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Elementary excitations for solitons of the nonlinear Schrödinger equation
PHYSICA 0
Physica D 53 (1991) 25-32 North-Holland
Elementary excitations for solitons of the nonlinear SchrGdinger equation V.M. Malkin Institute of Nuclear Physics, 630090 Nocosibirsk,
USSR
and E.G. Shapiro of Automation
institute
and Electrometry,
630090 Novosibirsk,
USSR
Received 15 January 1991 Accepted 2 May 1991 Communicated by V.E. Zakharov
The dynamics of a nonlinear wave field having a stable localized state (soliton) is essentially dependent on the spectrum of elementary excitations of the soliton. The spectrum is found below for a soliton of the two-dimensional Schrddinger equation with attracting cubic nonlinearity. The equation describes a number of physical phenomena, in particular the self-focusing of electromagnetic radiation.
1. Introduction
A field described by the equation
(
ii
+A+
(1.1)
1$12)9=0,
where A is a two-dimensional Laplace operator, has localized stationary states of the form $( r, t) =
KR(
KT)
exp(iK2t + icp).
(1.2)
The parameter ~~ can be called “the energy of a quantum drop out of the soliton”. The designation R(r) is used below for the monotone solution of equation (1.3) which was computed in ref. [l]. Eq. (1.3) has also
0167-2789/91/$03.50
0 1991-
Elsevier
Science
Publishers
localized solutions oscillating over r (see refs. [2-411, but those are exponentially unstable with respect to small perturbations. As for the Townes soliton [l], its small perturbation cannot grow faster than a power of t (see refs. [5-81 and below). If one includes in eq. (1.1) a small defocusing nonlinearity, proportional to the fifth order of ltJ1, then the Townes soliton undergoes minor changes in form while the instability disappears, and unstable modes turn into stable low-frequency modes. The dynamics of the thus stabilized soliton at strongly excited lowfrequency modes is of essential interest and depends on the spectrum of other eigenmodes (see ref. [9]). The determination of the spectrum is the purpose of present paper. Prior to starting to solve the above problem, it is useful to call all that is known of exact solutions and symmetries of eq. (1.1).
B.V. AI1 rights reserved
2. Exact solutions The stationary
and symmetries state (1.2) is a particular
the four parameter tions
of eq. (l.l),
These
solutions
ter set of exact solutions
family
case of
of axisymmetric
which were
found
can be represented
of eq. (1.1):
solu-
in ref. [lOI. in the follow-
ing form: This set exists due to the invariance
I dt + d(f) r’ ___ ___ f 11,a’( * 1 4u(t) +tcpll
!I
’
l/2
a(t)=
a;+$4 i
(2.1)
7
i
where N, a,,, f,,,cp,, are constants and HP,(U) is the monotone solution of the ordinary differential equation
-I+;&+r&R2-‘lar2 i R(r,a)=O. (2.2) At cy = 0 eq. (2.2) coincides with (1.3) and the soliton R(r, a) coincides with the Townes soliton:
=R(r).
R(r,O)
At (Y+ 0 the solution tionary state (1.2) cubically growing in eq. (l.l), linearized tionary state (1.2),
(2.1) differs
by the on and
from the sta-
an infinitely small term time t. This term satisfies a background of the stacan be interpreted as a
growing eigenmode in the corresponding stability problem. A little more complicated way of proceeding to the limit LY= 0, namely a + 0,
K -2
3
a,, -+ 0, cuti/ai
Cp= ‘PC,-
= const .,
$TKKt,,/U,,
allows one to obtain
+
K’t,,
=
COtlSt.,
the following
three
parame-
of eq. ( I.1 1
with respect to the gauge, scale and Talanov transformations (the latter of which was found explicitly in ref. [ 111). The two parameter set of stationary states ( 1.2) can be obtained from (2.3) as a particular case corresponding to an infinite value of the parameter t,,. At t,, + = the solution (2.3) differs from (1.2) by a small term (proportional to t,;’ --$ 01, which grows quadratically in time t. The relation between this unstable eigenmode and an infinitesimal Talanov transformation was pointed out in ref. [7]. Both of the cubically and quadratically growing eigenmodes can be stabilized by the inclusion of a small defocusing nonlinearity in eq. (1.1). as was already mentioned in the introduction. Infinitesimal gauge and scale transformations (acting inside the family of stationary states (1.2)) generate constant and linearly growing in the time eigenmodes, respectively. The latter mode increases due to the accumulation of the phase difference only, which does not imply the instability of the Townes soliton. Apart from the gauge, scale and Talanov transformations, eq. (1.1) is invariant with respect to a five parameter group of Galilei transformations, including translations, rotations and movements with a constant velocity in the plane x = r cos 0. y = r sin 8. A rotational symmetry of eq. ( 1.l) and of the Townes soliton allows one to introduce an azimuth quantum number m = 0, _t 1, + 2,. and to treat independently perturbations having different values of m. Infinitesimal translations and velocity variations of the soliton generate two constant dipole (lrnl = 1) eigenmodes and two dipole eigenmodes linearly growing in time, respectively. The behaviour of the latter pair of
KM. Malkin, E.G. Shapiro /Elementary
modes is explained by a growing shift of the soliton as a whole and does not imply the instability.
of the functions equations i,uw=6JL’w,
3. Basic equations
where
In order to simplify further formulae, it is convenient to study elementary excitations for the soliton (1.2) of unit size (K-’ = 1) and zero phase (cp = 0). This does not lead to any loss of generality due to the scale and gauge invariance of eq. (1.1). An elementary excitation of the frequency w looks like
i, =i,,-
S$J r, t) = [A,,(r)
27
excitations
A,(r)
and B,(r)
iouw=wu,,
(3.4)
2R=.
(3.5)
The pair of equations (3.4) is presentable in the form of a one matrix equation for a two component function:
e-‘“’ + Bz( r) eiw*‘] ei’
(3.6)
(3.1)
and satisfies the equations
In the invariant subspace em of functions having a definite azimuth quantum number m the Laplacian looks as
WA, =&,A,-R2(A,+B,), -wB,=&B,-R2(A,+B,),
(34
A=‘drA-!!!f r dr
where 1
2
L,,=l-A-R.
(3.3)
The perturbation (3.1) does not change at combined replacements w + -w*,
BE,,
+A,,
A_,,,, + B,*.
Beside this, eqs. (3.2) are invariant with respect to combined complex conjugation and replacements w-+w*,
A*,, +A”,
B;*+B,.
It follows that if o is an eigenvalue,
then -w*, w* and --o are also eigenvalues. Thus, the investigation of the whole spectrum reduces to that in the first quadrant of the complex plane o: RewrO,
dr
(3.7)
,.2
and i,,, i, are reduced to ordinary differential operators. The reductions are designated below by the same symbols as the complete operators. A regular solution of eq. (3.6) in the invariant subspace e, behaves at r + 0 as
x,(r)
Cl rm +Lqrm+2).
=
c i
2
I
(3.8)
Main terms of asymptoticses of functions A,(r) and B,(r) at r + co are given by following formulae: A,(r)
z r-1/2{ A+ exp[ (1 - o)“‘r]
+A_exp[-(1-ti)“‘r]},
Imw20.
The linear combinations u&l=A,
satisfy the
+ B,,
u, =A,
-B,
B,(r)
E~-‘/~{B+
exp[(l +o)“‘r]
+B_ exp[ -(I
+ W)“*r]).
(3.9)
For the solution
regular
r = 0, each
in the point
of coefficients A + and Bi depends linearly on two parameters: C, and C, (besides which it dependson
w and m). IfImw=Oand
Rew>
then only one of the four exponents at r + =. The
growing
the preexponential The condition
exponent
factor
B,
is absent
when
into a zero.
by a proper
choice
of ratio of coefficients C, and C?. At such a ratio the solution tends to a zero at r + r: and represents an eigenfunction corresponding to a given value of w. It follows that the real ray o > 1 belongs to the continuous spectrum of the operator i. For other values of w in the half-plane Re o 2 0 already two of the four exponents in formulae (3.9) grow at r + ~0 and. to obtain a solution which tends to zero at I’ + a, one should satisfy two conditions: A + = 0 and B, = 0. This is possible only when the matrix lld):“‘(w)ll in the linear relation
i
B+ = d\;“‘(w) d$y’( w) A+
H
is degenerate,
d\‘;“(w)
c,
d:;“‘( 0)
ii c,
1
(3.10)
i.e. when the determinant
D,,,(w) =dgyo)dgyW)
-clyp(w)dy:‘(o) (3.1 I)
turns into zero. Thus, all eigenvalues w in the half-plane Re w 2 0 without the ray w > 1 (which is excised) are roots of equations
Qn(w) = 01 and belong tor C.
LX
(‘1 =
(‘(‘I =
Lb’)= 0.
0,
correspond-
K.
(4.1)
in (3.9) grows
turns
can be satisfied
1,
e,f = e,, X e,, only one eigenfunction ing to the zero eigenvalue:
This eigenfunction
is generated
imal gauge transformation Besides
x (I), there
of heights
by the intinites-
of the Townes
exist root eigenfunctions
n = 2, 3 and 4, corresponding
zero eigenvalue
of operator
4. Point spectrum
spectrum
of isotropic
(4.2) ix(R)
=
I/(3) =
()
x’zI, (.(3) = Ir2/( s
(4.3)
)
ix (4) --x (II 1 J1) = i,
I p1,
(‘(1) = ().
(4.4)
The root eigenfunctions x”’ and xc3’ arc generated by infinitesimal scale and Talanov transformations of Townes soliton, while x”’ is connected with its perturbation cubically growing in time (this perturbation was described in section 2). The graph of function ~6~’ is shown in fig. 1. The operator i has no root eigenfunction of height 5 corresponding to the zero eigenvalue in the subspace e,:. Indeed, if such a function ,,$” f
uL4’
of the opera-
perturbations
For isotropic perturbations, i.e. in the subspace e,,, the operator i,, is degenerate, while i, is reversible. The operator i has in the subspace
x(“) to the
i:
m =O, * 1, _t2,...
to the discrete
soliton.
Fig.
I.
The graph of root eigenfunction
u”‘(r)
V. M. Malkin, E.G. Shapiro /Elementary
exists, it should satisfy the equation L r,$) = 0
excitations
29
by the formula
u(4)
L’* =-
1
ra I 32R* 0 drr”R2S
The solvability condition of this equation reads , u(4))
( c,(I)
s
d2,.U(1)*U(4)
=
where R, is the coefficient in the asymptotics of the Townes soliton at r + w
()
I
and is not fulfilled, since ([Jo),
U(4))
=
(&,C2),
=
(U
(3,
U(4))
Ls(39
=
=
(@,
&
/
&u(4))
dzrr2RZ
#
0.
An existence of a four-dimensional root subspace corresponding to the zero eigenvalue of operator i in the space e,f implies that the function D,,(w) has a fourth order zero in the point w = 0. To explain this, one can make in eq. (3.6) the following substitution:
The representation (4.5) implies similar representations for functions u,, I!~, A, and B,,,, as well as for coefficients C,, C,, A + and B, in formulae (3.8) and (3.9). Since the function x(n) is decreasing at r + co for n = 1,2,3,4, the coefficients A(:’ and B ‘$’ for the same n are equal to zero. Thus, one has A + = w4&‘,
B, = 04B’:’
at + w”u’~‘(O),
C, = u,(O) = -&R(O) xw = ;
ufl- ‘x(n) + &*C’.
(4.5)
n=l
The auxiliary function xc’ satisfies the equation ix;s’ =
$4)
+
wx1;5’.
(4.6)
C, = u,(O) = R(0).
Substitution of these expressions in (3.10) reveals the following relation between columns of the matrix IIdW I, w>I/. .
For an unambiguous definition of this function, its value at r = 0 is conveniently put to zero:
&i&4)( 0) R(0)
xl;“(O) = 0. At w = 0 eq. (4.6) reduces to i,@
= 0,
&@
=
u(4).
follows that z@ = 0 and the function u,) grows exponentially at r --) a:
This relation allows one to rewrite the determinant D,(w) in the form
It
D,(w)
= &(di’l’(+l”‘(co) -d$‘(
The function nb5) is easily expressed in terms of the Townes soliton (by the Green function method). In particular, the coefficient U, is given
w)B’:‘( w)).
(4.7)
At w + 0 a further simplification is possible, since A(5)(0)
+
=
-B(5)(0)
+
=
12~‘~
,
d!:‘(O) = d$?(O) = %A 2 *,
where
pi.+ is
the coefficient
in the asymptotics
Ul-,-x- =U*rY’/zeI of the solution the initial
of the equation
condition
that the operator value
i,
= 0 satisfying
has only one negative
in the subspace
coefficient
i,u
2
U(O) = 1. It is easy to show eigen-
e,, and, consequently,
I’ .+ is negative.
Taking
the above note, one can conclude
into
0.5
the
account
that
: /
0
A$(,
w-4D,,(w) =
0
;;;;) > 0. Fig.
0
0.5
Graphsof functions
It can be shown (see refs. [5, 6, 81 and below) that all eigenvalues of the operator L are real. Thus, the numerical study of the point spectrum has to be undertaken only in the interval 0 < w < 1. At o + 1 the function D,,(o) increases by the law numbered by vnlues of 111= 0. I. 2 in the interval
which follows from the logarithmical increase of the solution of eqs. (3.2) at w = 1, r + m. A numerically constructed graph of the function
in the interval (0,l) is presented by the curve labelled 0 in fig. 2. As seen from the figure, the function has no zeros inside the interval. The following simple arguments show that the ,. operator L, in fact, has only real eigenvalues in the space /,f. Let F(n) (n = 1,2.3,4) be the root eigenfunction corresponding to the zero eigenvalue of operator it (adjoint to i) in the space e,:. The from the x(“) by a function ?(‘I’ is obtained simple rearrangement of components:
0.
II = 1,3,
( U(“), I.,,) = 0,
II = 2,4.
F(r,) = &1)
(n = 1.1,3.4).
(4.9)
Let k be the orthogonal projection operator reflecting the space e,, at its subspace orthogonal to the function I’(” = K. The projection of eqs. (3.4) to the subspace
0
I:,, =
E,L,,.
Here
and relations $I’) = [,W)
I ),
Each eigenfunction corresponding to a nonzero eigenvalue of the operator i is orthogonal to the XC”) (n = 1,2,3,4). In terms of single component functions the orthogonality conditions read
(1.(‘I)) L1,,) = (4.8)
(0.
WLl<,,
Fe,,
=
has the form
ioF,,,.
(4.10)
V.M. Malkin, E.G. Shapiro /Elementary
generated by infinitesimal Galilei transformations, also exist. Root eigenfunctions of height 3 are absent, since the solvability conditions of equations
are taken into account. In the subspace P^e,, the operator i,, is positive definite (which follows from the monotonicity of the eigenfunction R corresponding to the zero Eigenvalue of & in the space e,,). The operator L, is degenerate in the subspace Fe,,, since
&‘;f’=
$2) = 0
are not fulfilled:
The function ~6’) has only one zero, while functions without zeros are not orthogonal to R and do not belong to the subspace Fe,,. This implies that the function uf2) coiresponds to the smajlest eigenvalue of operator L,, i.e. the operator L, is nonnegative definite in the subspace Be,. The above properties of operators Lo and i, guarantee that all eigenvalues of the operator L in the space e,f are real, since the formula
(u’;‘, ~‘2’) = +( R, R) # 0.
@2
Ez
(% GU”)
(4.11)
(% L,‘u,) ’
holds, which follows from eq. (4.10).
5. The point spectrum of nonisotropic perturbations
7
L’(I) =
(5.1)
0.
*
These eigenfunctions are generated by infinitesimal translations of the Townes soliton in the plane X, y. Root eigenfunctions ,$z”’
3
(5.2)
/,,(2) +
The radial part, dR/dr, of the functions u’:’ (corresponding to zero eigenvalues of the operator f., in subspaces e + > has no zeros in the range O
For dipole perturbations, i.e. in subspaces e + ,, the operator i, is positive definite, while the operator i, is degenerate. The operator i has in each of the subspaces e +, only one eigenfunction corresponding to thezero eigenvalue:
31
excitations
a (1 -d-3/4.
A numerically constructed B,(o)
graph of the function
=riTTw-2D,( w) (1 - w’)
3/4 (5.3)
in the interval 0 < w < 1 is presented by the curve labelled 1 in fig. 2. For multipole perturbations, i.e. in subspaces e,,, at lml 2 2, both operators i, and i, are positive definite. This implies that all eigenvalues of the operator i in subspaces 1: are real, as seen from eq. (4.11). Due to the mirror symmetry of the Townes soliton and basic equations, the functions D,,,(u) do not depend on the sign of m. At m s+ 1 the functions D,(w) are easily calculable analytically. Picking out the analytical depen-
32
dence,
V.M. Malkin,
one can introduce dT(
rs,( w)
1 _
auxiliary
E.G. Shapiro
functions
growing
excitations
separation
w2)“1:2+‘/4
(5.4)
Qn(W)~
m!)2
the process
is well forbidden
for the Hamiltonian at m --) 00. A numerically
tending
to unity
structed
graph of the function
by the curve labelled
between
the solitons.
On the
other hand, a small perturbation cannot cause the decay of a soliton for two other solitons, since
= 227
/ Elrmentary
by conservation
and the number
laws
of quanta.
con-
~,(cIJ) is presented
2 in fig. 2. At higher
m, the
functions fi,,
Acknowledgement The authors useful remark.
are grateful
to G.M. Fraiman
for a
6. Conclusion References The main conclusion to the above analysis consists in the following: solitons of the two-dimensional Schrodinger equation with attracting cubic nonlinearity have no elementary excitations, besides the continuous spectrum and eight discrete modes generated by well-known symmetries and exact solutions of the problem. This fact plays a significant role in the soliton’s adiabatic dynamics describing the self-focusing of radiation in different media (see ref. [9]). It is noteworthy that solitons of the one-dimensional Schrodinger equation with attracting cubic nonlinearity also have no localized excited states. As seen from the very scheme of finding solutions by the inverse scattering method (see, for instance, ref. [12]), an excited state, if one exists, can be treated as a bound state of two solitons moving with equal velocities. An arbitrarily small perturbation
breaking
the
equality
leads
to a
[II R.Y. Chiao,
E. Garmire and C.H. Townes. Phys. Rev. Lett. 13 (1964) 479. Izv. VUZov. Radiofizika 9 (1966) 412 121Z.K. Jankauskas, [Sov. Radiophys. 9 (1966) 261 [31 HA. Haus, Appl. Phys. Lett. 8 (1966) 128. V.V. Sobolev and VS. Synakh, Zh. Exp. [41 V.E. Zakharov, Teor. Fiz. 60 (1971) 136 [Sov. Phys. JETP 33 (1971) 771. [Sl E.W. Laedke, K.H. Spatschek and L. Stenflo, J. Math. Phys. 24 (1983) 2764. [hl M.J. Weinstein, Commun. Math. Phya. 87 (1983) 567. and S.K. Turitsyn. Phys. Lett. A 112 [71 E.A. Kuznetsov (1985) 273. [81 G.M. Fraiman, Zh. Exp. Teor. Fiz. X8 (1985) 390 [Sov. Phys. JETP]. LOIV.M. Malkin, On the analytical theory for self-focusing of radiation, to be published. [lOI V.I. Talanov, Zh. Exp. Teor. Fia. Pis’ma, 11 (1970) 303 [Sov. Phys. JETP Lett. 11 (1970) 1991. [Ill V.I. Talanov, Izv. VUZov. Radiotizika 9 (1966) 412 [Sov. Radiophys.]. S.V. Manakov, S.P. Novikov and L.P. iI21 V.E. Zakharov, Pitaevski, Theory of Solitons: Inverse Scattering Method, vol. 1 (Nauka, Moscow. 1981). Ch. 10.
Physica D 53 (1991) 25-32 North-Holland
Elementary excitations for solitons of the nonlinear SchrGdinger equation V.M. Malkin Institute of Nuclear Physics, 630090 Nocosibirsk,
USSR
and E.G. Shapiro of Automation
institute
and Electrometry,
630090 Novosibirsk,
USSR
Received 15 January 1991 Accepted 2 May 1991 Communicated by V.E. Zakharov
The dynamics of a nonlinear wave field having a stable localized state (soliton) is essentially dependent on the spectrum of elementary excitations of the soliton. The spectrum is found below for a soliton of the two-dimensional Schrddinger equation with attracting cubic nonlinearity. The equation describes a number of physical phenomena, in particular the self-focusing of electromagnetic radiation.
1. Introduction
A field described by the equation
(
ii
+A+
(1.1)
1$12)9=0,
where A is a two-dimensional Laplace operator, has localized stationary states of the form $( r, t) =
KR(
KT)
exp(iK2t + icp).
(1.2)
The parameter ~~ can be called “the energy of a quantum drop out of the soliton”. The designation R(r) is used below for the monotone solution of equation (1.3) which was computed in ref. [l]. Eq. (1.3) has also
0167-2789/91/$03.50
0 1991-
Elsevier
Science
Publishers
localized solutions oscillating over r (see refs. [2-411, but those are exponentially unstable with respect to small perturbations. As for the Townes soliton [l], its small perturbation cannot grow faster than a power of t (see refs. [5-81 and below). If one includes in eq. (1.1) a small defocusing nonlinearity, proportional to the fifth order of ltJ1, then the Townes soliton undergoes minor changes in form while the instability disappears, and unstable modes turn into stable low-frequency modes. The dynamics of the thus stabilized soliton at strongly excited lowfrequency modes is of essential interest and depends on the spectrum of other eigenmodes (see ref. [9]). The determination of the spectrum is the purpose of present paper. Prior to starting to solve the above problem, it is useful to call all that is known of exact solutions and symmetries of eq. (1.1).
B.V. AI1 rights reserved
2. Exact solutions The stationary
and symmetries state (1.2) is a particular
the four parameter tions
of eq. (l.l),
These
solutions
ter set of exact solutions
family
case of
of axisymmetric
which were
found
can be represented
of eq. (1.1):
solu-
in ref. [lOI. in the follow-
ing form: This set exists due to the invariance
I dt + d(f) r’ ___ ___ f 11,a’( * 1 4u(t) +tcpll
!I
’
l/2
a(t)=
a;+$4 i
(2.1)
7
i
where N, a,,, f,,,cp,, are constants and HP,(U) is the monotone solution of the ordinary differential equation
-I+;&+r&R2-‘lar2 i R(r,a)=O. (2.2) At cy = 0 eq. (2.2) coincides with (1.3) and the soliton R(r, a) coincides with the Townes soliton:
=R(r).
R(r,O)
At (Y+ 0 the solution tionary state (1.2) cubically growing in eq. (l.l), linearized tionary state (1.2),
(2.1) differs
by the on and
from the sta-
an infinitely small term time t. This term satisfies a background of the stacan be interpreted as a
growing eigenmode in the corresponding stability problem. A little more complicated way of proceeding to the limit LY= 0, namely a + 0,
K -2
3
a,, -+ 0, cuti/ai
Cp= ‘PC,-
= const .,
$TKKt,,/U,,
allows one to obtain
+
K’t,,
=
COtlSt.,
the following
three
parame-
of eq. ( I.1 1
with respect to the gauge, scale and Talanov transformations (the latter of which was found explicitly in ref. [ 111). The two parameter set of stationary states ( 1.2) can be obtained from (2.3) as a particular case corresponding to an infinite value of the parameter t,,. At t,, + = the solution (2.3) differs from (1.2) by a small term (proportional to t,;’ --$ 01, which grows quadratically in time t. The relation between this unstable eigenmode and an infinitesimal Talanov transformation was pointed out in ref. [7]. Both of the cubically and quadratically growing eigenmodes can be stabilized by the inclusion of a small defocusing nonlinearity in eq. (1.1). as was already mentioned in the introduction. Infinitesimal gauge and scale transformations (acting inside the family of stationary states (1.2)) generate constant and linearly growing in the time eigenmodes, respectively. The latter mode increases due to the accumulation of the phase difference only, which does not imply the instability of the Townes soliton. Apart from the gauge, scale and Talanov transformations, eq. (1.1) is invariant with respect to a five parameter group of Galilei transformations, including translations, rotations and movements with a constant velocity in the plane x = r cos 0. y = r sin 8. A rotational symmetry of eq. ( 1.l) and of the Townes soliton allows one to introduce an azimuth quantum number m = 0, _t 1, + 2,. and to treat independently perturbations having different values of m. Infinitesimal translations and velocity variations of the soliton generate two constant dipole (lrnl = 1) eigenmodes and two dipole eigenmodes linearly growing in time, respectively. The behaviour of the latter pair of
KM. Malkin, E.G. Shapiro /Elementary
modes is explained by a growing shift of the soliton as a whole and does not imply the instability.
of the functions equations i,uw=6JL’w,
3. Basic equations
where
In order to simplify further formulae, it is convenient to study elementary excitations for the soliton (1.2) of unit size (K-’ = 1) and zero phase (cp = 0). This does not lead to any loss of generality due to the scale and gauge invariance of eq. (1.1). An elementary excitation of the frequency w looks like
i, =i,,-
S$J r, t) = [A,,(r)
27
excitations
A,(r)
and B,(r)
iouw=wu,,
(3.4)
2R=.
(3.5)
The pair of equations (3.4) is presentable in the form of a one matrix equation for a two component function:
e-‘“’ + Bz( r) eiw*‘] ei’
(3.6)
(3.1)
and satisfies the equations
In the invariant subspace em of functions having a definite azimuth quantum number m the Laplacian looks as
WA, =&,A,-R2(A,+B,), -wB,=&B,-R2(A,+B,),
(34
A=‘drA-!!!f r dr
where 1
2
L,,=l-A-R.
(3.3)
The perturbation (3.1) does not change at combined replacements w + -w*,
BE,,
+A,,
A_,,,, + B,*.
Beside this, eqs. (3.2) are invariant with respect to combined complex conjugation and replacements w-+w*,
A*,, +A”,
B;*+B,.
It follows that if o is an eigenvalue,
then -w*, w* and --o are also eigenvalues. Thus, the investigation of the whole spectrum reduces to that in the first quadrant of the complex plane o: RewrO,
dr
(3.7)
,.2
and i,,, i, are reduced to ordinary differential operators. The reductions are designated below by the same symbols as the complete operators. A regular solution of eq. (3.6) in the invariant subspace e, behaves at r + 0 as
x,(r)
Cl rm +Lqrm+2).
=
c i
2
I
(3.8)
Main terms of asymptoticses of functions A,(r) and B,(r) at r + co are given by following formulae: A,(r)
z r-1/2{ A+ exp[ (1 - o)“‘r]
+A_exp[-(1-ti)“‘r]},
Imw20.
The linear combinations u&l=A,
satisfy the
+ B,,
u, =A,
-B,
B,(r)
E~-‘/~{B+
exp[(l +o)“‘r]
+B_ exp[ -(I
+ W)“*r]).
(3.9)
For the solution
regular
r = 0, each
in the point
of coefficients A + and Bi depends linearly on two parameters: C, and C, (besides which it dependson
w and m). IfImw=Oand
Rew>
then only one of the four exponents at r + =. The
growing
the preexponential The condition
exponent
factor
B,
is absent
when
into a zero.
by a proper
choice
of ratio of coefficients C, and C?. At such a ratio the solution tends to a zero at r + r: and represents an eigenfunction corresponding to a given value of w. It follows that the real ray o > 1 belongs to the continuous spectrum of the operator i. For other values of w in the half-plane Re o 2 0 already two of the four exponents in formulae (3.9) grow at r + ~0 and. to obtain a solution which tends to zero at I’ + a, one should satisfy two conditions: A + = 0 and B, = 0. This is possible only when the matrix lld):“‘(w)ll in the linear relation
i
B+ = d\;“‘(w) d$y’( w) A+
H
is degenerate,
d\‘;“(w)
c,
d:;“‘( 0)
ii c,
1
(3.10)
i.e. when the determinant
D,,,(w) =dgyo)dgyW)
-clyp(w)dy:‘(o) (3.1 I)
turns into zero. Thus, all eigenvalues w in the half-plane Re w 2 0 without the ray w > 1 (which is excised) are roots of equations
Qn(w) = 01 and belong tor C.
LX
(‘1 =
(‘(‘I =
Lb’)= 0.
0,
correspond-
K.
(4.1)
in (3.9) grows
turns
can be satisfied
1,
e,f = e,, X e,, only one eigenfunction ing to the zero eigenvalue:
This eigenfunction
is generated
imal gauge transformation Besides
x (I), there
of heights
by the intinites-
of the Townes
exist root eigenfunctions
n = 2, 3 and 4, corresponding
zero eigenvalue
of operator
4. Point spectrum
spectrum
of isotropic
(4.2) ix(R)
=
I/(3) =
()
x’zI, (.(3) = Ir2/( s
(4.3)
)
ix (4) --x (II 1 J1) = i,
I p1,
(‘(1) = ().
(4.4)
The root eigenfunctions x”’ and xc3’ arc generated by infinitesimal scale and Talanov transformations of Townes soliton, while x”’ is connected with its perturbation cubically growing in time (this perturbation was described in section 2). The graph of function ~6~’ is shown in fig. 1. The operator i has no root eigenfunction of height 5 corresponding to the zero eigenvalue in the subspace e,:. Indeed, if such a function ,,$” f
uL4’
of the opera-
perturbations
For isotropic perturbations, i.e. in the subspace e,,, the operator i,, is degenerate, while i, is reversible. The operator i has in the subspace
x(“) to the
i:
m =O, * 1, _t2,...
to the discrete
soliton.
Fig.
I.
The graph of root eigenfunction
u”‘(r)
V. M. Malkin, E.G. Shapiro /Elementary
exists, it should satisfy the equation L r,$) = 0
excitations
29
by the formula
u(4)
L’* =-
1
ra I 32R* 0 drr”R2S
The solvability condition of this equation reads , u(4))
( c,(I)
s
d2,.U(1)*U(4)
=
where R, is the coefficient in the asymptotics of the Townes soliton at r + w
()
I
and is not fulfilled, since ([Jo),
U(4))
=
(&,C2),
=
(U
(3,
U(4))
Ls(39
=
=
(@,
&
/
&u(4))
dzrr2RZ
#
0.
An existence of a four-dimensional root subspace corresponding to the zero eigenvalue of operator i in the space e,f implies that the function D,,(w) has a fourth order zero in the point w = 0. To explain this, one can make in eq. (3.6) the following substitution:
The representation (4.5) implies similar representations for functions u,, I!~, A, and B,,,, as well as for coefficients C,, C,, A + and B, in formulae (3.8) and (3.9). Since the function x(n) is decreasing at r + co for n = 1,2,3,4, the coefficients A(:’ and B ‘$’ for the same n are equal to zero. Thus, one has A + = w4&‘,
B, = 04B’:’
at + w”u’~‘(O),
C, = u,(O) = -&R(O) xw = ;
ufl- ‘x(n) + &*C’.
(4.5)
n=l
The auxiliary function xc’ satisfies the equation ix;s’ =
$4)
+
wx1;5’.
(4.6)
C, = u,(O) = R(0).
Substitution of these expressions in (3.10) reveals the following relation between columns of the matrix IIdW I, w>I/. .
For an unambiguous definition of this function, its value at r = 0 is conveniently put to zero:
&i&4)( 0) R(0)
xl;“(O) = 0. At w = 0 eq. (4.6) reduces to i,@
= 0,
&@
=
u(4).
follows that z@ = 0 and the function u,) grows exponentially at r --) a:
This relation allows one to rewrite the determinant D,(w) in the form
It
D,(w)
= &(di’l’(+l”‘(co) -d$‘(
The function nb5) is easily expressed in terms of the Townes soliton (by the Green function method). In particular, the coefficient U, is given
w)B’:‘( w)).
(4.7)
At w + 0 a further simplification is possible, since A(5)(0)
+
=
-B(5)(0)
+
=
12~‘~
,
d!:‘(O) = d$?(O) = %A 2 *,
where
pi.+ is
the coefficient
in the asymptotics
Ul-,-x- =U*rY’/zeI of the solution the initial
of the equation
condition
that the operator value
i,
= 0 satisfying
has only one negative
in the subspace
coefficient
i,u
2
U(O) = 1. It is easy to show eigen-
e,, and, consequently,
I’ .+ is negative.
Taking
the above note, one can conclude
into
0.5
the
account
that
: /
0
A$(,
w-4D,,(w) =
0
;;;;) > 0. Fig.
0
0.5
Graphsof functions
It can be shown (see refs. [5, 6, 81 and below) that all eigenvalues of the operator L are real. Thus, the numerical study of the point spectrum has to be undertaken only in the interval 0 < w < 1. At o + 1 the function D,,(o) increases by the law numbered by vnlues of 111= 0. I. 2 in the interval
which follows from the logarithmical increase of the solution of eqs. (3.2) at w = 1, r + m. A numerically constructed graph of the function
in the interval (0,l) is presented by the curve labelled 0 in fig. 2. As seen from the figure, the function has no zeros inside the interval. The following simple arguments show that the ,. operator L, in fact, has only real eigenvalues in the space /,f. Let F(n) (n = 1,2.3,4) be the root eigenfunction corresponding to the zero eigenvalue of operator it (adjoint to i) in the space e,:. The from the x(“) by a function ?(‘I’ is obtained simple rearrangement of components:
0.
II = 1,3,
( U(“), I.,,) = 0,
II = 2,4.
F(r,) = &1)
(n = 1.1,3.4).
(4.9)
Let k be the orthogonal projection operator reflecting the space e,, at its subspace orthogonal to the function I’(” = K. The projection of eqs. (3.4) to the subspace
0
I:,, =
E,L,,.
Here
and relations $I’) = [,W)
I ),
Each eigenfunction corresponding to a nonzero eigenvalue of the operator i is orthogonal to the XC”) (n = 1,2,3,4). In terms of single component functions the orthogonality conditions read
(1.(‘I)) L1,,) = (4.8)
(0.
WLl<,,
Fe,,
=
has the form
ioF,,,.
(4.10)
V.M. Malkin, E.G. Shapiro /Elementary
generated by infinitesimal Galilei transformations, also exist. Root eigenfunctions of height 3 are absent, since the solvability conditions of equations
are taken into account. In the subspace P^e,, the operator i,, is positive definite (which follows from the monotonicity of the eigenfunction R corresponding to the zero Eigenvalue of & in the space e,,). The operator L, is degenerate in the subspace Fe,,, since
&‘;f’=
$2) = 0
are not fulfilled:
The function ~6’) has only one zero, while functions without zeros are not orthogonal to R and do not belong to the subspace Fe,,. This implies that the function uf2) coiresponds to the smajlest eigenvalue of operator L,, i.e. the operator L, is nonnegative definite in the subspace Be,. The above properties of operators Lo and i, guarantee that all eigenvalues of the operator L in the space e,f are real, since the formula
(u’;‘, ~‘2’) = +( R, R) # 0.
@2
Ez
(% GU”)
(4.11)
(% L,‘u,) ’
holds, which follows from eq. (4.10).
5. The point spectrum of nonisotropic perturbations
7
L’(I) =
(5.1)
0.
*
These eigenfunctions are generated by infinitesimal translations of the Townes soliton in the plane X, y. Root eigenfunctions ,$z”’
3
(5.2)
/,,(2) +
The radial part, dR/dr, of the functions u’:’ (corresponding to zero eigenvalues of the operator f., in subspaces e + > has no zeros in the range O
For dipole perturbations, i.e. in subspaces e + ,, the operator i, is positive definite, while the operator i, is degenerate. The operator i has in each of the subspaces e +, only one eigenfunction corresponding to thezero eigenvalue:
31
excitations
a (1 -d-3/4.
A numerically constructed B,(o)
graph of the function
=riTTw-2D,( w) (1 - w’)
3/4 (5.3)
in the interval 0 < w < 1 is presented by the curve labelled 1 in fig. 2. For multipole perturbations, i.e. in subspaces e,,, at lml 2 2, both operators i, and i, are positive definite. This implies that all eigenvalues of the operator i in subspaces 1: are real, as seen from eq. (4.11). Due to the mirror symmetry of the Townes soliton and basic equations, the functions D,,,(u) do not depend on the sign of m. At m s+ 1 the functions D,(w) are easily calculable analytically. Picking out the analytical depen-
32
dence,
V.M. Malkin,
one can introduce dT(
rs,( w)
1 _
auxiliary
E.G. Shapiro
functions
growing
excitations
separation
w2)“1:2+‘/4
(5.4)
Qn(W)~
m!)2
the process
is well forbidden
for the Hamiltonian at m --) 00. A numerically
tending
to unity
structed
graph of the function
by the curve labelled
between
the solitons.
On the
other hand, a small perturbation cannot cause the decay of a soliton for two other solitons, since
= 227
/ Elrmentary
by conservation
and the number
laws
of quanta.
con-
~,(cIJ) is presented
2 in fig. 2. At higher
m, the
functions fi,,
Acknowledgement The authors useful remark.
are grateful
to G.M. Fraiman
for a
6. Conclusion References The main conclusion to the above analysis consists in the following: solitons of the two-dimensional Schrodinger equation with attracting cubic nonlinearity have no elementary excitations, besides the continuous spectrum and eight discrete modes generated by well-known symmetries and exact solutions of the problem. This fact plays a significant role in the soliton’s adiabatic dynamics describing the self-focusing of radiation in different media (see ref. [9]). It is noteworthy that solitons of the one-dimensional Schrodinger equation with attracting cubic nonlinearity also have no localized excited states. As seen from the very scheme of finding solutions by the inverse scattering method (see, for instance, ref. [12]), an excited state, if one exists, can be treated as a bound state of two solitons moving with equal velocities. An arbitrarily small perturbation
breaking
the
equality
leads
to a
[II R.Y. Chiao,
E. Garmire and C.H. Townes. Phys. Rev. Lett. 13 (1964) 479. Izv. VUZov. Radiofizika 9 (1966) 412 121Z.K. Jankauskas, [Sov. Radiophys. 9 (1966) 261 [31 HA. Haus, Appl. Phys. Lett. 8 (1966) 128. V.V. Sobolev and VS. Synakh, Zh. Exp. [41 V.E. Zakharov, Teor. Fiz. 60 (1971) 136 [Sov. Phys. JETP 33 (1971) 771. [Sl E.W. Laedke, K.H. Spatschek and L. Stenflo, J. Math. Phys. 24 (1983) 2764. [hl M.J. Weinstein, Commun. Math. Phya. 87 (1983) 567. and S.K. Turitsyn. Phys. Lett. A 112 [71 E.A. Kuznetsov (1985) 273. [81 G.M. Fraiman, Zh. Exp. Teor. Fiz. X8 (1985) 390 [Sov. Phys. JETP]. LOIV.M. Malkin, On the analytical theory for self-focusing of radiation, to be published. [lOI V.I. Talanov, Zh. Exp. Teor. Fia. Pis’ma, 11 (1970) 303 [Sov. Phys. JETP Lett. 11 (1970) 1991. [Ill V.I. Talanov, Izv. VUZov. Radiotizika 9 (1966) 412 [Sov. Radiophys.]. S.V. Manakov, S.P. Novikov and L.P. iI21 V.E. Zakharov, Pitaevski, Theory of Solitons: Inverse Scattering Method, vol. 1 (Nauka, Moscow. 1981). Ch. 10.