Symmetries and solutions of the vector nonlinear
Schrödinger equation (*)
A. SCIARRINO(1) and P. WINTERNITZ(2)
(1) Dipartimento di Scienze Fisiche
Università di Napoli Federico II and INFN, Sezione di Napoli - I-80125, Napoli, Italy
(2) Centre de Recherches Mathématiques, Université de Montréal
C.P. 6128, Succ. Centre-ville, Montréal, Québec, Canada H3C 3J7
(ricevuto il 20 Settembre 1996; approvato il 20 Novembre 1996)
Summary. — The Lie point symmetries of a coupled system of nonlinear Schrödinger equations with a cubic nonlinearity are used to obtain analytic solutions. The equation is integrable in 1 11 dimensions. The emphasis in this article is on the ( 2 11)-dimensional case, involving 2 or 3 coupled waves.
PACS 03.65.Fd – Algebraic methods.
PACS 02.30.Jr – Partial differential equations. PACS 02.30.Hq – Ordinary differential equations.
1. – Introduction
The purpose of this article is to obtain new solutions of the vector nonlinear Schrödinger equation (VNLSE), invariant under various subgroups of the symmetry group of this equation. The equation to be studied is
ict1 Dc 4 (c c) c , (1.1)
where c CN is an N component complex vector and D is the Laplace operator in n
dimensions. The bar over c denotes Hermitian conjugation. In much of the article we restrict to the case of two or three waves (N 42, 3) and two space dimensions (n 42).
Equation (1.1) plays an important role in many branches of physics. Thus, in nonlinear optics [1] it describes the interaction of electromagnetic waves with different polarizations propagating in nonlinear media, e.g. in an isotropic plasma.
More generally, eq. (1.1) describes the interaction of N waves, propagating in a regime that for one wave leads to the usual (scalar) nonlinear Schrödinger equation.
(*) The authors of this paper have agreed to not receive the proofs for correction.
These can for instance be water waves in a deep fluid [2-4], interacting on the surface or propagating at different levels (internal waves) [5].
In one space dimension (n 41) the VNLSE is known to be integrable via the inverse scattering transform and similar techniques [6-9]. Moreover, the nonlinear Schrödinger equation and its vector generalizations have been shown to be gauge-equivalent to the Heisenberg ferromagnet equations and their generalizations [9, 10]. Our approach does not rely on integrability and is hence not restricted to the case of one space dimension. We shall find the group G of local Lie point transformations leav-ing eq. (1.1) invariant. The subgroups of G will then be used to reduce the VNLSE to a system of algebraic equations, or ordinary differential equations. These will be compat-ible with specific types of boundary conditions, such as translational invariance, or a cylindrical geometry. The reduced equations will be solved whenever possible, in par-ticular when they have the Painlevé property [11-15]. In this sense the present article uses the same approach for the VNLSE that was used earlier for the scalar NLSE [16-18] and also for the VNLSE involving 2 waves [19].
In sect. 2 we find the Lie algebra L of the symmetry group of the VNLSE (1.1) as well as the corresponding symmetry group G. The subalgebras of L that will be needed to perform symmetry reduction are classified in sect. 3. In sect. 4 we give all solutions obtained by reducing the VNLSE to a set of algebraic equations. Translationally in-variant solutions, i.e. solutions that essentially come from a VNLSE in 1 11 dimen-sions, are discussed in sect. 5. Section 6 is devoted to cylindrically invariant solutions. Other symmetries are briefly discussed in sect. 7.
2. – The symmetry group and its Lie algebra
We shall look for the local Lie group of local Lie point transformations
tA 4Tg(t , x K , c , c) , xAi4 Xi , g(t , x K , c , c) , cA4 Cg(t , x K , c , c) (2.1)
such that cA(tA, xKA) is a solution, whenever c(t , xK) is one. The algorithm for finding the Lie group (2.1) is an old one, due to S. Lie and is presented in many books and reviews on the subject [20-23].
Instead of constructing the symmetry transformations directly, one looks for the Lie algebra L of the symmetry group G. The symmetry algebra is realized by vector fields of the form
v
× 4ji¯xi1 t¯t1 fm¯cm1 fm* ¯cm*.
(2.2)
The coefficients ji, t , fmand fm* are functions of x
K
, t , c and c *, to be determined from the equation [20-23]
pr v× QENE 404 0 . (2.3)
Here E 40 is the equation under study and pr v× is the k-th prolongation of v×, where
k is the order of the equations. Computer programs exist that realize this algorithm
and provide a system of “determining equations” for the coefficients of the vector field
v
×. The programs partly or completely [24] solve the determining equations (for a MACSYMA program and a review of other ones, see ref. [25]).
N 43 and introduce the moduli and phases of the components of c, putting
cm4 rmeifm, 0 GrmE Q , 0 G fmG 2 p , m 41, 2, 3 .
(2.4)
We rewrite the VNLSE (1.1) as a system of real equations, use the MACSYMA program SYMGROUP [25] and obtain a set of determining equations for the coefficients t , ji fm and f* in eq. (2.2). The obtained linear partial differentialm equations are easy to solve and we obtain the symmetry algebra. A convenient basis is given by the following operators:
(2.5a)
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/
´
P04 ¯t, Pi4 ¯xi, Ki4 t¯xi1 1 2 xi!
m ¯fm, Lik4 xk¯xi2 xi¯xk, M 4!
m ¯fm, (2.5b) D 4t¯t1 1 2!
i xi¯xi2 1 2!
rm¯rm, (2.5c) C 4t2 ¯t1 tk
!
i xi¯xi2!
m rm¯rml
1 1 4g
!
i x2 ih
!
m ¯fm, (2.5d ).
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/
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´
T1 , mn4 1 2{
sin (fm2 fn)(rn¯rm2 rm¯rn) 1cos (fm2 fn)u
rn rm ¯fm1 rm rn ¯fnv
}
, m En , T2 , mn4 1 2{
cos (fm2 fn)(rn¯rm2 rm¯rn)2sin (fm2 fn)u
rn rm ¯fm1 rm rn ¯fnv
}
, m En , T3 , mn4 1 2 (¯fm2 ¯fn) , n 4m11 ( mod 3) .The range of parameters above is 1 Gm, nG3, 1 Gi, kG2 and so for instance Lik
represents a single rotation operator L124 2L21. We also have
. / ´ T1 , mn4 T1 , nm, T2 , mn4 2T2 , nm, T3 , mn4 2T3 , nm, T3 , 121 T3 , 231 T3 , 314 0 , Tjmm4 0 ( no summation ) . (2.6)
The symmetry algebra L is hence easily identified as
L 4sch (2)5su (3) ,
(2.7)
i.e. the direct sum of the Schrödinger algebra [26-29] in 2 11 dimensions and the Lie
algebra of the group SU( 3 ). Thus ]P0, P1, P2, K1, K2, L12, M( generate the extended
Galilei group (translations, proper Galilei transformations; rotations and overall changes of phase); D generates dilations, whereas C provides nonrelativistic conformal
transformations. The operators Ta , mn, a 41, 2, 3, m, n41, 2, 3 generate SU(3),
realized by the non-Abelian constant gauge transformations
cA4 Uc , U1U 4I , U C3 33,
(2.8)
For n F3 and N arbitrary the symmetry algebra of the VNLSE generalizes from (2.7) to
L 4gs (n)5su (N) .
(2.9)
Here gs (n) denotes the Galilei-similitude algebra given by (2.5a) and (2.5b) with the corresponding values of n and N . The gauge group SU(N) is generated by the operators (2.5d ) with 1 Gm, nGN. The conformal transformations (2.5c) do not generalize to higher dimensions for eq. (1.1). We mention that in one space dimension the equation
ict1 cxx4 (c c)2c
is conformally invariant, i.e. the right-hand side is quintic, rather than cubic. The transformations corresponding to the Galilei-similitude algebra are global ones, namely
.
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´
tA 4el (t 2t0) , x A 4elO2]cos a [x 2 x 01 v1(t 2t0) ] 1sin a[y2y01 v2(t 2t0) ]( ,yA 4elO2]2sin a [x 2 x01 v1(t 2t0) ] 1cos a[y2y01 v2(t 2t0) ]( ,
r Am4 e2lO2r m, f A m4 fm1 1 2 [v1(x 2x0) 1v2(y 2y0) ] 1 1 4 (v 2 11 v22)(t 2t0) 1g . (2.10)
The conformal transformations, generated by C, are local and can be composed with (2.10). The transformation (for 1 2ptD0) is
.
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/
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´
t 84 t 1 2pt , x 84 x 1 2pt , y 84 y 1 2pt , r 8m4 rm( 1 2pt) , f 8m4 fm1 p(x2 1 y2) 4( 1 2pt) . (2.11)Above, t0, x0, y0, v1, v2, a , l , g and p are group parameters.
The VNLSE is also invariant under time reversal and under the reflection of each of the space coordinates:
. / ´ T : t K2t , Mj: t Kt , xiK xi, xjK 2xj, c Kc* , c Kc , 1 GjGn , (2.12)
where Mjis applied for each j separately. The elements T and Mj generate a discrete
finite group GD.
3. – Subalgebras of the symmetry algebra
To perform symmetry reduction for eq. (1.1) we need to classify the subalgebras of L into conjugacy classes. Subalgebras of a direct sum of two Lie algebras, L 4
L15 L2 are either “splitting”, or “nonsplitting”. The splitting ones are themselves
not such direct sums and they can be constructed via the “Goursat twist” method [30, 31].
The subalgebras of the extended Schrödinger algebra sch ( 2 ) are known [32], as are those of su ( 3 ). We need a classification of low-dimensional subalgebras L0%L , into
conjugacy classes under the group of transformations G , leaving eq. (1.1) invariant. The group G has the structure
G 4GD’ G0,
(3.1)
where ’ denotes a semidirect product, with G0as invariant subgroup. The group G0is
the connected component of G, obtained by integrating the vector fields (2.5), i.e. the Lie algebra (2.7). The group GDwas defined in eq. (2.12).
In order to reduce eq. (1.1) to an algebraic equation, an ordinary differential equation (ODE), or a partial differential equation (PDE) in two variables respectively,
we need to know all subalgebras L0 of dimension dim L04 3 , 2 , and 1, respectively.
Moreover, the corresponding subalgebras should not contain the element M, nor elements of su (3) (as separate basis elements), as long as we are interested in invariant solutions, rather than partially invariant ones [22, 33, 34].
A complete list of representatives of all SU (3) classes of nontrivial subalgebras of
su (3) is given in table I. To simplify the notation, we put
.
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/
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´
T3fT3 , 124 1 2 (¯f12 ¯f2) , T1fT1 , 12, T2fT2 , 12, V1fV1 , 13, V2fV2 , 13, U3fT3 , 234 1 2 (¯f22 ¯f3) , U14 T1 , 23, U2fT2 , 23, V34 T31 U3. (3.2)In table II we list representatives of all three-dimensional subalgebras that provide reductions to algebraic equations. When the corresponding subgroups act on the space ]t , x , y , c , c * ( they sweep out orbits, the projections of which onto the space of independent variables has dimension 3. Not included are subalgebras
L3 , 64 ]L31 a1M 1a2T31 a3U3, P1, P2( ,
L3 , 74 ]L31 eT 1 a1M 1a2T31 a3U3, P1, P2( ,
L3 , 84 ]L31 bD 1 a1M 1a2T31 a3U3, P1, P2( ,
L3 , 94 ]C , D , T( . TABLEI. – Proper subalgebras of su (3).
No Dimension Basis Comment
1 2 3 4 5 1 2 3 3 4
cos aT31 sin aU3 ]T3, U3( ]T1, T2, T3( ]T1, U1, V2( ]T1, T2, T3, U31 V3( 0 GaEp Abelian su (2) 0(3) U( 2 )
TABLE II. – Three-dimensional subalgebras of the symmetry algebra providing reductions to
algebraic equations. We have k 40, 61, ai, bi, ciR .
No. Type Basis
L3 , 1 L3 , 2 L3 , 3 L3 , 4 L3 , 5 3 A1 A15 A2 A3 , 3 A3 , 4 A1 O2 3 , 5 P01 kM 1 a1T31 a2U3, P11 b1T31 b2U3, P21 c1T31 c2U3 L31 a1M 1a2T31 a3U3, D 1b1M 1b2T31 b3U3, P0 D 1a1M 1a2T31 a3U3, P1, P2 D 1a1M 1a2T31 a3U3, K1, P2 D 1a1M 1a2T31 a3U3, P0, P2
TABLE III. – Two-dimensional subalgebras of the symmetry algebra providing reductions to
ordinary differential equations. The conventions for the constants are k 40, 61, e461, ak, bk,
c R , c c 0 .
No. Type Basis
L2 , 1 L2 , 2 L2 , 3 L2 , 4 L2 , 5 L2 , 6 L2 , 7 L2 , 8 2 A1 P11 kK11 ak¯fk, P21 bk¯fk K11 ak¯fk, P21 bk¯fk P01 ak¯fk, P21 bk¯fk P01 K11 ak¯fk, P21 bk¯fk P01 ak¯fk, L31 bk¯fk D 1ak¯fk, L31 bk¯fk C 1P01 ak¯fk, L31 bk¯fk L32 e(C 1 P0) 1ak¯fk, K11 eP21 bk¯fk L2 , 9 L2 , 10 A2 L31 cD 1 ak¯fk, P0 D 1ak¯fk, P2
The reason is that L3 , 6, L3 , 8 for b 40 and L3 , 9 lead to partially invariant
solutions [22, 33, 34], which are not treated in this article. The group corresponding to the algebra L3 , 7 has the same orbits as the one corresponding to L3 , 1 and hence gives
the same results. Similarly, L3 , 8 for b c 0 gives the same results as L3 , 3. The type of
algebra, listed in column 2 of table II, corresponds to the notation of ref. [35]. Thus,
L3 , 1is Abelian, L3 , 2non-Abelian, but decomposable. L3 , 3, L3 , 4 and L3 , 5are all solvable
with a diagonal action of the first generator on the Abelian ideal, given by the last two. In table III we represent all conjugacy classes of two-dimensional subalgebras, providing reductions to ODEs. The first 8 algebras are Abelian, the remaining two non-Abelian, solvable.
We do not need a classification of one-dimensional subalgebras, since they lead to PDEs, that we must reduce further (to the same ODEs mentioned above).
4. – Solutions obtained from reductions to algebraic equations
Let us now run through the subalgebras of table II, obtain the reduced equations and solve them. Throughout we shall assume that none of the amplitudes vanishes identically.
Algebra L3 , 1
The solutions of the VNLSE invariant under the three translations combined with changes of phase have the form
cm4 rmei(am1 bmt 1gmx 1dmy), (4.1)
where rm, am, bm, gmand dmare constants. Each component is a plane wave with constant amplitude rm. Equation (1.1) implies the following relations among the constants:
bm1 g2m1 d2m4 2
!
N a 41 r2 a, m 41, R, N , (4.2)so that we must have bmE 0 for all m.
Algebra L3 , 2
Invariance under the corresponding subgroup implies that the solution has the form cm4 Fm r e i(am2 bu 1 gmln r), (4.3)
where r and u are polar coordinates and am, b , gm and Fm are real constants. The
VNLSE (1.1) implies gm4 0 , b2f1 2
!
N a 41 F2 a, m 41, R, N . (4.4)The solution is hence
cm4
Fm
r e
i(am2 bu).
(4.5)
The solution is thus singular at the origin r 40 and satisfies cmK 0 for r K Q. It is
single-valued if b is an integer.
Algebra L3 , 3
An invariant solution has the form
cm4
Fm
kt e
i[am1 (pmO2 ) ln t],
(4.6)
where Fm, am and pm are real constants. However, the VNLSE (1.1) imposes the
constraint
(pm1 i) Fm4 22(Fa2) Fm (4.7)
Algebra L3 , 4
The invariant solution is
cm4
Fm
kt e
i [x2O4 t 2 (
!
aFa2) ln t],(4.8)
where Fm are real constants. We see that at time t 40 the amplitudes are singular and
they vanish for t K1Q.
Algebra L3 , 5
The invariant solution is
cm4 Fm x e iam,
!
m F 2 m4 2 (4.9)with Fmand amreal constants.
Each of the solutions of this section can be further generalized by applying the transformations (2.10), (2.11).
5. – Translationally invariant solutions
5.1. General comments. – We shall apply the term translationally invariant solutions somewhat loosely, namely it will refer to solutions invariant under a subgroup containing translations, by choice in the y direction, possibly combined with changes of phase. Thus, the invariance subalgebra will contain the element
A 4P21 bk¯fk, bkR .
(5.1)
The corresponding two-dimensional symmetry algebras are L2 , 1, R , L2 , 4 and L2 , 10 of
table III.
Invariance under the subgroup generated by the element (5.1) implies that the solution will have the form
ck(x , y , t) 4 cAk(x , t) ei(bky 2b
2
kt),
(5.2)
where cAk(x , t) satisfies the integrable ( 1 11)-dimensional VNLSE
i cAk , t1 cAk , xx4
!
N a 41N c A aN2cAk. (5.3)Its symmetry algebra is inherited from that of eq. (1.1) and is given by ]P0, P1, K1, M , D(5su (N) .
5.2. The subalgebra L2 , 1(k). – For k 40 we reobtain the solution (4.1). For
k 4e461 the invariant solution is ck4 Rk k1 1et exp
y
i{
bk(y 2bkt) 1 e(x 14ak) 2 4( 1 1et) 2 eg
!
a R 2 ah
ln ( 1 1et)}
z
, (5.5)where Rkare real constants.
5.3. The subalgebra L2 , 2. – The invariant solution in this case is
ck4 Rk kt exp
y
i{
bk(y 2bkt) 1 (x 12ak)2 4 t 2g
!
a R 2 ah
ln t}
z
. (5.6)Solutions (5.5) and (5.6) are seen to be plane waves, damped by a t-dependent factor. In optical applications t is usually not time, but a spacelike variable in the direction of propagation.
5.4. The subalgebra L2 , 3. – The invariant solution of eq. (1.1) has the form
cm(x , y , t) 4rm(x) ei[am(x) 1bmy 1bmt], (5.7)
where bmand bm are constants and the functions rm(x) and am(x) satisfy
a.m4 Sm r2 m , Sm4 const , (5.8) r O m2 (bm1 bm2) rm2 Sm2 r3m 4
g
!
a r2 ah
rm, m 41, R, N . (5.9)Equation (5.9) can be decoupled and solved by introducing a spherical coordinate
representation for the moduli rm(x). Let us do this for N 42 and N43.
For N 42 we put
r14 r cos x , r24 r sin x
(5.10)
and, in order to be able to decouple, restrict to the case
bm1 bm2fb , (m . (5.11)
We can then reduce eq. (5.9) to
rO2 r x.2 2 br 2 S 2 1 r3cos2x 2 S2 2 r3sin2x 4 r 3, (5.12) rxO1 2 r.x.1 S 2 1sin x r3cos3 x 2 S2 2cos x r3sin3 x 4 0 . (5.13)
Equation (5.13) can be integrated once and we obtain
r4x.2 1 S 2 1 cos2x 1 S2 2 sin2x 4 K1. (5.14)
Equation (5.12) then reduces to an equation for r alone. We integrate it once and obtain r.2 4 1 2 r 4 2 K1 r2 1 br 2 1 K2. (5.15) Putting r(x) 4
k
W(x) (5.16)we obtain an equation for elliptic functions
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W.24 2(W 2 W1)(W 2W2)(W 2W3) , W11 W21 W3422 b , W1W21 W2W31 W3W14 2 K2, W1W2W34 2 K1. (5.17)In view of eq. (5.16) we need W to be real and nonnegative. Such solutions can also be finite if all three roots Wiare positive (we have K1D 0). If all three roots are distinct
we have W 4W11 (W22 W1) sn2
u
o
W32 W1 2 x , kv
, k 2 4 W22 W1 W32 W1 (5.18)with 0 EW1G W G W2E W3, where sn (u , k) is a Jacobi elliptic function [36]. For W24
W3this solution reduces to an elementary one, namely a soliton given by
W 4W11 (W22 W1)
y
tanho
W22 W1 2 xz
2 . (5.19)Equation (5.17) also has real nonnegative singular solutions. When the roots Wiare all distinct we have W 4W11 W32 W1 sn2
(
k
(W32 W1) O2 x, k)
, k 4 W22 W1 W32 W1 . (5.20)Two different limits can be considered here. For W2K W1we obtain an elementary
singular and periodic solution
W 4W11 W32 W1 sin2
k
(W32 W1O2 x . (5.21)For W3K W2we obtain a “singular soliton”, namely
W 4W11
W22 W1
[ tanh
k
(W22 W1) O2 x]2. (5.22)
It is difficult to attribute a direct physical meaning to singular periodic or nonperiodic solutions of this type. However, the presence of a small amount of dissipation in the original VNLSE would tend to shift the singularities into the complex plane, so that the physical amplitudes would have finite peaks, rather than infinite ones.
Once W(x) and hence r(x) is known, eq. (5.14) can be solved for the “phase” x(x). We transform to a new variable, putting
x(x) 4 xA(z) , z 4
dx r2(5.23)
and integrate the obtained ODE. The solution of eq. (5.14) hence is
x(x) 4arcsin [ (A22 A1) sin2kK1(z 2z0) 1A1]1 O2 (5.24) with A1 , 24 1 2 K1
[
K2 12 S121 S226k
(K12 S121 S22)22 4 K1S22]
.The case N43 of three coupled waves can be treated in the same manner. We put
r14 r sin b cos g , r24 r sin b sin g , r34 r cos b ,
(5.25)
and restrict to the case (5.11). We substitute (5.25) into eq. (5.9) and obtain a system of three equations for r , b and g . One of them provides a first integral
r4g.2sin4 b 1 S 2 1 cos2g 1 S2 2 sin2g 4 K0. (5.26)
Using (5.26) to eliminate g. we obtain a further first integral, namely
r4b.2 1 S 2 3 cos2b 1 K0 sin2b 4 K1. (5.27)
Finally, we obtain an equation for r(x) that can be integrated once to reobtain
eq. (5.15). Thus, for N 43 we have a solution given by eq. (5.25), with r4kW and W as
in (5.18), R , (5.22). The function b is given by the right-hand side of eq. (5.24) with (S2
1, S22) replaced by (S32, K0). Finally, g(x) also has the form given in eq. (5.24) with K1
replaced by K0and
z 4
dx r2sin2b .(5.28)
5.5. The subalgebra L2 , 4. – Invariance under the corresponding subgroup implies
that the solution of eq. (1.1) has the form
cm(x , y , t) 4rm(j) ei[am(j) 1xtO22t 3 O6 1 bmt 1qmy], j 4x2 1 2 t 2, (5.29)
where bmand qmare constants.
The functions rmand amsatisfy (5.7) and
r O m2
k
1 2 j 1bm1 q 2 ml
rm2 Sm2 r3m 4g
!
a r2 ah
rm. (5.30)For N 42 we use (5.10) and (5.11) to obtain eq. (5.13) for x(j). The equation for r(j) in this case is r O 4
g
1 2 j 1bh
r 1 K1 r3 1 r 3. (5.31)We put r 4kW as in eq. (5.15) and obtain
W
O
4 W . 2 2 W 1 (j 1 2 b) W 1 2 K1 W 1 2 W 2. (5.32)This is one of the equations classified by Painlevé [11] and Gambier [12], and reproduced by Ince [13]. More specifically, by a rescaling of W and j, together with a translation of j, eq. (5.32) is transformed into eq. PXXXIV (ref. [13], p. 340). It can be integrated in terms of the second Painlevé transcendent PII.
For N 43 we use eqs. (5.25) and (5.11). The variables g(j) and b(j) again satisfy eq. (5.26) and (5.27), respectively. The function r(j) then satisfies eq. (5.31) and can hence be expressed in terms of the Painlevé transcendent PII.
5.6. The subalgebra L2 , 10. – The invariant solution in this case can be written as
cm(x , y , t) 4 1 kt rm(j) e i[am(j) 1bmln t], j 4 x kt , bm4 const . (5.33)
The functions rm(j) and am(j) satisfy
r O m2 rma .2 m1 1 2 rmj a . m2 bmrm4
g
!
a r 2 ah
rm, (5.34) 2 r.ma . m1 rma O m2 1 2 j r . m2 1 2 rm4 0 . (5.35)To solve eq. (5.35) we introduce new variables:
Mm(j) 4
r2m(j) dj (5.36) and obtain (5.37).
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a.m4 4 Sm1 j M . m1 Mm 4 M.m , MP
m4 WO
2 m 2 M.m 2 j 2 M.m 8 1 Mm2 8 M.m 1 2 S2 m M.m 1 SmMm M.m 12 bmM . m1 2g
!
a M . ah
M . m.In this case we have a system of N coupled third-order real ordinary differential equations to solve. We can decouple them, but this leads to a higher-order equation, even for N 42. We shall not pursue this matter here.
6. – Cylindrically invariant solutions
6.1. General comments. – Let us now consider reductions to ODEs, involving
cylindrical symmetry, i.e. invariance under the subgroup corresponding to the algebra
L31 bm¯fm. The invariant solution will have the form cm(x , y , t) 4 cAm(r , t) eibmu, (6.1) i cAm , t1 cAm , rr1 1 r c A m , r2 b2 m r2 c A m4
!
a N c A aN2cAm. (6.2)The symmetry algebra of eq. (6.2), inherited from that of eq. (1.1) is ]P0, D , C , M(5su (n) .
(6.3)
Correspondingly, further reductions of eq. (6.2) to ODEs will correspond to reductions of eq. (1.1) by means of the subalgebras L2 , 5, L2 , 6and L2 , 7 of table III.
6.2. The subalgebra L2 , 5. – The invariant solution of the VNLSE will in this case
have the form
cm(x , y , t) 4rm(r) ei(amt 1bmu 1am(r)), (6.4) a.m4 Sm rr2m , Sm4 const , (6.5) rOm1 1 r r . m2
g
am1 b2 m r2h
rm2 S2 m r2r3 m 4g
!
a r2ah
rm. (6.6)Let us now consider the cases of two and three interacting wave modes. For N 42 we use eq. (5.10), restrict to the case
am4 a , bm4 b , (m
(6.7)
and obtain two equations, similar to (5.12) and (5.13). The second one can be integrated to give r2r4x.21 S 2 1 cos2x 1 S2 2 sin2x 4 K1, (6.8)
where K1is a nonnegative constant.
The remaining equation is
rO1 r . r 2
g
a 1 b2 r2h
r 2 K1 r2r3 2 r 3 4 0 . (6.9)To simplify eq. (6.9) we put
.
/
´
r(r) 4ka r21 O3k
H(j) , j 4br2 O3, a D0 , b R, a , b 4const (6.10) and obtain HO4 H .2 2 H 1 1 2g
9 b2 2 1 j2 1 9 aj b3h
H 1 9 a 2 b2H 2 1 9 K1 2 a2b2H . (6.11) For b2c1 O9 eq. (6.11) does not have the Painlevé property and we are not able to solve in terms of known functions. For
9 b2
2 1 4 0 , a c 0
(6.12)
we put b3
4 (29 aO2 ), a 4 3kK1(29aO2)21 O3 and obtain eq. PXXXIV of Ince [13]
which can be integrated in terms of the second Painlevé transcendent PII.
For 9 b2 2 1 4 0 , a 40 (6.13) we put a 4 (2K1)1 O3, b 43(2K1)1 O6Ok2
and integrate eq. (6.11) once to obtain
H.2
4 (H 2 H1)(H 2H2)(H 2H3)
(6.14a) with
(6.14b) H11 H21 H34 0 , H1H21 H2H31 H3H14 K2, H1H2H34 1 .
Two of the roots must be negative, one positive and this implies that all positive solutions H(j) satisfy
H3G H2G 0 G H1G H .
(6.15)
When all three roots are distinct we obtain singular periodic solutions:
H 4H31 H12 H3 sn2
(
(kH 12 H3O2 )(j 2 j0), k)
, k2 4 H22 H3 H12 H3 . (6.16)For H34 H2we obtain elementary periodic singular solutions:
H 4H31 H12 H3 sin2
(
kH 12 H3O2)
(j 2j0) . (6.17)For N 43 the results are quite similar. We again introduce the spherical
angle g is found from the equation r2r4g.2sin4 b 1 S 2 1 cos2g 1 S2 2 sin2g 4 K0. (6.18)
Equation (6.18) is solved just like eq. (5.14), once r and b are known. In turn, b satisfies r2r4b.21 S 2 3 cos2 b 1 K0 sin2 b 4 K1, (6.19)
also solved like eq. (5.14) was, once r is known. Finally, r(r) satisfies eq. (6.9), with solutions (6.16) and (6.17).
These solutions for r(r) are singular at r 40 and then on concentric circles, given in the case of solution (6.16) by
r 4
{
r2 O3 0 1 ( 2 n 11)p kH12 H2 k2 3( 2 K1)1 O6}
3 O2 , n 40, 1, 2, R . (6.20)Such singular waves are at face value quite nonphysical; however, as usual, the addition of some dissipation in the original VNLSE (1.1) will reduce the “peaks” to finite ones.
6.3. The subalgebra L2 , 6. – We write the solution of the VNLSE as
cm(x , y , t) 4 1 kt rm(j) e i[am(j) 1bmu 1amln t], j 4 r kt . (6.21)
The phase amand modulus rmsatisfy
a.m4 Sm jr2 m 1 j 4 , Sm4 const , (6.22) r O m1 r.m j 1
g
j2 16 2 am2 b2 m j2h
rm2 S2 m j2r3m 4g
!
a r2ah
rm. (6.23)For N 42 we again use the spherical components (5.10), put a14 a24 a , b14 b2fb and
obtain j2r4x.2 1 S 2 1 cos2x 1 S2 2 sin2x 4 K1, (6.24)
which can easily be integrated, once r(j) is known. The function r(j) satisfies r O 1 r . j 1
g
j2 16 2 a 2 b2 j2h
r 2r 3 2 K1 j2r3 4 0 . (6.25)Putting, as usual, r 4kW , we obtain
W
O
4 W . 2 2 W 2 W. j 2 2g
j2 16 2 a 2 b2 j2h
W 12W 2 1 2 K1 j2W . (6.26)This equation does not have the Painlevé property for any values of a , b and K1, as can
be verified by setting
W(j) 4a0j22 O3H(z0j2 O3) , a0, z04 const
(6.27)
and comparing the result with the equations given by Ince [13] (canonical equations of type iii, p. 336).
We have thus reduced the problem of solving the VNLSE to that of solving the nonlinear ODE (6.26), but are not able to proceed further analytically.
The results for N 43 are quite analogous: everything is again reduced to eq. (6.26), plus equations for b and g that are easy to solve.
6.4. The subalgebra L2 , 7. – We write the solution as
cm4 1
k
t2 1 1 rm(j) ei]bmu 1amarctan t 1r 2tO[4(t21 1 ) ] 1 a i(j)(, j 4 rk
t2 1 1 (6.28) and obtain a.m4 Sm jr2 m , rOm2 S2 m j2r3 m 1 1 j r . m2g
am1 j2 4 1 b2 m j2h
rm4 r 2 rm. (6.29)For N 42 we use the representation (5.10) and find that the “angle” x satisfies eq. (6.24) (with j as in (6.28). The quantity W 4r2
11 r22then satisfies W
O
4 W . 2 2 W 2 W. j 1 2g
a 1 j2 4 1 b2 j2h
W 12W 2 1 2 K j2W . (6.30)This equation can be reduced to a standard form (with no first derivative and a constant coefficient of the quadratic term) by putting
.
/
´
W(j) 4j22 O3H(z) , z 4j2 O3, HO
4 H .2 2 H 1 2 H 2 1 9 K 2 1 H 1 9 4k
g
2 9 1 2 b 2h
z22 1 29 z 1 1 2z 4l
H . (6.31)This equation does not have the Painlevé property, so we would have to proceed further numerically.
7. – Other symmetries
7.1. The subalgebras L2 , 8. – The reduction formula in this case is a rather
complicated one, namely
(7.1)
.
`
`
/
`
`
´
cm4 rm(j)k
t21 1 ei[sm1 am(j) ],sm4 2eam arctan t 1bm xt 1ey
t2 1 1 1 1 1 4(t2 1 1 )2 [x 2t(t2 1 3 ) 2 2 exy(t22 1 ) 1 y2t(t2 2 1 ) ] , j 4 x 2eyt t2 1 1 ,
where the functions am(j) and rm(j) satisfy
a.m(j) 4 Sm r2 m(j) , rOm2 Sm2 r3 m 1 [eam2 (j 1 bm)2] rm4 r2rm. (7.2)
In spherical components r , x of eq. (5.10) we have
r4x.21 S 2 1 cos2x 1 S22 sin2x 4 K , r O 2 K r3 1 [ea 2 (j 1 b) 2 ] r 4r3. (7.3)
Again, the equation for r does not have the Painlevé property and we cannot solve it analytically.
7.2. The subalgebra L2 , 9. – The reduction formula is
cm4 1 rrm(j) e i[am(j) 1bmln r], j 4u1A ln r , A 4 2 c , c c 0 . (7.4)
These are static solutions and they correspond to spiral structures. Indeed, j 4const is a logarithmic spiral.
The equations we obtain are
.
/
´
(rOm2 rma .2 m)( 1 1A 2 ) 22Ar.m2 2 Abmrma . m1 rm( 1 2b 2 m) 4r 2 rm, ( 2 rma . m1 rma O m)( 1 1A2) 12Abmr . m2 2 Arma . m2 2 bmrm4 0 . (7.5)In order to solve eq. (7.5) we restrict to the case bm4 0 . We then have a.m4 Sm r2 m e 2 A 1 1A2j. (7.6)
For two waves, N 42 we use eq. (5.10) and find that x and W4r2 11 r22satisfy x.2W2e24 AjO( 1 1 A2) 1 S 2 1 cos2x 1 S22 sin2x 4 K1, (7.7) W
O
4 W . 2 2 W 1 2 A 1 1A2 W . 2 2 1 1A2 (W 2W 2 ) 1 2 K1 W e 4 AjO(11A2) . (7.8)This again is an equation that does not have the Painlevé property.
8. – Conclusions
One of the possible applications of this study is in the context of high-temperature superconductivity. Indeed, Birman and Lu [37] have proposed a model in which the superconducting order parameter is a two-component one. In the framework of a Landau-Ginzburg theory they obtain a coupled system of Schrödinger equations that can be approximated by the VNLSE of this article. Additional terms can then be treated perturbatively, or exactly, whenever they are compatible with some subalgebra of the symmetry algebra of the VNLSE.
In this condensed-matter context the fact that we are considering eq. (1.1) in 2 11 dimensions is physically significant.
In sect. 4 we have obtained several rather simple exact solutions, depending on all three variables x , y , and t . They are, by construction, invariant under three-dimensional subgroups of the symmetry group. The reductions of sect. 5 also lead to exact solutions. The phases of the amplitudes depend on all three variables, the moduli only on x and t . The amplitudes are expressed in terms of elementary functions, elliptic
functions, or the second Painlevé transcendent PII. Some elementary solutions are
localized (solitary waves), others are periodic. Most of the reductions involving rotational invariance (cylindrical symmetry) lead to equations that do not have the Painlevé property. An exception is the reduction (6.4) leading to periodic solutions in terms of elliptic and trigonometric functions.
Even in those cases when we did not obtain explicit solutions, the reductions lead to relatively simple ordinary differential equations. They can be used for numerical studies.
* * *
We thank Prof. J. BIRMAN for helpful discussions. The authors thank each other’s
institutions for hospitality during mutual visits necessary for the completion of this work. The research of PW was partly supported by research grants from NSERC of Canada and FCAR du Québec.
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