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Further examples of integrable systems in two dimensions

R S KAUSHAL and S C MISHRA*

Department of Physics, Ramjas College, University of Delhi, Delhi 110007, India

*Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India MS received 18 June 1985; revised 25 November 1985

Almtraet. The construction of the second constant of motion of second order for two- dimensional classical systems is carried out in terms ofz -- qi + k/2 and ~ = qt -/q2. As a result a class of Toda-type potentials admitting second order invariants is explored.

Keywm'ds. Second constant of motion; integrable systems; Toda-potential.

PACS No. 03.20;, 03.40, 11.30

In recent times, there have been several efforts in exploring both time-independent (Holt 1982; Inozemtsev 1983; DoriTzi et al 1983) and time-dependent (Katzin and Levine 1982, 1983; Kaushal et a11984; Mishra et al 1984) integrable classical d y , amical systems (see e.g. Whittaker 1972) in two dimensions. The invariants ffconstructed for such systems have utility from several points of view particularly in reducing some nonlinear dynamical problems to a quadrature, in solving several problems of plasma physics and hydrodynamics, in the study of classical analogue of Yang-Mills field equation (Chang 1984). The propagation of wave trains and solitary waves in a lattice was studied by Toda (1967) in terms of an exponential potential of the type

V(ql, q2) = ~'+ exp (q2 + ~/~-ql ) + ~- exp (q2 -- ~P3-q 1) + flexp ( -- 2q2). (1) The integrability of this system was further studied by Ford et al (1973) and an exact second constant of motion (the first being the Hamiltonian) has been obtained (Berry 1978; Holt 1982; Inozemtsev 1983; Hall 1983; Ford et a11973) which involves the third orders of momenta namely

I = 3ct + (p, - x/~P2)exp (q2 + x/~ql) + 3~t_ (p~ + x/~P2) exp (q2 - x/~ql)

- 6[3pl exp ( - 2q2) + Px (P~ - 3p~). (2)

Using Lax pair formalism Olshanetsky and Perelomov (1981, 1983) have discussed classical and quantum integrability of a large class of trigonometrical and exponential potentials in one-dimension. The unequal mass case of the free-end-Toda system has been studied recently by Dorizzi et al (1984). Other generaliTations of the Toda system have also been done by taking recourse to Lie algebraic structures (Bogoyavlensid 1976;

Mikhailov et a11981; Gutkin 1985). Now the question is whether we really have to go to third order invariants (cf. equation (2)) for this system or whether there is an alternative 109

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110 R S Kaushal and S C Mishra

to Toda potential which admits second order invariants? In this note we present a positive answer to this question and show that there exists a class of Toda-type potentials in two dimensions which admits second order invariants.

The method which we adopt here is the one used earlier for the study of both time- dependent (Kaushal et a11984; Mishra et al 1984) and time-independent (Kaushal et al 1985) systems in two dimensions. In fact, the complexitication of the two dimensions has not only led to the reproduction of known results but also provides some new integrable systems. While the earlier work (Kaushal et al 1985) was mainly concerned with the study of third and fourth order invariants, here we extend our analysis to rather simpler second order case.

We consider a dynamical system described by the Lagrangian

L = ~2-~- V(z, z-), (z = ql +iqz; z = p~ + iP2), (3)

with the concomitant equations of motion+

~V _ 2 0 V

= - 2 z = a z " ( 4 )

We assume the existence of the second constant of motion (in the following called invariant), l, up to second order in momenta in a general form as*

I = ao +X2aq ~i ~j, (5)

where i, j = 1, 2, ~1 = z, ~e = ~, and the coeffÉcients ao and a~j are functions ofz and only. The coefficient a~j is symmetrized with respect to any interchange of its indices.

Using d l / d t = O, we find from (5),

ao,, ¢, + ½a, + ½a,j( ,¢j + = o. (6)

After accounting for the proper symmetrization of the coefficients and since (6) must hold identically in ~'s, we obtain the following relations:

a o , k -F a j k J + aki,j --'-- O, (7)

ao,i + aij ~i = 0. (8)

Equations (7) and (8) after using (4) yield the following set of partial differential equations,

+ Although from the applications point of view the Lagrangian L and hence the potential V need to be real functions of z and ~, it will turn out later that this reality condition is not necessary because the arbitrary constants of integration which occur in the solutions (cf. equations (16-18)) can suitably be chosen to yield real V and the invariant L

* Here we assume that the invariant contains either only even powers or only odd powers of momenta. This is mainly because it can be seen (Holt 1982; Inozemtsev 1983 and Kaushal et a11985) from a general form of I that there does not exist any coupling between the corresponding coefficients in the resultant set of partial differential equations for the case of time-independent systems. This is, however, not the case for time- dependent systems (Katzin and Levine 1982, 1983; Kaushal et al 1984; Mishra et al 1984; Kaushal 1985).

Further justification to this assumption is based on the time-reversal symmetry of the corresponding Lagrangian.

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3all = 0 , aa22

~z ---~- = 0, (9, 10)

c3axl /~ ~ - 2 ~ 2 = 0 , cqa22 + 2 coal2 = ~ ~ 0, (11,12)

~ao

~V ~V

~--~- = 2 all -~z + 2 at2 ~ z ' (13)

~ao aV ~V

= 2a22 ~-z + 2at2 ~ - z . (14)

To solve these equations we notice from (9) and (10) that

all = a l l ( ~ = ~l(2-') and a22 = a22(z) -- el(Z) •

N o w differentiating (I I) and (12) w.r.t. ~ and z, respectively and subtracting the results we obtain

d2~,i/d~ 2 = d2$i/dz 2 = constant cl (say), (15) whose solutions are the polynomials,

all = ~/11 = ½Cl Z 2 + C 2 ~'+C3, a22 = {#l = ½Cl 22 + C4 Z + C 5.

(16) (17)

Using these expressions for at t and a22 once again in (11) and (12) and integrating the resultant equations, we obtain an expression for al2 as

at2 = - ½ c t

z-~-½c2 z -½c 4 ~ + c6.

(18)

In (16)-(18) c{s are some arbitrary constants of integration except ct, which is a separation constant. The solution for ao is not trivial unless the form of V is known.

Therefore, we eliminate ao at this stage by differentiating (13) and (14) w.r.t. $ and z, respectively and noticing that

(~2ao/a-Z.~z)= (692ao/c?~.Oz).

This would lead to a

"potential" equation of the form

3(ctz + c 4 ) ~ + ( ½ c l z 2 +c4z + c s ) ~

_ ~V -2 a2V

= ~ (clz

+c2)~z-z +

(½clz

+c2~+c3) ~z2. (19)

As such the solution of this potential equation is difficult; therefore, we assume

V(z~z-~

to be separable in the form

V(z, z-) = U(z)'w(z--),

(20)

which reduces (19) to a pair of equations

d2U

(clz+c,)d-~ -U

U = 0, (21a)

(½c1: + c , z . z -Co

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112 R S Kaushal and S C Mishra

_-2 - d 2 w 3 dw

(½clz +c2z +c3)-d-~+~(c I e +c2)-d-i-CoW = 0, (21b), where Co is another separation constant. Now, we consider the following two cases:

Case (a): Let cl = c2 = c4 = 0 in (21a) and (21b), then the general solutions o f the resultant equations can be easily obtained as

U(z) = ~1 exp (~1z)+[31 exp (-81z), w(z--) = ~2 exp (e2~)+f12 exp (-82~), which lead to the form of V as

V(z, z-) = vt exp (elz + ez~) + v2 exp (81z - e2~) + v3 exp ( - 81z + 82~)

+ v4 exp ( - ~ l z - e 2 ~ ) , (22)

where el = ~ e2 = ~ a n d vt = 0t10~2, V 2 = 0~1~2, V 3 = O t 2 ~ l , V 4 = ~lfl2.In this case, the coefficients all from (16)-(18) become

a l l -~- c3, a 2 2 ~- c5, a 1 2 ~ c6,

and the coefficients ao from (13), (14) and (22), takes the form

ao = To + 2 (c6 + cx/~ac5) [v l exp (81z + e2~) + v,t exp ( - e t z - ~-2~)]

+ 2(c6 - cx/~acs) Iv2 exp (etz - e2~) + v a exp ( - ~tz + 82~)],

where ][o is a constant of integration. With these expressions for the coefficients a 0 and ao the invariant (5) corresponding to the potential (22) can be written as (since g = q t -I- iq2 and ~ = P l -I- ip2)

I =-~0 + 2(c6 + cx/~3cs){vt exp [(et + e2)ql + i(et -e2)q2]

+ v4 exp [ - (el + e2)ql - i(81 - £ 2 ) q 2 ] }

+ 2(c6 - cx/~acs){v2 exp [(et + 82)ql + i(et + 82)q2]

+ v3 exp I- - (el - g2)ql - - i (~1 q- g 2 ) q 2 ] }

+½(2c6 +ca +cs)p 2 +½(2c6 -ca -cs)p 2 +i(c3 -cs)p~p2. (23) In order to have an analogy with Toda potential (1) we relate the arbitrary constants el and ~2 by

i(8~ -82) = 1, 8~ +82 = x/~, which will imply that

c,,/7~3cs = Co; c3 + c 5 = Co; c3 - c 5 =

-,¢/3iCo.

As a result the invariant (23), corresponding to the potential

V(ql, q2) = V1 exp (q2 + x/~qx)+ v2 exp (-- i(ql -- x/~q2))

+va exp(i(qt--x/~q2))+v4 exp(--q2--x/~qx), (24)

(5)

can be written as

I = ~ o +2(c~ + Co){½p~ + v l exp

(q2 4-x/~ql)+V+

exp ( - q 2 - x/~ql}

+ 2(c6

--Co){½p~

+ v2 exp ( - i ( q t

- v/3q2))

+ v3 exp (i(ql - x//3q2)) } x/~copl P2 -½Co(P 2 -P22) (25) Note that only for Co = ](0 = 0 and ca = ½, I takes the form of Hamiltonian; otherwise the following four special cases of (24) and (25) by setting-£o = 0 are of interest. Before discussing these special cases a few remarks about the potential (24) are in order. While the potential (24) is of exponential type it consists of four terms unlike the Toda potential (I) which has only three exponential terms. In fact, the standard Toda potential has been deduced (Ford et al 1973) for a linear chain of three particles in a periodic lattice. Further using an appropriate canonical transformation for the change of variables, form (I) is obtained. O n the other hand, the potential (24) can be visualized as corresponding to four particles in a similar lattice (Gutldn 1985) but with a choice that two relative coordinates are n o w made cyclic.

(I) For vl = v, = Vo and v2 = v3 = Vo, V and I are given by V(ql, q2) = 2[Vo cosh(q2 + x/~q0 +V0 cos (ql - x//3qe)],

I = (c6 + Co) [p2 + 4Vo cosh (q2

+ %/~ql)]

+ tc~ - Co) [P~ + 4Vo cosfql - ~/3q2)]

+ v/3copl P2

- 1 (pl - p2). 2 (26a)

(ii) For vl = v, = Vo and -v2 = v3 =Vo, V a n d I are given by

V(q,,

q2) = 2[Vo cosh (q2 + v/3q, ) + i~o sin (qt - v/3q2)],

I = (c 6 + Co) [.p~ + 4Vo cosh(q2 + v / 3 q l ) ]

+ (c6 - Co) [p~ + 4i~o sin (ql - v/3q2)] +

v/3CoP,P2 - ~ o ( P 2 - P #

(26b)

(iii) F o r v I = - v + = Vo and v2 = v3 = Vo, V and I are given by

V = 2[Vo sinh(q2 + v/3q~) +To cos(ql - v/3q2)], (26c) I = (c 6 + Co)[p 2 + 4Vo sinh(q2

+ %//3ql

)]

+ (C~ -- CO) [P~ + 4VO Cos(q, -- x/~q2)] +

v/3cop, P2 - ~o (P~ - P~).

(iv) F o r vl = - v4 = Vo and - v2 = v3 = Vo, V and I are given by

V = 2[Vo sinh(q2 +. x / ~ q , ) + i~o sin (q, - x/~q2)], (26d) I = (ca + Co)[p 2 + 4Vo sinh(q2 + v / 3 q , ) ]

+ (c+ - Co)LV + 4, o sintq, - + J eop,p - - pI).

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114 R S Kaushal and S C Mishra

Case (b): Setting c2 = ca = c4 = cs = 0 in (21a) and (21b), the solution of the resultant equations becomes simple and is given by

U(z) = A12 -l±q, w(z--) = A2~ - l ± q , which imply the form of V(z, z-) as

V (z, z-) = A(z~) -l ±7, (27)

or

V([zl) = A r - 2 ± 2~ = A(q21+ q2) -l ±rt,

where A = AIA2, r 2 = z~, and r/is a real number given by t / = (1 + 2 Co~C1) t/2. For this case, the coefficients aij from (16)-(18) reduce to

atl = ½ctz2,a22 = ½cl z2, a12 = -½ctz-f-l-Cr, and the coefficient ao, after using (27) in (13) and (14) becomes

ao = 2 Acr(zz-)-l±~+ko,

where ko is a constant o f integration. Finally, the invariant (5) turns out to be I = ko +

2Ac6

r -2±2~ --et(qtP2 --q2Pl) 2 + Cr(Pl 2 +p~), (28) which is, in fact, not a new invariant but represents a linear combination of the Hamiltonian and the square of angular momentum. In this way we recover a known integrable system.

If we assume the separability of V(z,z-) in the form, V(z, z--) = f l (z) +f2tz-),

then (19) still reduces to a pair of equations which in a compact form, can be expressed as

d ( d f , ~ d { 3/2df2~

dz xal/2 =

- _ -dT /

where Xt = ½Cl z2 + c4z + cs, X2 = ~ t ~2 + c2g+ c3 and 2 is a separation constant. For

= 0, the solution of these equations further yields an integrable system of the type

2At (cl z + c4) 2A2 (ct~+ c2)

" ( z , l ) = (2c, cs - c 2) (½ct z z + c 4 z + c s ) "~ (2cl c3 - c 2) (~1~2+ czZ+ c3) ' where At and A2 are the constants of integration. It may be noted that the structure of

V(z,-~) turns out to be more complicated when we take 2 4: 0.

Here we have considered only a few particular solutions of the "potential" equation (19) and this could lead to various integrable systems (cf. (26a)-(26d)). To obtain other integrable systems which admit second order invariants it may be of interest to explore more general solutions of this equation in the sense of Olshanetsky and Perelomov (1981). In fact, to some extent the integrable systems obtained here are the extension of their results to two dimensions. Alternatively, if for a given system V(q t,q2) the arbitrary

(7)

constants ci's in (19) can be determined uniquely, then also it is possible to construct the corresponding invariant (rationalization method (Kaushal et al 1984; Mishra et al 1984)). Further, it is worthwhile to study the wave propagation in a lattice through the potentials (26a)--(26d) which have Toda-type features but admit now second order invariants.

To summarize, our method based on the complexification of the two dimensions of a two-dimensional system leads to quite a few new integrable systems which to the best of our knowledge has not been discovered earlier.

Acknowledgements

The authors are thankful to Dr K C Tripathy for discussions. One of us (SCM) gratefully acknowledges the financial help from CSIR, New Delhi. The authors also thank the referees for several useful suggestions. A part of this work was carded out when one of the authors (RSK) was in West Germany as visiting Av. H. Fellow.

References

Berry M V 1978 AlP Conf. Proc. 46, 16

Bogoyavlenski O I 1976 Commun. Math. Phys. 51 201 Chang S J 1984 Phys. Rev. D29 259

Dorizzi B, Grammaticos B and Ramani A 1983 J. Math. Phys. 24 2282 Dorizzi B e t al 1983 J. Math. Phys. 24 2289

Doriz~ Bet al 1984 3. Math. Phys. 25 2200

Ford J, Stoddard S P and Turner J S 1973 Proor. Theor. Phys. 50 1547 Gutkin E 1985 Physica D I 6 398

Hall L S 1983 Physica D8 90 Holt C R 1982 d. Math. Phys. 23 1037 lnozemtsev V I 1983 Phys. Lett. A96 447

Katzin G H and Levine J L 1982 J. Math. Phys. 23 552 Katzin G H and Levine J L 1983 J. Math. Phys. 24 1761 Kaushal R S 1985 Pramana (3. Phys.) 24 663

Kaushal R S, Mishra S C and Tripathy K C 1984 Phys. Lett. AI02 7 Kaushal R S, Mishra S C and Tripathy K C 1985 J. Math. Phys. 26 420 Mikhailov A V e t al 1981 Commun. Math. Phys. 79 473

Mishra S C, Kaushal R S and Tripathy K C 1984 2. Math. Phys. 25 22t7 Olshanetsky M A and Perelomov A M 1981 Phys. Rep. C71 313 Oishanetsky M A and Perelomov A M 1983 Phys. Rep. C94 313 Toda M 1967 J. Phys. Soc. Jpn 23 501

Whittaker E T 1972 Analytical dynamics (London: Cambridge Univ. Press) p. 332

References

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