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Hydrodynamic symmetries for the Whitham equations for the nonlinear Schrödinger equation (NSE)
Volume 154, number 9
PHYSICS LETTERS A
22 April 1991
Hydrodynamic symmetries for the Whitham equations for the nonlinear Schrodinger equation (NSE) V.R. Kudashev Department ofMathematics, Bashkirian State University, FrunzeStreet 32, Ufa 450074, USSR
and S.E. Sharapov
1
I. V. Kurchatov Institute ofAtomic Energy, Ploshad’ Kurchatova 46, Moscow 123182, USSR Received 7 January 1991; accepted for publication 20 February 1991 Communicated by D.D. Hoim
The structure of the infinite series of the hydrodynamic symmetries forthe Whitham equations, derived for one-zone solutions ofthe NSE, is presented. It is shown that the structure is determined up to an arbitrary function. Explicit formulae for a number ofexact solutions of the Whitham equations are obtained.
1. There has been a significant progress in the study of the Whitham hydrodynamic equations [1] (i.e. the equations, which are derived under Whitham averaging [2], see eq. (5) below) during the last years. An analysis of exact solutions of the Whitham equations by the generalized hodograph method [3] (see also refs. [1,4—6]) is based on the use of hydrodynamic type symmetries. The purpose of this Letter is to describe a special infinite series (which is determined up to an arbitrary function) of the hydrodynamic symmetries for the Whitham equations, derived for one-zone (one-phase) solutions of the NSE. The formulae for the hydrodynamic symmetries obtained in this Letter can be used for the construction of a large number of exact solutions of the Whitham equations. These solutions can be used for the problem of nondissipative shock waves [7] (described by “defocusing” NSE) and for the problem ofthe nonlinear stage of the long-wavelength limit of the modulation instability of the simplest periodic solutions (described by “focusing” NSE [8—10]).
To whom correspondence should be addressed. Elsevier Science Publishers B.V. (North-Holland)
2. The nonlinear Schrodinger equation, iq1 + ~ + 2crI qI 2q= 0, a= ±1 has one-zone solutions [9—111
,
q= ,,jR~exp(iq), ~p=yi+h(6), O=i~—wt, yi=icx—Qt,
(1)
(2) (3)
wheref(O) and h(O) are elliptic functions. A complete description of the solutions (2), (3) and their relation with Riemann invariants is beyond the scope of this Letter (see refs. [9—11]). But for the sake of simplicity we can use all the necessary relations in the case of “defocusing” NSE (~= 1) in the following form [7,11]: —
f=b 2((b 2(x— Ut) s2) + (b2 —b3) sn 2/f, 1 —b3)” ~ =3(1/2 (b 1 b2b3) “ = U2/4 b 2/f, (4) 1 b2 b3 + U( b1 b2b3)” where s2=(b 2—b3)/(b1—b3) is the modulus of the Jacobi elliptic function sn, b>~b2 ~ b3. The Whitham equations for (1)—(3) in the Rie—
~
—
—
—
—
445
Volume 154, number 9
mann invariants r, (i= 1, 2, 3, 4) were obtained in ref. [11]. One can write these equations in the form (cf. refs. [11,12]) r,, +(Q1U)r1~=0, U=r1 +r2 +r3 +r4, Q1 =
1+2
r1 1
—
a ( r2
22 April 1991
PHYSICS LETTERS A
—
(5)
8
r2 r4) / (r1 —
—
r4) Or1
‘
~
(11)
=
—
or,
Ork
Let us show that the symmetries (9), where V is defined by (10), are defined up to an arbitrary function. In fact, if we apply a Laplace-type integral transformation to the system of equations (10),
Q2=1+2 1—~(r~—r3)/(r2—r3) r2—r1 Or2’ a
V(r)=JW(p)exp(_~pjrj)dpidp2d~3dp4,
Q3=1+2
(12) we obtain a system of first-order equations for W,
—
p~( r2
r3—r4 r4) / (r2 —
—
0 r3) Or3’
Q4=l+2 r4—r3 0 (6) 1 —~(r~ —r3)/(r~—r4) Or4’ 2= (r s 1 —r2) (r3 —r4)/(r~—r3) (r2 —r4) 2)/K(s2) (7) ~i_—E(s where E and K are the complete elliptic integrals of the second and first type. The variables b. in (4) are related with r, as follows [7,11]: ,
b~=(r
2/4, 1+r2—r3—r4) b 2/4, 2 = (r1 +r3 —r2 —r4) = (r 2/4. (8) 1 +r4 —r2 —r3) Let us note that in the case of focusing NSE (a= + 1) the explicit formulae (4), (7), (8) are different, but eqs. (2), (3), (5) and (6) remain the same. Following ref. [1], we call eq. (5) (and eq. (9)) hydrodynamic type equations. 3. Let us search for the series of hydrodynamic symmetries for (5) in the form
law
P~_2p1p1
ow —
+
(p1—ps) W=0,
(13)
and the differential algebraic relations forrelations (13) of(11) the following are replaced type:by
Pj~ik—Pi~jk=Pk~~~
(14)
The system (13) has solutions W=w(p
2, (15) 1+p2+p3+p4)(p1p2p3p4)~ where w(z) is an arbitrary function of the argument z. So one can see that (12) and (15) determine the solutions of eq. (10) up to the arbitrary function w(z). Eq. (10) admits three one-parameter point transformation groups: the translation group, characterized by the generator T, T= ~
(16)
the scaling group, characterized by the generator S,
a
r 1~+(Q1V)r1~=0,V=V(r1, r2, r3, r4).
(9)
This form is analogous to the form of the symmetries for the Whitham equations, derived in ref. [121 for the case of cnoidal waves of the Korteweg— de Vries equation. To have (9) and (5) commute (i.e. the condition r.1~=r,1,to be fulfilled) the following defining equations for V must be satisfied: 02V 8V OV + =0. (10) —
There exists a differential relation for eq. (10), 446
S= ~ r1~
(17)
and the group i~= r,/ (1 2ar,) —
V(i’,)=V(r1)[(l—2ar1)(l—2ar2)(l—2ar3) 2, X(l—2ar4)]” characterized by the generator L= ~(2r~_~_ +r,’~. Or 1 /
(18) (19)
Volume 154, number 9
PHYSICS LETTERS A
Eqs. (10) are also invariant with respect to the transformations (20) P~=l/r1, V(?,)=V(r1)(r1r2r3r4)”2 and
22 April 1991
act on the solution characterized by (27) and (29), one can obtain a solution that is analogous to (27), but with the function v defined by v=K(l—s2) (30) .
It is interesting to note that the solution (27), (29) r~—~r 2, r2 —~r3, r3 —~r~ r~ —~r4 r1—~—r1
(21)
coincides with the period of the function f (see eq. (4)) up to a constant.
Relations (16)—(21) can be used for the construetion of a separate series of solutions V. One can see that (20) can generate a solution ~ defined by
4. The solutions of the Whitham equations (5), which are invariant with respect to the combination ofthe symmetry (9) and the symmetry generatedby the scaling group i=at,.~=ax, are defined by the
V( r.) = (r1 r2 r3 r4)
—
I /2{
V( f,) } I
i Ir,
,
(22)
system of equations (see refs. [6,121)
where Vis a solution ofeq. (10). For example, ifwe substitute the obvious solution
[(Q,V)t(Q,U)+x]r1~0 (31) By imposing the constraint R=0 (R is the rank of
V0=const
into the right side of eq. (22), we obtain a new solution
the matrix [(Q1V)—t(Q1U)+x]), r1~~0 we obtain an algebraic construction of the solution of eq. (5), the so-called “generalized hodograph method” [1,3— 6]:
Vo=(rir2r3r4)~~/2.
(Q1V)=t(Q1U)—x.
(23)
(24)
The solution (24) corresponds to the choice w ( z) = const in (15). To obtain a new series of explicit solutions one can let the generators L and T operate on the “seed” solutions V0 and V0, characterized by eqs. (23) and (24): V~=L~V0,
(25)
P~~=T~V0.
(26)
.
(32)
By imposing the constraint R=N (where N=0, 1, 2, 3), we obtain the “N-rank solutions” (this term was proposed by A.B. Shabat). It is obvious that the case R=4 corresponds to r1=const. By analogy with the case of the KdV equation [1], eq. (5) has the scale-invariant solutions r1=t~l1(x/i’~’) ,
(33)
It is interesting to note the existence of solutions of eq. (10) of the type
characterized by(31), the power y. Using thethe construction of the solutions one can see that symmetry defined by (9) and (25) gives the explicit scale-in-
(27) ~.J(r1 —r3)(r2—r4) 2, and v(~)is a solution of the hypergeowhere metric ~=s equation
variant solutions of the Whitham equations (5) in the form (33), where ‘= l/(n— 1) (34) The symmetry defined by (9) and (26) gives the solutions characterized by .
(28) (35)
Eq. (28) has a particular solution of the form v=K(~)=K(s2).
(29)
The solution characterized by (27) and (29) is a compatible invariant of the operators Tand L (i.e. TV 1 = L V~= 0). By letting the transformation (21)
An additional analysis shows that the solutions (33) and (34) are of interest to the problem of nondissipative shock waves in the case of defocusing NSE [7] and the solutions (33) and (35) are of interest to the problem of the nonlinear stage of instability 447
Volume 154, number 9
PHYSICS LETTERS A
of the simplest periodic solutions of focusing NSE [8—10]. 5. Earlier [7] the reduction ofeq. (5) to the Whitham equations for KdV cnoidal waves [1,21 was considered. But the procedure of the reduction was not entirely correct. The correct reduction has, for example, the following form:
22 April 1991
where Ck=const, and vk(~1) is a solution of the hypergeometric equation dvk d2vk ~( 1 + [1—(~—k)~] —~)
—~~-—
+ k(1 k) Vk/4 = 0. (42) The analysis shows that the solutions of the Whitham equations defined by (41) and (42) in ac—
r
—~
0 = const, I
r0 I
x — r~ ~,
~
>> max (I ra I)
,
ra
—
~= t,
ra/3
,
a = 1, 2, 3
.
(36)
Under this reduction the system of equations (5) passes to the Whitham equations for the KdV equation [1,2] in the (~, 1) coordinate system. Of special interest is the reduction of eq. (5) of the type [7,13] = const,
r4 = const
.
(37)
In this case eq. (5) describes the so-called “quasisimple waves” in the NSE hydrodynamics and, by taking into account (36), in the KdV hydrodynamics too. With condition (37) one obtains for the symmetry (9) one equation only: 2V OV Ov 2(r 8 3—r2) Or2 +~—~—=0. (38) Or3
=r3 —r2
,
X2
=r2 +r3
(39)
reduces to the Euler—Poisson—Darboux equation 2V O2V+I~±1 (40) 0 = 04 x~Ox~ which allows separation of variables in different coordinate systems [141. Let us demonstrate, for example, the solutions of eq. (40) in the form V= ~ Ckx~vk(i~),,~xf/x~, k
448
[13]. We intend to analyse the details of the physical consequences of these results in future publications. References [11BA.
Dubrovin and S.P. Novikov, Usp. Mat. Nauk 44
(1989) 29, and references therein, [21 GB. Whitham, Proc. R. Soc. A 283 (1965) 238. [3] 5.P. Tsarev, Doki. Akad. Nauk SSSR 282 (1985) 534. [4] I.M. Krichever, Funkt. Anal AppI. 22 (1988) 37. [5] Y. Kodama, Phys. Lett. A 135 (1989) 171; Integrability of hydrodynamic type equations, in: Proc. 4th mt. Workshop, Vol. 2,Kudashev Kiev(l989) 115. Sharapov, Teor. Mat. Fiz. 85 [6]V.R. andp. S.E. (1990)205; Preprint IAE-51 16/6, Moscow (1990). [7]A.V. Gurevich and A.L. Krulov, Zh. Eksp. Teor. Fiz. 92 (1987) 1684.
Eq. (38) with the aid of the transformation X1
cordance with (31) (two-rank solutions) give examples of explicit solutions of the problem of quasisimple waves, analysed earlier (numerically) in ref.
(41)
[8] G. Rowlands, J. Inst. Math. AppI. 13 (1974) 367. [9]Y.C. Ma and M. Ablowitz, Stud. AppI. Math. 65 (1981) 113. [10] 255. E. Infeld and J. Ziemkiewicz, Acta Phys. Polon. A 59 (1981) [11]M.V.Pavlov,Teor. Mat. Fiz. 71 (1987) 351. [12] yR. Kudashev and S.E. Sharapov, The legacy of KdV symmetries under Witham averaging and hydrodynamic symmetries of the Whitham equation, preprint IAE-522 1 / 6, Moscow (1990), to be published in Teor. Mat. Fiz. [13] A.V. Gurevich, A.L. Krulov and N.G. Mazur, Zh. Eksp. Teor. Fiz. 95 (1989) 1674. [14] E.G. Kalnins and M. Miller Jr., J. Math. Phys. 17 (1976) 369.
PHYSICS LETTERS A
22 April 1991
Hydrodynamic symmetries for the Whitham equations for the nonlinear Schrodinger equation (NSE) V.R. Kudashev Department ofMathematics, Bashkirian State University, FrunzeStreet 32, Ufa 450074, USSR
and S.E. Sharapov
1
I. V. Kurchatov Institute ofAtomic Energy, Ploshad’ Kurchatova 46, Moscow 123182, USSR Received 7 January 1991; accepted for publication 20 February 1991 Communicated by D.D. Hoim
The structure of the infinite series of the hydrodynamic symmetries forthe Whitham equations, derived for one-zone solutions ofthe NSE, is presented. It is shown that the structure is determined up to an arbitrary function. Explicit formulae for a number ofexact solutions of the Whitham equations are obtained.
1. There has been a significant progress in the study of the Whitham hydrodynamic equations [1] (i.e. the equations, which are derived under Whitham averaging [2], see eq. (5) below) during the last years. An analysis of exact solutions of the Whitham equations by the generalized hodograph method [3] (see also refs. [1,4—6]) is based on the use of hydrodynamic type symmetries. The purpose of this Letter is to describe a special infinite series (which is determined up to an arbitrary function) of the hydrodynamic symmetries for the Whitham equations, derived for one-zone (one-phase) solutions of the NSE. The formulae for the hydrodynamic symmetries obtained in this Letter can be used for the construction of a large number of exact solutions of the Whitham equations. These solutions can be used for the problem of nondissipative shock waves [7] (described by “defocusing” NSE) and for the problem ofthe nonlinear stage of the long-wavelength limit of the modulation instability of the simplest periodic solutions (described by “focusing” NSE [8—10]).
To whom correspondence should be addressed. Elsevier Science Publishers B.V. (North-Holland)
2. The nonlinear Schrodinger equation, iq1 + ~ + 2crI qI 2q= 0, a= ±1 has one-zone solutions [9—111
,
q= ,,jR~exp(iq), ~p=yi+h(6), O=i~—wt, yi=icx—Qt,
(1)
(2) (3)
wheref(O) and h(O) are elliptic functions. A complete description of the solutions (2), (3) and their relation with Riemann invariants is beyond the scope of this Letter (see refs. [9—11]). But for the sake of simplicity we can use all the necessary relations in the case of “defocusing” NSE (~= 1) in the following form [7,11]: —
f=b 2((b 2(x— Ut) s2) + (b2 —b3) sn 2/f, 1 —b3)” ~ =3(1/2 (b 1 b2b3) “ = U2/4 b 2/f, (4) 1 b2 b3 + U( b1 b2b3)” where s2=(b 2—b3)/(b1—b3) is the modulus of the Jacobi elliptic function sn, b>~b2 ~ b3. The Whitham equations for (1)—(3) in the Rie—
~
—
—
—
—
445
Volume 154, number 9
mann invariants r, (i= 1, 2, 3, 4) were obtained in ref. [11]. One can write these equations in the form (cf. refs. [11,12]) r,, +(Q1U)r1~=0, U=r1 +r2 +r3 +r4, Q1 =
1+2
r1 1
—
a ( r2
22 April 1991
PHYSICS LETTERS A
—
(5)
8
r2 r4) / (r1 —
—
r4) Or1
‘
~
(11)
=
—
or,
Ork
Let us show that the symmetries (9), where V is defined by (10), are defined up to an arbitrary function. In fact, if we apply a Laplace-type integral transformation to the system of equations (10),
Q2=1+2 1—~(r~—r3)/(r2—r3) r2—r1 Or2’ a
V(r)=JW(p)exp(_~pjrj)dpidp2d~3dp4,
Q3=1+2
(12) we obtain a system of first-order equations for W,
—
p~( r2
r3—r4 r4) / (r2 —
—
0 r3) Or3’
Q4=l+2 r4—r3 0 (6) 1 —~(r~ —r3)/(r~—r4) Or4’ 2= (r s 1 —r2) (r3 —r4)/(r~—r3) (r2 —r4) 2)/K(s2) (7) ~i_—E(s where E and K are the complete elliptic integrals of the second and first type. The variables b. in (4) are related with r, as follows [7,11]: ,
b~=(r
2/4, 1+r2—r3—r4) b 2/4, 2 = (r1 +r3 —r2 —r4) = (r 2/4. (8) 1 +r4 —r2 —r3) Let us note that in the case of focusing NSE (a= + 1) the explicit formulae (4), (7), (8) are different, but eqs. (2), (3), (5) and (6) remain the same. Following ref. [1], we call eq. (5) (and eq. (9)) hydrodynamic type equations. 3. Let us search for the series of hydrodynamic symmetries for (5) in the form
law
P~_2p1p1
ow —
+
(p1—ps) W=0,
(13)
and the differential algebraic relations forrelations (13) of(11) the following are replaced type:by
Pj~ik—Pi~jk=Pk~~~
(14)
The system (13) has solutions W=w(p
2, (15) 1+p2+p3+p4)(p1p2p3p4)~ where w(z) is an arbitrary function of the argument z. So one can see that (12) and (15) determine the solutions of eq. (10) up to the arbitrary function w(z). Eq. (10) admits three one-parameter point transformation groups: the translation group, characterized by the generator T, T= ~
(16)
the scaling group, characterized by the generator S,
a
r 1~+(Q1V)r1~=0,V=V(r1, r2, r3, r4).
(9)
This form is analogous to the form of the symmetries for the Whitham equations, derived in ref. [121 for the case of cnoidal waves of the Korteweg— de Vries equation. To have (9) and (5) commute (i.e. the condition r.1~=r,1,to be fulfilled) the following defining equations for V must be satisfied: 02V 8V OV + =0. (10) —
There exists a differential relation for eq. (10), 446
S= ~ r1~
(17)
and the group i~= r,/ (1 2ar,) —
V(i’,)=V(r1)[(l—2ar1)(l—2ar2)(l—2ar3) 2, X(l—2ar4)]” characterized by the generator L= ~(2r~_~_ +r,’~. Or 1 /
(18) (19)
Volume 154, number 9
PHYSICS LETTERS A
Eqs. (10) are also invariant with respect to the transformations (20) P~=l/r1, V(?,)=V(r1)(r1r2r3r4)”2 and
22 April 1991
act on the solution characterized by (27) and (29), one can obtain a solution that is analogous to (27), but with the function v defined by v=K(l—s2) (30) .
It is interesting to note that the solution (27), (29) r~—~r 2, r2 —~r3, r3 —~r~ r~ —~r4 r1—~—r1
(21)
coincides with the period of the function f (see eq. (4)) up to a constant.
Relations (16)—(21) can be used for the construetion of a separate series of solutions V. One can see that (20) can generate a solution ~ defined by
4. The solutions of the Whitham equations (5), which are invariant with respect to the combination ofthe symmetry (9) and the symmetry generatedby the scaling group i=at,.~=ax, are defined by the
V( r.) = (r1 r2 r3 r4)
—
I /2{
V( f,) } I
i Ir,
,
(22)
system of equations (see refs. [6,121)
where Vis a solution ofeq. (10). For example, ifwe substitute the obvious solution
[(Q,V)t(Q,U)+x]r1~0 (31) By imposing the constraint R=0 (R is the rank of
V0=const
into the right side of eq. (22), we obtain a new solution
the matrix [(Q1V)—t(Q1U)+x]), r1~~0 we obtain an algebraic construction of the solution of eq. (5), the so-called “generalized hodograph method” [1,3— 6]:
Vo=(rir2r3r4)~~/2.
(Q1V)=t(Q1U)—x.
(23)
(24)
The solution (24) corresponds to the choice w ( z) = const in (15). To obtain a new series of explicit solutions one can let the generators L and T operate on the “seed” solutions V0 and V0, characterized by eqs. (23) and (24): V~=L~V0,
(25)
P~~=T~V0.
(26)
.
(32)
By imposing the constraint R=N (where N=0, 1, 2, 3), we obtain the “N-rank solutions” (this term was proposed by A.B. Shabat). It is obvious that the case R=4 corresponds to r1=const. By analogy with the case of the KdV equation [1], eq. (5) has the scale-invariant solutions r1=t~l1(x/i’~’) ,
(33)
It is interesting to note the existence of solutions of eq. (10) of the type
characterized by(31), the power y. Using thethe construction of the solutions one can see that symmetry defined by (9) and (25) gives the explicit scale-in-
(27) ~.J(r1 —r3)(r2—r4) 2, and v(~)is a solution of the hypergeowhere metric ~=s equation
variant solutions of the Whitham equations (5) in the form (33), where ‘= l/(n— 1) (34) The symmetry defined by (9) and (26) gives the solutions characterized by .
(28) (35)
Eq. (28) has a particular solution of the form v=K(~)=K(s2).
(29)
The solution characterized by (27) and (29) is a compatible invariant of the operators Tand L (i.e. TV 1 = L V~= 0). By letting the transformation (21)
An additional analysis shows that the solutions (33) and (34) are of interest to the problem of nondissipative shock waves in the case of defocusing NSE [7] and the solutions (33) and (35) are of interest to the problem of the nonlinear stage of instability 447
Volume 154, number 9
PHYSICS LETTERS A
of the simplest periodic solutions of focusing NSE [8—10]. 5. Earlier [7] the reduction ofeq. (5) to the Whitham equations for KdV cnoidal waves [1,21 was considered. But the procedure of the reduction was not entirely correct. The correct reduction has, for example, the following form:
22 April 1991
where Ck=const, and vk(~1) is a solution of the hypergeometric equation dvk d2vk ~( 1 + [1—(~—k)~] —~)
—~~-—
+ k(1 k) Vk/4 = 0. (42) The analysis shows that the solutions of the Whitham equations defined by (41) and (42) in ac—
r
—~
0 = const, I
r0 I
x — r~ ~,
~
>> max (I ra I)
,
ra
—
~= t,
ra/3
,
a = 1, 2, 3
.
(36)
Under this reduction the system of equations (5) passes to the Whitham equations for the KdV equation [1,2] in the (~, 1) coordinate system. Of special interest is the reduction of eq. (5) of the type [7,13] = const,
r4 = const
.
(37)
In this case eq. (5) describes the so-called “quasisimple waves” in the NSE hydrodynamics and, by taking into account (36), in the KdV hydrodynamics too. With condition (37) one obtains for the symmetry (9) one equation only: 2V OV Ov 2(r 8 3—r2) Or2 +~—~—=0. (38) Or3
=r3 —r2
,
X2
=r2 +r3
(39)
reduces to the Euler—Poisson—Darboux equation 2V O2V+I~±1 (40) 0 = 04 x~Ox~ which allows separation of variables in different coordinate systems [141. Let us demonstrate, for example, the solutions of eq. (40) in the form V= ~ Ckx~vk(i~),,~xf/x~, k
448
[13]. We intend to analyse the details of the physical consequences of these results in future publications. References [11BA.
Dubrovin and S.P. Novikov, Usp. Mat. Nauk 44
(1989) 29, and references therein, [21 GB. Whitham, Proc. R. Soc. A 283 (1965) 238. [3] 5.P. Tsarev, Doki. Akad. Nauk SSSR 282 (1985) 534. [4] I.M. Krichever, Funkt. Anal AppI. 22 (1988) 37. [5] Y. Kodama, Phys. Lett. A 135 (1989) 171; Integrability of hydrodynamic type equations, in: Proc. 4th mt. Workshop, Vol. 2,Kudashev Kiev(l989) 115. Sharapov, Teor. Mat. Fiz. 85 [6]V.R. andp. S.E. (1990)205; Preprint IAE-51 16/6, Moscow (1990). [7]A.V. Gurevich and A.L. Krulov, Zh. Eksp. Teor. Fiz. 92 (1987) 1684.
Eq. (38) with the aid of the transformation X1
cordance with (31) (two-rank solutions) give examples of explicit solutions of the problem of quasisimple waves, analysed earlier (numerically) in ref.
(41)
[8] G. Rowlands, J. Inst. Math. AppI. 13 (1974) 367. [9]Y.C. Ma and M. Ablowitz, Stud. AppI. Math. 65 (1981) 113. [10] 255. E. Infeld and J. Ziemkiewicz, Acta Phys. Polon. A 59 (1981) [11]M.V.Pavlov,Teor. Mat. Fiz. 71 (1987) 351. [12] yR. Kudashev and S.E. Sharapov, The legacy of KdV symmetries under Witham averaging and hydrodynamic symmetries of the Whitham equation, preprint IAE-522 1 / 6, Moscow (1990), to be published in Teor. Mat. Fiz. [13] A.V. Gurevich, A.L. Krulov and N.G. Mazur, Zh. Eksp. Teor. Fiz. 95 (1989) 1674. [14] E.G. Kalnins and M. Miller Jr., J. Math. Phys. 17 (1976) 369.