
y F � F � D2xF � F � a2GG� ¼ kFF
ð3:5Þ
where k is a real constant and the Hirota bilinear operators Dmx Dk
t are defined as follows [9]
Dmx Dk
t a � b ¼ o
ox� o
ox0
� �mo
ot� o
ot0
� �k
aðx; tÞbðx0; t0Þjx0¼x;t0¼t:
Assume G and F as the following
G ¼ 1þ ðb1eip1xþip2y þ b2e�ip1x�ip2yÞeXtþc þ b3e2Xtþ2c
F ¼ 1þ b4ðeip1xþip2y þ e�ipx�ip2yÞeXtþc þ b5e2Xtþ2cð3:6Þ
where a, p, X, c, b4, b5 are real and b1, b2, b3 are complex. Substitution (3.6) into (3.5) leads to the following relationsamong these constants [10]
k ¼ �a2; b1 ¼ b2 ¼iXþ p2
1 þ p22
iX� p21 � p2
2
b4
b3 ¼iXþ p2
1 þ p22
iX� p21 � p2
2
� �2
b5; b5 ¼X2 þ ðp2
1 þ p22Þ
2
X2b2
4
X1 ¼ðp2
1 þ p22Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a2 � ðp2
2 � p21Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
2 � p21
p ; X2 ¼ �X1
p1 ¼ k sin w0; p2 ¼ sin w0
ð3:7Þ
where w0 is constant. From2a2�ðp2
2�p2
1Þ
p22�p2
1
> 0, we get p2 <ffiffi2pjajffiffiffiffiffiffiffi
1�k2p . Then (1.1) has homoclinic solutions like
qj ¼ aeia2 t 1þ b1ðeip1xþip2y þ e�ip1x�ip2yÞeXj tþc þ b3e2Xjtþ2c
1þ b4ðeip1xþip2y þ e�ip1x�ip2yÞeXj tþc þ b5e2Xjtþ2c
uj ¼ �8p2 þ 2b2
4p2ðeip1xþip2y þ e�ip1x�ip2yÞðe�Xj t�c þ b5eXj tþcÞ½e�Xjt�c þ b4ðeip1xþip2y þ e�ip1x�ip2yÞ þ b5eXjtþc�2
j ¼ 1; 2
ð3:8Þ
where a, b1, b3, b4, b5, p1, p2, X1, X2 satisfy (3.4) [Fig. 1]. h
We show the further structures property of homoclinic solutions in Theorem 2.
020
4060
800
20
40
60
80
100
0.80.91
1.1
020
4060
80100
a
0 20 40 60 80 1000
20
40
60
80
100
b
Fig. 1. (a) The graph of q1(x,y) with 0 < x,y < 100. (b) The contour of q1(x,y).
0 0.5 1 1.5 2
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
Fig. 2. The polar coordinates phase of homoclinic tubes.
J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002 1001
Theorem 2. There exists a spatial shift between homoclinic solutions (q1,u1) and (q2,u2), i.e., if (q1(x0,y0, t),u1(x0,y0, t)) is
a homoclinic solution, ðq1ðx0 þ pp1; y0 þ p
p2; tÞ;u1ðx0 þ p
p1; y0 þ p
p2; tÞÞ is also a homoclinic solution. Thus (q1,u1) and (q2,u2)
form a pair of symmetric homoclinic orbits and all of these symmetric homoclinic orbit pairs construct the homoclinic tubes.
Proof. In the follows, we will investigate the homoclinic tubes structure of solutions. The solutions given by (3.7) rep-resent homoclinic solutions converging to the fixed periodic cycles respectively, i.e.,
ðq1;u1Þ !iXþ p2
1 þ p22
iX� p21 � p2
2
� �2
aeia2t; 0
!; ðq2;u2Þ ! ðaeia2 t; 0Þ; t ! þ1;
ðq1;u1Þ ! ðaeia2 t; 0Þ; ðq2;u2Þ !iXþ p2
1 þ p22
iX� p21 � p2
2
� �2
aeia2 t; 0
!; t! �1:
From
b3
b5
¼ iXþ p21 þ p2
2
iX� p21 � p2
2
� �2
¼ e2ih
we get
h ¼ arctg2X1 p2
1 þ p22
� �ðp2
1 þ p22Þ
2 � X21
ðfor q1Þ
and
h ¼ arctg2X1 p2
1 þ p22
� �X2
1 � ðp21 þ p2
2Þ2ðfor q2Þ:
Therefore, we find that there exists a phase shift between homoclinic solutions (q1,u1) and (q2,u2). If
(q1(x0,y0, t),u1(x0,y0, t)) is a homoclinic solution, then q1 x0 þ pp1; y0 þ p
p2; t
;u1 x0 þ p
p1; y0 þ p
p2; t
is another homo-
clinic solution. Thus (q1,u1) and (q2,u2) form a pair of symmetric homoclinic orbits respectively and all of these con-struct the homoclinic tubes [Fig. 2]. h
4. Conclusion
In this paper, by the novel method as in [4] and [5], we have proved the existence of homoclinic solutions for DSI.Moreover, by the Hirota’s bilinear method, we have get a new class of exact homoclinic solutions which forms homo-clinic tubes. At the same time, we also give some numerical simulations which validate these analytical results. Thesehomoclinic solutions will be important to the dynamics investigation of DSI equations in both theoretic and practicalareas. However, we still do not know whether DS equations including DSI and DSII have heterocilinc solutions.Whether the perturbed DS equations still have homoclinic solutions. These questions are our coming researches inthe future.
1002 J. Huang, Z. Dai / Chaos, Solitons and Fractals 35 (2008) 996–1002
Acknowledgement
This work is supported by the National Natural Science Fund of China No.10361007 and Yunnan Natural ScienceFoundation Grant No.2004A0001M.
References
[1] Li Y. Smale horseshoes and symbolic dynamics in perturbed nonlinear Schrodinger equation. J Nonlinear Sci 1999;9(4):363.[2] Shatah J, Zeng C. Homoclinic orbits for the perturbed Sine-Gordon equation. Comm Pure Appl Math 2000;53(3):283–99.[3] Gao Ping, Guo Boling. Homoclinic orbits for the coupled nonlinear Schrodinger system and long-short wave equation. Phys Lett
A 2005;340(1–4):209–11.[4] Dai ZD, Huang J. Homoclinic tubes for the Davey-Stewartson II equation with periodic boundary condition. Chinese J Phys
2005;43(2).[5] Dai ZD, Huang J, Jiang MR, Wang SH. Homoclinic orbits and periodic solitons for Boussinesq equation with even constraint.
Chaos, Solitons & Fractals 2005;26(4):1189–94.[6] Davey A, Hoching LM, Stewartson K. On the nonlinear evolution of three-dimensional disturbances in plane Poiseuille flow. J
Fluid Mech 1974;63(3):529–36.[7] Ablowitz MJ, Hernst BM. SIAM J Appl Math 1990;50:339.[8] Ablowitz MJ, Herbst BM, Schober CM. J Comput Phys 1996;126:299.[9] Hirota R, Satsuma J. Prog Theor Phys Suppl 1976;59:64.
[10] Dai ZD, Huang J, Jiang MR. Homoclinic solutions and structure for Davey-Stewartson I Equation, working paper. 2005.