On symmetries of KdV-like evolution equations

0.17 MB·English

by  Artur Sergyeyev

Vol. XX (199X) REPORTS ONMATHEMATICAL PHYSICS No. X
7
0
0
2 ON SYMMETRIES OF KDV-LIKE EVOLUTION EQUATIONS

n
a Artur Sergyeyev∗
J
Institute of Mathematics of NAS of Ukraine,
9
Tereshchenkivs’ka Str. 3, 252004 Kyiv, Ukraine

] e-mail: [email protected],[email protected]
P
A
Thex-dependenceofthesymmetriesof(1+1)-dimensionalscalartranslationally
.
h invariant evolution equations is described. The sufficient condition of (quasi)poly-
t nomialityintimetofthesymmetriesofevolutionequationswithconstantseparant
a
is found. The general form of time dependenceof thesymmetries of KdV-likenon-
m
linearizable evolution equations is presented.
[

3
1. Introduction
v
5 Itiswellknownthatprovidedscalar(1+1)-dimensionalevolutionequation(EE)with
0
time-independentcoefficientspossessestheinfinite-dimensionalcommutativeLiealgebra
1
of time-independent non-classical symmetries, it is either linearizable or integrable via
5
0 inverse scattering transform [1, 2]. This algebra is usually constructed with usage of
8 the recursion operator [2], but it may be also generated by the repeated commuting of
9 mastersymmetry with few time-independent symmetries [3]. In its turn, to possess the
/
h mastersymmetry,EEinquestionmusthave(atleastone)polynomialintimetsymmetry.
t This fact is one of the main reasons of growing interest to the study of whole algebra of
a
m time-dependent symmetries of EEs [4, 5, 6].
However,itisverydifficulttodescribethisalgebraeveninthe simplestcaseofscalar
:
v (1+1)-dimensionalEE.Tothebestofauthor’sknowledge,intheclassofscalarnonlinear
i
X EEs the complete algebrasoftime-dependent localsymmetries were foundonly for KdV
equationbyMagadeevandSokolov[7]andforKdVandBurgersequationsbyVinogradov
r
a etal. [8]. In[8]therewerealsoprovedtwono-gotheorems,whichshow,whenthesymme-
triesofthirdorderKdV-likeandsecondorderBurgers-likeEEsareexhaustedbyLieones.
Surprisinglyenough,using onlythe invarianceofgivenEEunder∂/∂tand∂/∂xand
somesimpleobservationsontheexplicitformofsymmetries,wecanshow(videTheorem
6below)thatanysymmetryofKdV-likenon-linearizableEEasafunctionoftisalinear
combination of quasipolynomials (i.e. of the products of exponents by polynomials).
Moreover,fortheclassofEEswithconstantseparantourapproachallowstofindasimple
sufficientconditionforsuchasituationtotakeplace(Theorem3),andforthegenericEE
(1)itenablesustodescribethedependenceofitssymmetriesonvariablex(Theorem1).
∗Current address (as of January 9, 2007): Silesian University in Opava, Mathematical Institute,
NaRybn´ıˇcku1,74601Opava,CzechRepublic. E-mail: [email protected]
[1]
[Authorandtitle] 2
2. Some general properties of symmetries of evolution equations
Consider the scalar (1+1)-dimensional EE
∂u/∂t=F(t,u,u ,...,u ), n≥2, (1)
1 n
where u =∂lu/∂xl,l∈N, u ≡u,andits symmetries, i.e.the righthandsides Gof EEs
l 0
∂u/∂τ =G(x,t,u,u ,...,u ), (2)
1 k
compatible with equation(1). The biggestnumberk suchthat ∂G/∂u 6=0is calledthe
k
order of symmetry and is denoted as k = ord G. If G is independent of u,u ,..., then
1
∞
we assume that ordG=0. We set S(k) ={G∈S|ordG≤k} and S = S(k).
j=0
For any sufficiently smooth function h(x,t,u,u ,...,u ) introduce tShe quantities [1]
1 r
r ∞ ∞
∂
h∗ = ∂h/∂uiDiand∇h = Dj(h)∂/∂uj, whereD = + ui+1∂/∂ui.
∂x
X X X
i=0 j=0 i=0
Now we can define the Lie bracket, which endows S by the structure of Lie algebra, as
{h,r}=h∗(r)−r∗(h)=∇r(h)−∇h(r).
Thisdefinitiondiffersfromtheconventionalone[1,2,9]bythesign,butismoresuitable
for our purposes. Note that S(1) is Lie subalgebra in S.
Equation (2) is compatible with (1) if and only if
∂G/∂t={F,G}. (3)
From (3) one may easily derive [2] that
∂G∗/∂t≡(∂G/∂t)∗ =∇G(F∗)−∇F(G∗)+[F∗,G∗], (4)
∞ ∂2G
where ∇F(G∗) ≡ Dj(F) Di and similarly for ∇G(F∗); [·,·] stands for the
∂u ∂u
i,Pj=0 j i
usual commutator of linear differential operators.
Let ordG=k. Then equating the coefficients at Dl on both sides of (4) yields
∂2G n ∂2F k ∂2G
= Dm(G) − Dr(F)
∂u ∂t ∂u ∂u ∂u ∂u
l mP=0 m l rP=0 r l
+ k n Ci+j−l∂F Di+j−l ∂G (5)
j=max(P0,l+1−n)i=max(Pl+1−j,0)h i ∂ui (cid:16)∂uj(cid:17)
−Ci+j−l∂GDi+j−l ∂F , l =0,...,n+k−1,
j ∂uj (cid:16)∂ui(cid:17)i
q! 1
where Cp = and we assume that =0 for s∈N.
q p!(q−p)! (−s)!
[Authorandtitle] 3
If k ≥2, one easily obtains from (5) with l=n+k−1,...,n+1 the formulas
∂G/∂u =c (t)(∂F/∂u )k/n, (6)
k k n
k [np−−1i] ∂qc
∂G/∂u =c (t)(∂F/∂u )i/n+ χ (t,x,u,...,u ) p,i=2,...,k−1, (7)
i i n pq k ∂tq
X X
p=i+1 q=0
where c (t) are arbitrary functions of t (cf. [1, 2]).
j
Furthermore, we see that by virtue of (4)
[∇F −F∗,G∗]=0 (mod Dp), p=max(k,n) (8)
Equating the coefficients at powers of D in (8) yields the equations (5) with l = p+
1,...,n +k −1, from which we may find ∂G/∂u , i = max(k −n +1,2),...,k. By
i
(7), the only arbitraryelements, which they may contain,are functions c (t), while their
i
dependence on x,u,u ,... is uniquely determined from (8).
1
On the other hand, since F is x-independent, the existence of the solution F∗ of (8)
with p = n guarantees the solvability of the equations for ∂G/∂u , i = max(k −n+
i
2,2),...,k, in terms of functions of u,...,u and t only (cf. [10]). Therefore, ∂G/∂u ,
k i
i=max(k−n+2,2),...,k,arex-independent. Inparticular,anyG∈S(n) hasthe form
G=g(t,u,...,u )+Φ(t,x,u,u ). (9)
k 1
Thus, for any symmetry G∈S ∂G/∂x∈S and ord∂G/∂x≤max(1,ordG−n+1).
ApplyingthisresulttoG˜ =∂G/∂xandsoon,weobtainthat∂rG/∂xr ∈S(n) andhence
is of the form (9), if r=r , where for q =0,1
k,n,1
k
fork 6≡0,...,q (mod n−1),
rk,n,q = mhnax−(01,i k −1)fork ≡0,...,q (mod n−1), andrk,n,−1 =hn−k 1i;
hn−1i
[s] denotes here the integer part of the number s. The integration of ∂rG/∂xr r times
with respect to x, taking into account the above, yields the following result:
Theorem 1. Any symmetry G of order k of (1) may be represented in the form
s
G=ψ(t,x,u,u1)+ xjgj(t,u,...,uk−j(n−1)), s≤rk,n,1. (10)
X
j=0
Remark 2. In complete analogy with the above, one may show that if
∂F/∂un−i =φi(t), i=0,...,j, (11)
whereφ (t)arearbitraryfunctionsoft,thenin(8)p=max(k,n−1−j)anditispossible
i
to find from (8) ∂G/∂u , i=max(k−n+2,max(1−j,0)),...,k, which again turn out
i
to be x-independent, and hence (cf. [10]) in (10) s≤rk,n,−min(1,j) and ψ satisfies
∂ψ/∂u =0, r =max(1−j,0),...,1. (12)
r
Notethatif(11)holdstrue,(6),(7)holdfork ≥max(1−j,0),i=max(1−j,0),...,k−1.
[Authorandtitle] 4
3. Symmetries of the equations with constant separant
Let us turn to the particular case, when ∂F/∂t=0 and F has the form
F =un+f(u,...,un−1), (13)
i.e. when F has a constant separant,equal to unity [10]. Note that any F with constant
separant, different from unity, may be reduced to the form (13) by rescaling of time t.
For the sake of brevity we shall refer to EE (1) with F (13) as to EE with constant
separant. Let us also mention that if ∂F/∂t=0, then in (7) ∂χ /∂t=0.
pq
Assume that ordG≡k>n−1. Then, by (13) and (6), (5) with l =k reads
nD(∂G/∂uk−n+1)=∂ck(t)/∂t+R, (14)
whereRstandsforthetermswhichdependonlyonF anditsderivativesandon∂G/∂u ,
i
i=k−n+2,...,k. Moreover,R=D(K) for some x-independent K, as it follows from
the fact that if F has a constant separant, F∗ is solution of (8) with p = n−1. Really,
if the term ∂G∗/∂t in (4) would be absent, G∗ would satisfy (8) with p = n−1 and
the equation for ∂G/∂uk−n+1 would be solvable in terms of functions of t,u,u1,... (cf.
the proof of Theorem 1 and [10]). But the only term in (14), generated by ∂G∗/∂t, is
∂ck(t)/∂t, while R is the same as if it would be in absence of ∂G∗/∂t in (4). Hence,
R=D(K), ∂K/∂x=0, and
∂G/∂uk−n+1 =(x/n)∂ck(t)/∂t+K+ck−n+1(t). (15)
Since ∂G/∂u , i = k − n + 2,...,k, are x-independent by Theorem 1, by (15)
i
ord∂G/∂x=k−n+1 and ∂2G/∂uk−n+1∂x=(1/n)∂ck(t)/∂t.
Iterating this process shows that for r =r Q=∂rG/∂xr ∈S(n−1) and
k,n,0
∂Q/∂u =(1/nr)∂rc (t)/∂tr, q ≡ordQ. (16)
q k
Theorem 3. If the symmetries from S(n−1) of the equation (1) with constant sepa-
rant either are all polynomial in t or are all linear combinations of quasipolynomials1 in
t, then so does any symmetry of this equation.
Proof. If the conditions of theorem are fulfilled, then by (16) for any symmetry G,
k ≡ ordG≥ 1, the function c (t) ≡∂G/∂u is either polynomial or linear combination
k k
of quasipolynomials in t. Hence, there exists a differential operator Ω = m a ∂l/∂tl,
l=0 l
a ∈ C, such that Ω(c (t)) = 0. In particular, if c (t) is polynomial in tPof order p, we
l k k
may choose Ω =∂p+1/∂tp+1 as Ω.
0
Nowassumethatthe theoremisalreadyprovedforthe symmetriesfromS(k−1) (itis
obviouslytruefork ≤n). ForEE(1)withF (13)S(k) isclosedunder∂/∂t,andtherefore
Ω(G) ∈ S(k). Moreover, since Ω(c (t)) = 0, ord Ω(G) ≤ k −1 and hence, by our as-
k
sumption,G˜ ≡Ω(G)iseitherpolynomialorlinearcombinationofquasipolynomialsint.
Obviously,sodoes anysolutionR ofthe equationΩ(R)=G˜, including R=G. Ifallthe
1Wecallquasipolynomialstheproducts exp(λt)P(t), whereλ∈CandP isapolynomial.
[Authorandtitle] 5
elements of S(n−1) are polynomial in t, then so does c (t) and hence the polynomiality
k
of G in t is guaranteed, because we may take Ω = Ω and because G˜ is polynomial in t
0
by our assumption. The induction by k, starting from k =n, completes the proof. (cid:3)
Theorem3isanaturalgeneralizationoftheresultof[7]onpolynomialityintofsym-
metriesofKdVequation. Itgivesaverysimplesufficientconditionforallthesymmetries
ofagivenEEwithconstantseparanttobe polynomialintimet. Notethatinsuchasit-
uation all the time-dependent symmetries of EE in question may be constructed via the
so-called generators of degree s for different s∈N, using the results of Fuchssteiner [3].
4. Symmetries of KdV-like equations
Now let us consider the equations with constant separant, whose f satisfies
∂f/∂un−1=const. (17)
We shall call the EEs (1) with F (13), satisfying (17), KdV-like, since the famous Ko-
rteweg – de Vries equation has the form (1) with F (13), where n = 3 and f = 6uu
1
obviously satisfies (17).
LetGbe the symmetry ofKdV-likeEE(1). Analyzingthe leadingtermof∂rG/∂xr,
r = ordG , like in the proof of Theorem 3, we obtain the following statement:
n−1
h i
Corollary 4. If the symmetries from S(n−2) of KdV-like equation (1) either are
all polynomial or are all linear combinations of quasipolynomials in t, then so does any
symmetry of this equation.
Example 5. Consider third order formally integrable nonlinear KdV-like EEs [11]:
u =u +uu ,
t 3 1
u =u +u2+c,
t 3 1
u =u +u2u +cu ,
t 3 1 1
u =u +u3+cu +d,
t 3 1 1
u =u −u3/2+(aexp(2u)+bexp(−2u)+d)u ,
t 3 1 1
where a,b,c,d∈C. All the symmetries of these EEs are polynomial in t by Corollary 4,
since so do their symmetries of orders 0 and 1.
Nowletusanalyzeinmoredetailthe generalformoftime dependence ofsymmetries
of KdV-like EE (1). Assume that the EE in question may not be linearized by means of
contact transformations (for the sake of brevity we shall call it non-linearizable). Then
dim Φ ≤ n [9], where Φ = {ϕ(x,t)|ϕ(x,t) ∈ S}. Let us show that in such a case
dimS(k) <∞ for any k =0,1,2,....
Let G∈S(k)/S(k−1), k >n−2. Then, obviously, it is completely determined by its
leading term h (t) ≡ ∂G/∂u . Like the above, but taking into account Remark 2, we
k k
may show that for KdV-like EE (1) Q=∂rG/∂xr ∈S(n−2), if r = k , and
hn−1i
∂Q/∂u ≡c (t)=(1/nr)∂rh (t)/∂tr,q ≡ordQ. (18)
q q k
[Authorandtitle] 6
Since h satisfies (18), for k >n−2
k
k k
dimS(k)/S(k−1) ≤dimS(k0)+ ,k =k− (n−1), (19)
0
hn−1i hn−1i
whence dimS(k) <∞ for k =0,...,n−2 implies the same result for any k.
By Theorem 1 and Remark 2 for KdV-like EE (1) (6) and (7) for k ≤n−2 read
k
∂G/∂u =c (t)+ χ (u,...,u )c (t), i=0,...,k−1, (20)
i i p k p
X
p=i+1
∂G/∂u =c (t), (21)
k k
and thus any symmetry G of order k ≤n−2 has the form
G=ψ(t,x)+g (t,u,...,u ). (22)
0 k
Without loss of generality we can assume that the function g is completely determined
0
by ∂G/∂u , i=0,...,k. Since ∂ψ/∂x∈Φ, we have
i
dimΦ x
ψ(t,x)=γ(t)+ a dy ϕ (y,t), (23)
pZ p
Xp=1 0
where a ∈ C, γ(t) is arbitrary function of t, and ϕ (x,t), p = 1,...,dim Φ, stand for
p p
some basis in Φ.
ThesubstitutionofG(22)withψ(23)intoequations(5)withl=0,...,n−1andinto
(3) yields in final account the system of first order linear ordinary differential equations
in t (and, possibly, algebraic equations) for c (t), i = 0,...,k and γ(t). Note that we
i
mustuse(3)inordertoobtainanODEoftheform∂γ(t)/∂t=···,allowingtofindγ(t).
Obviously, the general solution of this system of ODEs for k ≤n−2 may contain at
most N =dimΦ+k+2 arbitrary constants (including a , p=1,...,dimΦ).
k,n p
Hence, dimS(k) ≤N <∞ for k ≤n−2 and thus by (19) for any k=0,1,...
k,n
k
dimS(k) =dimS(k0)+ dimS(j)/S(j−1) <∞. (24)
X
j=k0+1
Thus, for any k the space S(k) is finite-dimensional. Since in addition this space is
invariant under ∂/∂t, the dependence of its elements on t is completely described by
Theorem3.1[12]. Namely, anysymmetry oforderk ofKdV-like non-linearizableEE(1)
is a linear combination of dimS(k) linearly independent symmetries of the form
m
H =exp(λt) tjh (x,u,...,u ),λ∈C,m≤dimS(k)−1. (25)
j k
X
j=0
[Authorandtitle] 7
Theorem 6. Foranynon-linearizableKdV-likeEE(1)dimS(k) <∞,k =0,1,2,...
and any symmetry Q of order k is a linear combination of the symmetries (25).
Thus, all the symmetries of non-linearizable KdV-like EEs are linear combinations of
quasipolynomialsint. ThispartiallyrecoverstheresultofCorollary4,butthiscorollary
still remains of interest, providing the convenient sufficient condition of polynomiality of
symmetries in time t.
It is interesting to note that some general properties of time-dependent symmetries,
which are linear combinations of the expressions (25), were studied by Ma [6]. However,
while he considered this form as given a priori, we have proved that all the symmetries
of non-linearizable KdV-like EE (1) indeed have this form.
Moreover,acting onany symmetry (25) by (∂/∂t−λ)m for λ6=0 orby ∂m−1/∂tm−1
for λ = 0, we obtain the symmetry which is either linear or exponential in t. Hence,
there is a very simple test of existence of any time-dependent symmetries for given
non-linearizable KdV-like EE (1). Namely, it suffices to check whether there exist the
symmetries of the form
G=exp(λt)Q ,λ∈C,λ6=0 (26)
0
or of the form
G=G +tG ,G 6=0, (27)
0 1 1
where Q , G and G are time-independent. If non-linearizable KdV-like EE (1) (with
0 0 1
time-independent coefficients!) has no time-dependent symmetries of the form (26) or
(27), then it has no time-dependent symmetries at all (but of course it may have time-
independent symmetries).
The substitution of (26) and (27) into (3) yields
{F,Q }=λQ , (28)
0 0
{F,G }=G , {F,G }=0. (29)
0 1 1
In the first case F is called scaling symmetry (or conformal invariance [13]) of Q .
0
However, known scaling symmetries F of integrable hierarchies, such as KdV, depend
usually only on x,u,u but not on u and higher derivatives [13] and hence do not gen-
1 2
erate EEs of the form (1), which we consider here. We guess that if KdV-like EE (1)
is non-linearizable and integrable, there exist no functions Q , which satisfy (28) with
0
λ6=0. Moreover,itisbelieved[4]thatinsuchacasetheonlypolynomialintsymmetries
(2) that EE (1) may possess are those linear in t.
Now let us consider the second case. Assume that there exists some commutative
algebra Alg of time-independent symmetries of KdV-like non-linearizable EE (1), such
that for any K ∈ Alg the Lie bracket {G ,K} ∈ Alg. Then G is mastersymmetry
0 0
of (1), and hence (1) possesses (under some extra conditions, vide [3]) the infinite set
of time-independent symmetries and is probable to be integrable via inverse scattering
transform. Let us mention that the condition of commutativity of Alg may be rejected
if G is scaling symmetry of F, i.e. {F,G }=µF for some µ∈C, µ6=0 [13].
0 0
Conjecture 7. For any KdV-like non-linearizable evolution equation (1) either all
its symmetries are polynomial in t or all they are linear combinations of exponents in t.
[Authorandtitle] 8
Acknowledgements
It is my pleasure to express deep gratitude to Profs. A.G. Nikitin and R.Z. Zhdanov
and Dr. R.G. Smirnov for the fruitful discussions on the subject of this work. I would
also like to thank the organizers of XXX Symposium on Mathematical Physics, where
this work was presented, for their hospitality.
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