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New Exact Solutions of the CDGSK Equation Related to a Non-local Symmetry
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1994 Chinese Phys. Lett. 11 593 (http://iopscience.iop.org/0256-307X/11/10/001) View the table of contents for this issue, or go to the journal homepage for more
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New Exact Solutions of the CDGSK Equation Related to a Non-local Symmetry* LOU Senyue Fudan-T. D. Lee Physics Laboratory, Fudan University, Shanghai 200433; and Lnstitute of Modern Physics, Ningbo Normal College, Ningbo 315211 (mailing address)
RUAN Hangyu, CHEN Weizhong, WANG Zhenli, CHEN Lili Institute of Modern Physics, Ningbo Normal College, Ningbo 315211
(Received 6 January 1994) A non-local symmetry of the Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK) equation has been used for finding exact solution in two different ways. Firstly, using the standard prolongation approach, we obtain the finite Lie BIcklund transformation and the single soliton solution. Secondly, combining some local symmetries and the nonlocal symmetry, we get the group invariant solution which is described by the Weierstrass elliptic function and is deduced to the so-called interacting soliton for a special parameter.
PACS: 03.40. -t, 02.20. -a Symmetries, including both local and non-local ones, play an important role in the construction of solutions of nonlinear partial differential equations (PDEs).' They transform given solutions of PDEs to new ones. Also one can find the invariant solutions under the symmetry transformations, Furthermore, using these symmetries, one can easily obtain the redutions of PDEs by both classical' and non-classical2 Lie group approaches. For the sake of obtaining the symmetry of PDE, the recursion operator @ and its inversion 0-l are widely applied as they generate the new symmetry by acting them on a given one. Recently, a lot of explicit inverse recursion operators have been fortunately found for some well known equations, such as the KdV,3 mKdV,4 AKNS5 and Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK) equations.6 Acting these @-' on the trivial symmetry g o = 0, which corresponds to the identity transformation, and taking D-lO / Odx = constf 0, a great number of new non-local ~ y m m e t r i e s ~ ~ ~ ~ ~ have been written in the wake of new inverse recursions for these equations. Now an important question is: can we get some new information, say,exact solutions, for the integrable models corresponding to these non-local symmetries? In Refs.7 and 8, the authors have get the single soliton solutions by means of the finite Lie Backlund transformation corresponding to some non-local symmetries for the KdV. Harry Dym and non-local Schrodinger equations. In this letter,we devote not only the finite transformation but also the similarity reductions for the CDGSK equationg ut
5 + -uzuzr +5 2
5 = Ko(u) = ~ z z z z z -2~ u z z z
+
4212%
,
* Supported by the Natural Science Foundation of Zhejiang Province, and the National Natural Science Foundation of China. @by the Chinese Physical Society
(1)
LOU Senyue et al.
594
Vol. 11
using its simplest non-local symmetry6
MO = Dg, g
G
1 exp(-D-lf) 2
,
(2)
where the pseudopotential, f ( x ,t)l is determined by the Riccati equations:
The integrability condition of Eqs. (3) and (4), fit = ft,, is just the CDGSK equation (1). It is fortunate that the prolongation system corresponding to the non-local symmetry (2)can be easily closed if the other two pseudopotentials p ( z , t ) and q(z,t ) are introduced as follows
1
Px=Zf
,
4+=9.
(5)
Now the symmetry (2) for single equation (1) should be replaced by
1 1 MO = ( 2-fexPp, - 3-expp, - -4, 6
- 124
2 T
)
,
for the prolonged equation system ( l ) ,(3) and (5), where the superscript T means the transposition of the matrix. That is to say, the prolongation corresponding t o symmetry (2)has the form
To obtain the finite transformations corresponding to the symmetry (6), we should solve the “initial” problem:
’.’
with the ”initial” condition (ii,f,p , q ) e = o = ( u l f,p , q ) . After solving Eq. (8), the finite transformations of U and pseudopotentials under G o can be written out
-
f = f - 4expp-
E
Eq
+ 12 ’
ij;
= p - 21n-
Eq+
12
12
,
-
4=-
1% 12 E q
+
(9)
Now starting from any initial solution, we can get a new solution using the finite Lie Backlund transformation (9). For instance, starting from the finite transformation (9) and the trivial solution of U , U = const = -2k2, with the corresponding initial pseudopotentials, f = 2ktanhC p = In cosh( and q = k-’sinhC(C = k x k ’ t ) , we get the single soliton solution of the CDGSK equation (1):
+
U
= -2k2
+ 12k~tanhC12sechC +1Ek-ltanhC
+
1 12E2(Ek-1tanh( 12sechC l2
+
(10)
LOU Senyue et al.
No. 10
with
E
595
and IC being two arbitrary constants.
On the other hand, starting from any one symmetry U of an equation system, we can get a corresponding group invariant solution (similarity reduction) by solving the symmetry constraint condition U = 0 and the original equation system. Now using the non-local symmetry MO and the two local symmetries U, and ut, we would like to search the group invariant solutions by solving the symmetry constraint condition
To solve Eq. (11) is equivalent to solve the following characteristic equation system du df (1/2)fexpp -(1/3)exPP where
a0
and
a1
-
dP dq -(1/6h -(1/12)q2
- -dx- --a0
dt ---a1 '
(12)
are constants.
After finishing the integrations, the solutions of Eq. (12) can be written as
where Ci = Ci((), (i = 1,2,3,4) and ( = a1x - aot are five group invariants. Substituting = 1,2,3,4) satisfy the ordinary Eq.(13) into Eqs. (l),(3) and (5), we see that Ci = Ci((),(i differential equation system
C1 = a,'(F - 1) ,
C2
= 12aoF, C3 = 2al(lnCz)t
,
C4 = -alCq
1 + -Ci . 4
(14)
In other words, we have obtained a kind of reduction equations (14) for the CDGSK equation (I), in which four group invariants C,(()(i = 1,2,3,4) instead of only one in the direct reduction approachlo are included. Whence the first of Eq. (14) is solved, the group invariants Ci(()(i= I,&,3,4) follow immediately from the other equations of Eq. (14). Fortunately, the F equation & can be solved by means of the Weierstrass elliptic function:
where the Weierstrass elliptic function P(77,92,93) with two arbitrary constants 92 and g3 is defined as: Pi = 4P3 - g2P - 93 . (16)
LOU Senyue et al.
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Vol. 11
Especially, when we fix the constants g2 and g3 as 92 = 12A2 and g3 = -8A3, the Weierstrass function becomes P ( q ,12A2,-8A3) = A - 3Asech2&q , (17) where A is an arbitrary positive constant, while the general solution (13) with Eqs.(14) and (15) deduces to the so-called interacting soliton solution'' of the CDGSK equation. In Summary, using the non-local symmetry MO = D-'g and the corresponding closed prolongation structure for the CDGSK equation, we can obtain single soliton solution from a trivial initial solution through the finite transformation. And a generalized nontravelling solution described by the Weierstrass elliptic function and the socalled interacting soliton solution are obtained by solving a similarity reduction equation.
REFERENCES P. J . Olver, Applications of Lie Groups to Differential Equations (Berlin, Springer, 1986). G . W . Bluman and I. D.Cole, J.Math. Mech. 1 0 (1969) 1025. S-y Lou, Proc. XXI diff. Geom. Methods in Theor. Phys. Tianjing, China, June 5-9, 1992, edited by C. N . Yang, M. L. Ge and X . W . Zhou, Int. J. Mod. Phys. A (Proc. Suppl.) 3A (1993) 531; J . M a t h . Phys. 35 (1994) 2390; Phys. Lett. A 181 (1993) 13. S-y Lou, Phys.Lett.B302 (1993) 261. S-y Lou and W-z Chen, Phys. Lett. A 179 (1993) 271. S-y Lou, Phys. Lett. A 175 (1993) 23. S-y Lou and X-b Hu, Chin. Phys. Lett. 10 (1993) 577. I F. Galas, J. Phys. A 25 (1992) L981. [9] P. J. Caudry, R. K . Dodd and J. D.Gibbon, Proc. R. Soc. A 351 (1976) 407; K. Sawada and T. Kotera, Proc. Theor. Phys. 51 (1974) 1355. [lo] P. A . Clarkson and M. D.Kruskal, J . Math. Phys. 30 (1989) 2201; S-y Lou, Phys. Lett. A 151 (1990) 133; S-y Lou, J. Phys. A 2 3 (1990) L649; S-y Lou et al., J.Phys. A 24 (1991) 1455. [ll] B.Fuchssteiner, Prog. Theor. Phys. 78 (1987) 1022.
