Nonlocal symmetries related to Bäcklund transformation and their applications

0.15 MB·English

by  S. Y. Lou

Nonlocal symmetries related to Ba¨cklund transformation
and their applications
2
1 SY Lou1,3, XiaoruiHu2,1 andYong Chen2,1
0
2 1FacultyofScience,NingboUniversity,Ningbo,Zhejiang315211,People’sRepublicof
n China
a 2ShanghaiKeyLaboratoryofTrustworthyComputing,EastChinaNormalUniversity,
J
Shanghai200062,China

7 3DepartmentofPhysics,ShanghaiJiaoTongUniversity,Shanghai200240,China
1
E-mail:[email protected], [email protected]
]
h
p
Abstract. Starting from nonlocal symmetries related to Ba¨cklund transformation (BT),
-
h many interesting results can be obtained. Taking the well known potential KdV (pKdV)
t
a equationasanexample,anewtypeofnonlocalsymmetryinelegantandcompactformwhich
m comesfromBTispresentedandusedtomakeresearchesinthefollowingthreesubjects: two
[ setsofnegativepKdVhierarchiesandtheircorrespondingbilinearformsareconstructed;the

nonlocal symmetry is localized by introduction of suitable and simple auxiliary dependent
1
v variablestogeneratenewsolutionsfromoldonesandtoconsidersomenovelgroupinvariant
9 solutions;someothermodelsbothinfinitedimensionsandinfinitedimensionsaregenerated
0
by comprising the original BT and evolution under new nonlocal symmetry. The finite-
4
3 dimensionalmodelsarecompletelyintegrableinLiouvillesense,whichareshownequivalent
. totheresultsgiventhroughthenonlinearizationmethodforLaxpair.
1
0
2
1
PACSnumbers:02.30.Ik,11.30.Na,04.20.Jb
:
v
i
X
1. Introduction
r
a
With the development of integrable systems and solion theory, symmetries [1–3] play the
more and more important role in nonlinear mathematical physics. Thanks to the classical or
nonclassicalLiegroupmethod,Liepointsymmetriesofadifferentialsystemcanbeobtained,
from which one can transform given solutions to new ones via finite transformation and
construct group invariant solutions by similarity reductions. However, little importance is
attached to the existence and applications of nonlocal symmetries [2,3]. Firstly, seeking for
nonlocal symmetries in itself is a difficult work to perform. One of our authors (Lou) has
made some efforts to get infinite many nonlocal symmetries by inverse recursion operators
[4,5] the conformal invariant form (Schwartz form) [6] and Darboux transformation [7,8].
Moreover, it appears that the nonlocal symmetries are rarely used to construct explicit
solutions since the finite symmetry transformations and similarity reductions can not be
directly calculated under the nonlocal symmetries. Naturally, it is necessary to inquire as
to whether nonlocal symmetries can be transformed to local ones. The introduction of
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 2
potential[3]andpseudopotentialtypesymmetries[9–11]whichpossessescloseprolongation
extends the applicability of symmetry methods to obtain solutions of differential equations
(DEs). In that context, the original given equation(s) can be embedded in some prolonged
systems. Hence, thesenonlocalsymmetrieswith closeprolongationareanticipated[12–14].
On the other hand, to find new integrable models is another important application of
symmetry study. A systematic approach have been developed by Cao [15–17] to find finite-
dimensional integrable systems by the nonlinearization of Lax pair under certain constraints
between potentials and eigenfunctions. Especially in the study of (1+1)-dimensional soliton
equations, various new kinds of confocal involutivesystems are constructed by the approach
of nonlinearization of eigenvalue problems or constrained flows [18,19]. It has also been
pointed that by restricting a symmetry constraint to the Lax pair of soliton equation, one can
not only obtain the lower dimensional integrable models from higher ones, but also embed
the lower ones into higher dimensional integrable models [6,8,20]. Here, alternatively, we
are inspiredto act thegivennonlocalsymmetryon theBa¨cklund transformation(BT) instead
ofLaxpairtogeneratesomeothernewsystemsviasymmetryconstraintmethod. Therelated
work maybeadventurousbut fullofenormousinterest.
In this paper, taking the well known potential KdV equation (pKdV) for a special
example, we will study the nonlocal symmetry defined by BT. Since the BT reveals a finite
transformation between two exact solutions of DEs, it must hint some symmetry. For pKdV
equation, a new class of nonlocal symmetries are derived from its BT, which may give
more interesting applications than those nonlocal symmetries only including potentials and
pseudopotentials. The prolongation of the new nonlocal symmetries are found close after
extending pKdV equation to an auxiliary system with four dependent variables. The finite
symmetry transformation and similarity reductions are computed to give exact solutions of
KdV equation. What we want to mention is the process can once lead to two exact solutions
from onegivenresult duetotheBa¨cklund transformation. Moreover,for thepKdV equation,
someothermodelsbothinfinitedimensionsand infinitedimensionsare obtained. Thefinite-
dimensional systems obtained here are found equivalent to the results given by Cao [16],
which have been verified completely integrable in Liouville sense. This discovery confirms
that these obtained infinite-dimensionalmodels should havemany nice integrableproperties,
whichneeds ourfurtherstudy.
The paper is organized as follows. In section II, we present a detailed description
about the new nonlocal symmetry with BT of pKdV equation. Two kinds of flow equations
corresponding to the given nonlocal symmetry, i.e. the negative pKdV hierarchies, are
obtained and their corresponding bilinear forms are also given out. In section III, we extend
the nonlocal symmetry to be equivalent to a Lie point symmetry of a auxiliary prolonged
system admitting pKdV equation and its BT. Then the finite symmetry transformation and
similarity reductions are made to produce exact solutions of pKdV and then KdV equation.
Section IV is devoted to constructing various integrable systems by means of symmetry
constraintmethod. Conclusionsand discussionsaregivenin Section V.
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 3
2. Nonlocalsymmetries andflow equations related to BT
2.1. BTfor thepKdV equation
Thewell-knownKdV equationreads
w +w 6ww =0, (1)
t xxx x
−
where subscripts x and t denote partial differentiation. For convenience to deal analytically
with a potential function u, introduced by setting w =u , it follows from equation (1) that u
x
wouldsatisfytheequation
u +u 3u2 =0, (2)
t xxx x
−
whichis called potentialKdV (pKdV)equation.
Forequation(2), thereexiststhefollowingBT [21]
(u u )2
u +u = 2l + − 1 , (3)
x 1,x
− 2
u +u =2u2+2u2 +2u u (u u )(u u ) (4)
t 1,t x 1,x x 1,x 1 xx 1,xx
− − −
withl beingarbitrary parameter.
Equations (3) and (4) show that if u is a solution of equation (2), so is u , that is to say,
1
theyrepresent afinitesymmetrytransformationbetween twoexact solutionsofequation(2).
On the other hand, equations (3) and (4) can also be viewed as a nonlinear Lax pair of
equation(2). For
(u u )2
u = u 2l + − 1 , (5)
1,x x
− − 2
u = u +2u2+2u2 +2u u (u u )(u u ), (6)
1,t t x 1,x x 1,x 1 xx 1,xx
− − − −
its compatibilitycondition u =u is exactly equation (2). In fact, both equation (3) and
1,xt 1,tx
equation(4)hintthattheyareallRiccatitypeequationsaboutuoru ,whichcanbelinearized
1
by thewell knownCole-Hopftransformation
y y
x 1,x
u= 2 , or u = 2 . (7)
y 1 y
− −
1
Moreover, by virtue of the dependent variables transformation (7), one can convert
equation(2)intothefollowingbilinearform
(D4+D D )y y =0, (8)
x x t
·
meanwhileitleads equations(3)and (4)to
(D2 l )y y =0, (9)
x 1
− ·
(D +D3+3l D )y y =0, (10)
t x x 1
·
wheretheHirota’sbilinearoperatorDmDn is defined by
x t
¶ ¶ m ¶ ¶ n
DmDna b= a(x,t)b(x,t ) .
x t · ¶ x−¶ x ¶ t −¶ t ′ ′
(cid:18) ′(cid:19) (cid:18) ′(cid:19) (cid:12)x′=x,t′=t
(cid:12)
(cid:12)
(cid:12)
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 4
2.2. The nonlocalsymmetryfromBa¨cklundtransformation
Forequation(2)withitsBT (3)and (4), consideringtheinvariantpropertyunder
l l +ed , u u+es , u u +es ,
1 1 ′
→ → →
we may find substantial possible nonlocal symmetries and a special case is presented and
studiedas follows.
Proposition1. ThepKdVequation(2)has anewtypeofnonlocalsymmetrygivenby
s =exp( u u dx), (11)
1
−
Z
where u and u satisfy BT (3) and (4). That means s given by (11) satisfies the following
1
symmetryequation
s +s 6u s =0. (12)
t xxx x x
−
Proof: By directcalculation.
On the other hand, we let the bilinear pKdV equation (8) be invariant under the
transformationy y +es y , which producesthecorrespondingsymmetryequation
→
(D4x+DxDt)s y y =0. (13)
·
y
The Cole-Hopf transformation u = 2 x between equation (2) and its bilinear equation (8)
y
−
determinesasymmetrytransformationfors and s y , saying
s = 2y xs y 2s y ,x. (14)
y 2 − y
Taking equations (11) and (7) into equation (14), we obtain a class of nonlocal symmetry for
equation(8)
y y 2
s y = 1dx. (15)
−2 y 2
Z
Correspondingly,itgivesthefollowingpropositionforequation(8).
Proposition2. The bilinearpKdV equation (8)has thenonlocal symmetryexpressedby
(15), wherey andy satisfybilinearBT (9)and(10).
1
Proof: One can directly check that s y given by (15) satisfies symmetry equation (13) under
theconsideration(9)and (10).
2.3. Two setsofnegativepKdV hierarchies
Theexistenceofinfinitelymanysymmetriesleadstothetheexistenceofintegrablehierarchies
and withthehelpofinfinitelymanynonlocalsymmetries,onecan extendtheoriginalsystem
to its negativehierarchies [22,23]. Here, starting from the nonlocal symmetry(11) related to
BT of equation (2), we would like to present two sets of negativepKdV hierarchies and their
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 5
correspondingbilinearformsare alsoconstructed onlybythetransformation(7).
Case1. Thefirst kindofnegativepKdVhierarchy can beobtained,reading
N
(cid:229)
u = exp( u u dx), (16)
t N − − i
− i=1 Z
(u u)2
u +u = 2l + − i , i=1,2,...,N, (17)
x i,x i
− 2
wherel isarbitrary constant.
i
Inparticular,whenN =1,onehasthefirstequationofnegativepKdVhierarchy,namely
2u u 4u u2 u2 4l u2 =0. (18)
xxt t x t xt 1 t
− − −
Here we have insteadt witht for simplicity. It is well known that the first negativeflow in
1
−
the KdV hierarchy is linked to the Camassa-Holm equation via a hodograph transformation
[24] or can be reduced to the sinh-Gordon/sine-Gordon/Liouville equations [25]. Here we
transformequation(18)intosine-Gordonand Liouvilleequations.
In fact, by settingb b (x,t)= u , wecan rewriteequation(18)intheform
t
≡ −
b b 2
b = xx + x , (19)
x −2b 4b 2
(cid:18) (cid:19)t
whichcan beintegratedoncewithrespect to xto give
b (lnb ) +b 2 =b (t), (20)
xt 0
whereb (t)isan arbitrary functionoft.
0
AsitisreportedinRef.[24],fornon-zerob (t),onecanrescaleb to b (t)b ,redefine
0 0
t ast/ b (t)and set b =exp(ih ) togivethesine-Gordonequation
0 p
p h =sinh , (21)
xt
whileforb (t)=0,by settingb = exp(h ),equation(20)becomes theLiouvilleequation
0
−
h =eh . (22)
xt
b b 2
Remark 1. The quantity xx + x A in the right hand side of equation (19) can be given
−2b 4b 2 ≡
interms ofaMiuratransformation
b
A= q q 2, q = x . (23)
− x− 2b
Furthermore, byvirtueofthedependent variabletransformation
y y
x i,x
u= 2 , u = 2 , (i=1,2,...,N) (24)
y i y
− −
i
thenegativepKdV hierarchy(16)-(17)isdirectly transformedintoitsbilinearform
N
D D y y = (cid:229) y 2, (25)
x t N · i
− i=1
(D2 l )y y =0, i=1,2,...,N. (26)
x i i
− ·
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 6
Case 2. For the nonlocal symmetry (11) being dependent with parameter l , we may derive
the second kind of negative pKdV hierarchy by expanding the dependent variable in power
series ofl . In thiscase, wehave
1 ¶ Nexp( u u dx)
1
u = − , (27)
t N −N! ¶l N
− (cid:18) R (cid:19)(cid:12)l =0
(u u )2 (cid:12)
ux+u1,x = 2l + − 1 . (cid:12)(cid:12) (28)
− 2
Under the transformation u= 2y x and u = 2y 1,x, the negativepKdV hierarchy (27)-(28)
y 1 y
− − 1
becomes
1 ¶ Ny 2
D D y y = 1 , (29)
x t N · N! ¶l N
− (cid:18) (cid:19)(cid:12)l =0
(D2x l )y y 1 =0. (cid:12)(cid:12) (30)
− · (cid:12)
Let y =y (l )haveaformalseries form
1 1
¥
y = (cid:229) y¯ l i, (31)
1 i
i=0
wherey¯ is l independent. Then (29)-(30)can berewrittenas
i
N
D D y y = (cid:229) y¯ y¯ , (32)
x t−N · k=0 k N−k
D2y y¯ =y y¯ (k=0,1,...,N) (33)
x k k 1
· −
withy¯ =0.
1
−
The negative pKdV hierarchy in bilinear form (32)–(33) is just the special situation of
thebilinearnegativeKPhierarchyfor¶ =0inRef. [22]. Fromthisobservation,wehavethe
y
followingremark:
Remark 2. The second negativepKdV hierarchy shown by (27)-(28) is a potential form of a
knownnegativeKdVhierarchygivenbyothermethods,say,theinverserecursionoperator[4],
Lax operator[23], and theGuthrie’sapproach [26].
3. Localizationofthe nonlocal symmetries
We know that the Lie point symmetries [2,3] can be applied to construct finite symmetry
transformationandgroupinvariantsolutionsforDEs,whereasthecalculationsareinvalidfor
the nonlocal symmetries. So it is anticipant to turn the nonlocal symmetries into local ones,
especially into Lie point symmetries. In order to make the nonlocal symmetry localized, one
may extend theoriginal system to a closed prolonged systemby introducingsome additional
dependablevariables[12–14]toeliminateintegrationand differentiation.
Fortunately, starting from the nonlocal symmetry (11), the prolongation is found to be
closedwhen anothertwodependent variablesv v(x,t)and g g(x,t)areintroducedby
≡ ≡
v =u u , v =2(u u )(u 2l ) 2u ,
x 1 t 1 x xx
− − − − (34)
g =ev, g = ev[2u +8l (u u )2].
x t x 1
− − −
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 7
Now the prolonged equations (2), (3), (4) and (34) contain four dependable variables u,
u , vand g, whosecorrespondingsymmetriesare
1
1
s =ev, s =0, s =g, s = g2. (35)
u u v g
1 2
Remark 3. What is more interesting here is that the symmetry s g shown in (35) implies the
auxiliarydependentvariablegsatisfies
g 3g2
g = g;x g +6l g , g;x xxx xx, (36)
t { } x x { }≡ g −2 g2
x x
whichisjusttheSchwartzformoftheKdV(SKdV)equation(1). Thismayprovideuswitha
newwaytoseekfortheSchwartzformsofDEs,especiallyforthediscreteintegrablemodels,
withoutusingPainleve´ analysis.
Dueto(35), thesymmetryvectoroftheprolongedsystemhastheform
¶ ¶ ¶ 1 ¶
V =ev +0 +g + g2 . (37)
¶ u ¶ u ¶ v 2 ¶ g
1
Then, bysolvingthefollowinginitialvalueproblem
du¯ du¯ dv¯ dg¯ 1
=ev¯, 1 =0, =g¯, = g¯2,
de de de de 2 (38)
u¯ e =0 =u, u¯1 e =0 =u1, v¯ e =0 =v, g¯ e =0 =g,
| | | |
thefinitetransformationcan bewrittenoutas follows
2e 2 2
u¯=u+ ev, u¯ =u , v¯=v+2ln , g¯= g. (39)
2 e g 1 1 2 e g 2 e g
− − −
Remark 4. The original BT (3) and (4) in itself suggests a finite transformation from
one solution u to another one u and then the new BT (39) obtained via (11) will arrive
1
at a third solution u¯. Actually, the finite transformation (39) is just the so-called Levi
transformation [27]. The result of this paper shows the fact that two kinds of BT possess
thesameinfinitesimalform (11).
Nowbyforce ofthefinitetransformation(39), onecan get newsolutionfromany initial
solution. Forexample,it iseasy to solvean initialsolutionofprolongedequationsystem(2),
(3), (4)and (34), namely
u=c, u =c+2√l tanhz , v= ln(tanh2z 1),
1
− −
sinh(2z ) x (40)
g= +6l t+c , z = √l ( x+4l t),
0
4√l −2 −
Wherel , cand c arethreearbitrary constants.
0
Starting fromthisoriginalsolution(40), anewsolutionof equation(2)can bepresented
immediatelyfrom (39):
8√le cosh2z
u¯=c , (41)
−8√l e [sinh(2z ) 2√l (x 12l t 2c )]
0
− − − −
whichthen givesthecorrespondingsolutionofKdV equation
[cosh(2z )+1]e + √l sinh(2z )[4+e (x 12l t 2c )]
w¯ =u¯ =16le − − 0 (42)
x
· [8√l e (sinh(2z ) 2√l (x 12l t 2c ))]2
0
− − − −
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 8
withz = √l ( x+4l t).
−
Besidesobtainingnewsolutionsfromoldones,symmetriescanbeappliedtogetspecial
solutions that are invariant under the symmetry transformations by reducing dimensions of
a partial differential equation. To find more similarity reductions of equation (2), we will
study Lie point symmetries of the whole prolonged equation system instead of the single
equation (2). Suppose equations (2), (3), (4) and (34) be invariant under the infinitesimal
transformations
u u+es , u u +es , v v+es , g g+es ,
1 1 1 2 3
→ → → →
with
s =X(x,t,u,u ,v,g)u +T(x,t,u,u ,v,g)u U(x,t,u,u ,v,g),
1 x 1 t 1
−
s =X(x,t,u,u ,v,g)u +T(x,t,u,u ,v,g)u U (x,t,u,u ,v,g),
1 1 1,x 1 1,t 1 1
− (43)
s =X(x,t,u,u ,v,g)v +T(x,t,u,u ,v,g)v V(x,t,u,u ,v,g),
2 1 x 1 t 1
−
s =X(x,t,u,u ,v,g)g +T(x,t,u,u ,v,g)g G(x,t,u,u ,v,g).
3 1 x 1 t 1
−
Then substituting the expressions (43) into the symmetry equations of equations (2), (3), (4)
and (34)
s +s 6u s =0,
t xxx x x
−
s +s (s s )(u u )=0,
1,x x 1 1
− − −
s s +2(u u )s +2(s s )u [4l +(u u )2 2u ]s
1t xxx 1 xx 1 xx 1 x x
− − − − − −
+2(s s )(u u )(2l u )=0,
1 1 x
− − −
s s +s =0, (44)
2,x 1
−
s +2s +2(u u)s +2(s s )(u 2l )=0,
2t xx 1 x 1 x
− − −
s evs =0,
3,x 2
−
1
s +2ev[s +(u u)(s s ) (u u )2s +(4l +u )s ]=0,
3t x 1 1 1 2 x 2
− − −2 −
andcollectingtogetherthecoefficientsofpartialderivativesofdependentvariables,ityieldsa
systemofoverdeterminedlinearequationsfortheinfinitesimalsX,T,U,U ,V andG, which
1
can besolvedbyvirtueofMapletogive
X(x,t,u,u ,v,g)=c (x+12l t)+c ,
1 1 5
T(x,t,u,u ,v,g)=3c t+c ,
1 1 2
U(x,t,u,u ,v,g)= c (2l x+u)+2c ev+c ,
1 1 4 3
− (45)
U (x,t,u,u ,v,g)= c (2l x+u )+c ,
1 1 1 1 3
−
V(x,t,u,u ,v,g)= c +2c g+c ,
1 1 4 6
−
G(x,t,u,u ,v,g)=c g2+c g+c ,
1 4 6 7
where c(i=1...7)are seven arbitrary constants. When c =c =c =c =c =c =0, the
i 1 2 3 5 6 7
reduced symmetryisjust(35).
To give the group invariant solutions, we would like to solve symmetry constraint
conditions s = 0 and s = 0(i = 1,2,3) defined by (43) with (45), which is equivalent to
i
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 9
solvethefollowingcharacteristicequation
dx dt du
= =
c (x+12l t)+c 3c t+c c (2l x+u)+2c ev+c
1 5 1 2 1 4 3
− (46)
du dv dg
1
= = = .
c (2l x+u )+c c +2c g+c c g2+c g+c
1 1 3 1 4 6 4 6 7
− −
Two nontrivial similar reductions under consideration c , 0 are presented and substantial
4
groupinvariantsolutionsarefound inthefollows.
Case1: c ,0 and c2 4c c ,0.
1 6− 4 7
Withoutlossofgenerality,weletc 1. Forsimplicity,weintroducearbitraryconstants
1
≡
a and a to replace c and c by a2 = c2 4c c and a = a2/(16c ), then after solving
4 7 4 7 4 6− 4 7 7 − 4 4
equation(46), wehave
u= l x+3l 2t+c3+c5l 3c2l 2+(3t+c2)−31[U(x )
− −
a
4 exp(V(x ) G(x ))tanhB],
−4a −
7
U (x )
u = l x+3l 2t+c +c l 3c l 2+ 1 ,
1 − 3 5 − 2 (3t+c2)13 (47)
1
v= ln(3t+c ) G(x )+V(x ) 2lncoshB,
2
−3 − −
8a c
7 6
g= [tanhB+ ]
a a
4 4
withB=a4(3G(x )+ln(3c1t+c2))/6and x =(x 6l t+c5 6c2l )/(3t+c2)31.
− −
Here, U(x ), U (x ), V(x ), G(x ) and x represent five group invariants and substituting
1
(47)intotheprolongedequationssystemgivesthefollowingreduced equations
1Hx2 a2
Hxx = 2 H +4a7H2−x H−32a42H, (48)
7
a7Hx2 x 2 a2
U (x )= 4a2H2+2a x H 4
1 H − 7 7 − 4 −16a H
7 (49)
Hx 1
U(x )=U1(x ) , V(x )=G(x ) ln(H), Gx (x )=
− H − 4a H
7
with H H(x ). One can see that whence H is solved from equation (48), two new group
≡
invariantsolutionsuandu ofequation(2)wouldbeimmediatelyobtainedthroughequations
1
(47)and (49).
Moreover,bymakingafurthertransformation[28]
1 x
H(x )= (Px +P2+ ), P P(x ), (50)
2a 2 ≡
7
equation(48)can beconvertedintothesecondPainleve´ equationP , reading
II
Pxx =2P3+x P+a , (51)
with a = (a +1)/2. Now, every known solution of P (51) will generate two new group
4 II
−
invariant solutions of equation (2), and then two new solutions of KdV equation (1) denoted
Nonlocalsymmetriesrelated toBa¨cklundtransformationandtheirapplications 10
as w andw can begivendirectlyafteronederivativewithrespect to xforu andu
1 2 1
1
w 1 = (Px +P2) l , (52)
(3t+c2)32 −
1 a2 2a P a2 2a P
w 2 = (3t+c2)32[−2F42sech2R1+( F4 −F42)tanhR1+ F4 +Px −P2]+l , (53)
where
1 1
F ≡F(x )=2Px +2P2+x , R1 = 6a4[ln(3t+c2)+3G(x )], Gx (x )= 2Px +2P2+x ,
and P satisfiesP (51)witha = (a +1)/2.
II 4
−
It is known that the generic solutions of P are meromorphic functions and more
II
information about P is provided in Ref. [29], saying: (1) For every a =N Z, there exists
II
∈
a unique rational solution of P ; (2) For every a =N+ 1, with N Z, there exists a unique
II 2 ∈
one-parameter family of classical solutions which are expressiblein terms of Airy functions;
(3)Forallothervaluesofa , thesolutionofP istranscendental.
II
For example, when a = 1 (a = 3), P (51) possesses a simple rational solution
4 II
−
P(x )= 1/x ,which leadsthesolutions(52) and(53)to
−
2
w˜ = l , (54)
1 (x 6l t+c 6c l )2 −
5 2
− −
and
w˜ = [x6 36tx5+(540t2 6)x4 (4320t3 168t 2)x3+36t(540t3 48t 1)x2
2
− − − − − − − −
(46656t5 7776t3 216t2 144t 12)x+46656t6 12960t4 432t3
− − − − − − −
720t2 48t+1]/[x3 18x2t+108xt2 (6t+1)(36t2 6t 1)]2 (55)
− − − − − −
In the formulation (55), we have made c = 0, c = 0 and l = 1 because the original
2 5
expression is much too complicated. The simplerational solutionsof PII will yield abundant
rationalsolutionsofKdVequation.
When a = 1 (a = 2),P (51)has asolutionexpressedby Airyfunction
2 4 − II
P(x )=2−133Ai(1,−2−31x )− √3Bi(1,−2−13x ). (56)
3Ai( 2−13x ) √3Bi( 2−31x )
− − −
Forsimplicity,weconvertequation(56)intotheequivalentform
√2x 32J(4, √2x 32) 2J(1, √2x 32)
P(x )= 3 3 − 3 3 , (57)
x J(1, √2x 32)
3 3
where J(n,x ) is the first kind of Bessel function. Substituting (57) into (52) and (53) with
c = c = 0 and l = 1 (or else the formulae are too long to written down here), two exact
2 5
solutionsofKdVequationare obtainedas follows:
x 12t x 6t J2
w = − + − 2, (58)
1′ 6t 3t J2
1
w = Y /W (59)
2′
−