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On discretizations of the vector nonlinear Schrödinger equation
29 March 1999
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 253 ( 1999) 287-304
On discretizations of the vector nonlinear SchrGdinger equation Mark J. Ablowitz”, Yasuhiro Ohtab, A. David Trubatch a a Department of Applied Mathematics, Universify ofColorado-Boulder, Boulder. CO 80309, USA h Department ofApplied Mathematics, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, H&ash&Hiroshima 739-8527. Japan Received 27 April 1998; revised manuscript
received 3 November 1998; accepted for publication Communicated
14 January
1999
by A.P. Fordy
Abstract Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric system are presented. Formulae for a constrained class of solutions of the symmetric system may be obtained. Numerical studies support the hypothesis that the symmetric system has general soliton solutions. @ 1999 Elsevier Science B.V. PAC.‘+ 03.40.Kf;
46IO.+z;
42.81.D~;
42.81.G
1. Introduction In recent years there has been wide interest in the study of solitons and integrable systems. Researchers have found that not only are continuous systems (i.e. PDEs) integrable via the inverse scattering transform (IST) but also that interesting classes of discrete systems (semi-discrete as well as partial difference equation) are integrable (cf. Ref. [l] for an early review). In this Letter, we discuss soliton solutions and the integrable nature of certain discrete systems associated with the vector extensions of the nonlinear Schrodinger (NLS) equation: iq, = qXx + 2&12.
(1)
NLS is a centrally important and physically significant nonlinear equation which possesses solitons and is integrable via IST [ 21. Furthermore, NLS arises in many areas of physics, such as the evolution of small amplitude slowly varying wave packets in: deep water, nonlinear optics and plasma physics (see e.g. Ref. [ I] ) . In 1974, Manakov [ 31 showed that the vector NLS (VNLS), iq, = 4xX + 211ql12q~ where q is an N-component integrated via IST (actually,
(2) vector and )I . 1) denotes the vector norm, also possessed solitons and could be in Ref. [3] only the case N = 2 was studied in detail; however the extension to
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00048-1
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Letters A 253 (1999) 287-304
the Nth order vector system is straightforward). The second order (N = 2) VNLS equation is relevant in the study of electromagnetic waves in optical media in which the electric field has two nontrivial components. In optical fibers, the components of q in Eq. (2) correspond to components of the electric field transverse to the direction of wave propagation. These components of the transverse field compose a basis of the polarization states. In optical fibers, the equations governing the field are more generally non-integrable variations of vector NLS [ 451, although there are circumstances in which Eq. (2) is the appropriate model [ 61. A related model with experimental significance is . (‘) = qg ‘41
+ 2(lq”‘[2
‘4t(*) = qL2,’+ 2(Blq(‘)j2
+ Blq(*y)q(‘),
(3a)
+ )q’*y)q’*‘,
(3b)
which is equivalent to (2) when B = 1 [ 61. In contrast to the Manakov equation (2), the system (3a)-( 3b) is not integrable when B # 1 (cf. Ref. [7] ) and cannot be solved by IST. This system further illustrates that integrability depends specifically on the form of the equation - i.e. the integrability of (3a)-(3b) depends on the precise value of B. In 1976, Ablowitz and Ladik [ 81 found that the following discrete system is integrable via IST: i$q,
= i(qn-t
-2q,
+qrl+t)
+ lqn12(qn+i ++I).
(4)
The continuum (h + 0) limit of this discrete system (4) is NLS ( 1) hence this system can be referred to as integrable discrete NLS (IDNLS). The discrete system itself is also useful in physical applications (see e.g. Refs. [9-121). IDNLS provides an excellent numerical scheme to solve NLS ( 1) (cf. Ref. [ 131) . Computations demonstrate that, by simply replacing the nonlinear term in (4) with 21q,(*q,, one obtains a poor numerical scheme. In contrast, the preservation of the integrable structure in IDNLS (4) plays a key role in its utility as a numerical scheme. This points out that care must be taken in the choice of discretization of the nonlinear terms for the purpose of numerical simulations. It is natural to look for useful discretizations of VNLS (2). However, the special character of IDNLS as compared to other discretizations of NLS, suggests that different discretizations of VNLS may have very different dynamics, Therefore, the choice of discretization merits close analysis. In this Letter, we discuss two discretizations of VNLS which we refer to as the symmetric and asymmetric discretizations. These discretizations are as follows: Symmetric discretization: i$q,
= ;(4,-t
- 2q, + qn+t) +
where q, is an N-component Asymmetric discretization: d iz4” -i$,,
1 = -((4+i h* = $(r,_l
where, as before, qn Both systems (5), take r = -q*, where (6a)-(6b), however,
l14nl12(4n-, + %+I))
(5)
vector.
- 2% + 4n+l) - ezndz, - 2r, + r,+l)
- (Cq,)r,-l
- a?,)%,,~ - (C+,q,)rn9
(6a) (6b)
is an N-component vector as is r,. The superscript T denotes the transpose. (6a)-(6b) reduce to VNLS (2) in the continuum limit (h -+ 0) where in (6a)-(6b) we * denotes the complex conjugate, after taking the continuum limit. The asymmetric system does not admit the symmetry r, = -4; and hence remains “asymmetric” for h finite.
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289
These systems have a number of interesting properties. Both systems (5), (6a) -( 6b) reduce to the integrable discrete scalar case (4), just as VNLS (2) reduces to NLS ( 1) . The asymmetric system (6a) -( 6b) is associated with a linear operator pair (see Eqs. (17a)-( 17b)). Although there is no known linear operator pair for the symmetric system (5), under the reduction qn = eiO’u,, where U, is independent of t, this equation reduces to an N-dimensional difference equation which is known to be integrable (cf. Ref. [ 141). The results in Ref. [ 141 suggest that the linear operator for the symmetric system (5) may be closely related to the matrix generalizations discussed below (cf. ( 17a) ) . Both the symmetric (5) and asymmetric (6a)-(6b) systems possess a class of soliton solutions. The asymmetric system has a class of soliton solutions which correspond to all the soliton solutions of the continuum limit NLS (2). However, the exact multi-soliton solutions found so far for the symmetric system (5) only reduce to a subset of those associated with Eq. (2). Since the known exact soliton solutions of Eq. (5) are only a subset of those known in the continuum limit (2)) we numerically examined the interactions of solitary waves associated with the symmetric system which lie outside this class. The numerical evidence indicates that, in the symmetric system, the solitary waves interact elastically and are therefore true solitons. However, as we discuss below, when (5) is modified, solitary-wave interactions are not elastic. These findings strongly suggest that the symmetric system (5) is indeed integrable. The generalization of (5) where the nonlinear term in the jth component, (C,“=, lq,$k) I*) (St>, + q$ ), is replaced by (cf=, of the form
Bj,klqLk)(*) (q;!,
+ ql;‘:{) is an interesting
discrete system. For example,
+q::),)+(lq~1’12+Blq~2)12)(q~l),
+q;:‘,),
+ q:2+)1)+ (Blq:“l’+
+q;$.
lq:*)j*)(q~,
the N = 2 system
(7b)
where B is a parameter, is a discretization of (3a)-( 3b). We concretely contrast the two-component symmetric system (i.e. (5) with N = 2) with (7a)-( 7b). In numerical simulations of the system (7a)-(7b) with B # 1, described below, solitary waves do not, in general, interact elastically. Rather, the solitary-wave interactions are qualitatively analogous to those of the continuum limit - i.e. the PDEs (3a)-( 3b). These results suggest that: (i) the discretization (7a)-( 7b) retains the qualitative properties of the PDE and (ii) the elastic soliton interactions observed in the symmetric system (5) depend on the precise structure of that system (e.g. B,i,k= 1) . We conclude that the systems (5)) (7a) -( 7b) - and generalizations - are extremely useful as numerical schemes for the study of their continuous analogs (2)) (3a)-( 3b) - and generalizations.
2. NLS and integrable
discrete NLS
We briefly review the relationship between NLS and IDNLS in order to illustrate the way that solitons are solutions preserved by an integrable discretization. NLS ( 1) can be obtained as compatibility condition of a linear operator pair (e.g. a Lax pair). The linear pair can be used to solve ( 1) by IST. The one-soliton solution of NLS is q(x, t) = ae -i(bx-(b’-J)+d)
sech(ax _ abr _ 0) = ei+gx,
t),
(8)
where a, b, 0 and C$ are arbitrary parameters. Either IST or Hirota’s method can be used to obtain the multisoliton solutions. The existence of an associated linear operator pair and IST are hallmarks of integrability. Therefore, in the effort to construct an integrable discretization of NLS, it is natural to look for a discrete version of the operator
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Letters A 253 (1999)
287-304
pair and the compatibility condition. In fact, the integrable discretization of NLS (4) was obtained in exactly this manner [S]. The extent to which the integrable structure of NLS is preserved under the integrable discretization is significant: The IST for IDNLS is analogous to the theory for NLS. In particular, the one-soliton solution of IDNLS (4) given by qn(t)
= Ae-i(bkn-w+)
sech( ahn - ut - 0) = e’“&(t)
,
(9)
with A = sinh(ah) h
’
6J=
2( 1 - cosh(ah) h2
cos(bh))
u = 2sinh(ah)
sin(bh) h2
’
converges (as h -+ 0) to the one-soliton solution of NLS (8) with the same four free parameters. More generally, solutions of IDNLS converge to the solutions of NLS with an error of O( h2). For both NLS and IDNLS, generic rapidly-decaying initial data resolves itself into some number solitons plus radiation that vanishes as t -+ 00 (in the sup norm). Even though the solitons interact nonlinearly upon collision, it is well-known that they retain their shape and speed after the collision. This can be determined by comparing the forward (t + +oo) and backward (t --+ -oa) long-time limits of the solutions. In these long-time limits, solitons traveling at different speeds are well-separated. Although this collision is elastic, the position and overall complex phase of the individual solitons are shifted between the forward and backward long-time limits. This elastic interaction of solitons up to a shift in phase (position and overall complex phase) is typical of ( 1 + I)-dimensional integrable systems.
3. Vector NLS Now we consider Vector NLS (2) and illustrate those properties which we seek to preserve under discretization. In particular we give the Hirota formula for the general M-soliton solution and describe soliton interaction in the vector system. VNLS can be obtained by substituting the vector q for q in NLS (1) and replacing complex conjugation with the Hermitian conjugate. Similarly, the linear operator pair for VNLS can be obtained by appropriately making the same substitution of scalars by vectors in the linear operator pair for NLS (cf. Ref. [ 31) . When considered as a model for the propagation of electromagnetic waves in optical fibers, the components of VNLS play the role of a basis for the polarization vector. In the derivation of VNLS by Manakov [3] the choice of basis is arbitrary. Therefore, the vector system (2) ought to be, and is, invariant under a change of basis for the polarization. Mathematically, a change of basis is obtained by multiplying the independent variable q by a unitary matrix, U. Therefore, the freedom in the choice of basis is reflected in the fact that the vector system (2) is invariant under the transformation
q+uq. This symmetry is an important feature in distinguishing among discrete versions of vector NLS (notably, the versions of discrete vector NLS discussed in this Letter (5)) (6a)-( 6b) retain this symmetry). Under the reduction q = cli,
(10) both
(11)
where c is a constant, N-component vector such that ]lc112= 1 and 6 is a scalar function of x and t, VNLS (2) reduces to NLS ( 1). This is a manifestation of the fact that the vector system is a generalization of NLS obtained by allowing the polarization to be non-constant. When the solution of VNLS has a constant
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291
polarization, NLS is recovered. As a consequence, any solution of scalar NLS has a corresponding family of solutions of VNLS. We call a solution of the form ( 11) a reduction solution. Because the vector c is arbitrary (up to the constraint that it is of unit length), a reduction solution has correspondingly more parameters than the solution of scalar NLS to which it reduces. We refer to the vector c in Eq. ( 11) as the polarization of the reduction solution. In particular we show below that the one-soliton solution of vector NLS is the reduction solution (i.e. of the form ( 11) where 9 is the one-soliton solution of NLS). 3.1. Soliton solutions The soliton solutions of vector NLS can be found by IST [3] or by Hirota’s method. The Hirota formula for the general M-soliton solution suggests the formulae for the soliton solutions associated with the discrete systems (5) and (6a)-(6b). The derivation of the general multi-soliton solution by Hirota’s method has not been previously reported in the literature. (The Hirota formula for the two-soliton solution is given in Ref. [ 151 and the derivation by IST is described in Ref. [3].) Here, we give the general Hirota formula, for both (i) completeness and (ii) as a motivation for the soliton solutions of the discrete vector systems. In order to obtain the Hirota formula, we rewrite VNLS (2) in bilinear form. With the independent variable transformation
where g is a vector and iD,f
.g=
Dif
f is a scalar, solutions of the bilinear equations
-g,
are solutions of VNLS. In particular, the M-soliton
D:f
- f = llgl12~
solution of VNLS is given by the determinant
formulae:
(13) where Z is the M x M identity matrix and (i) A, and B are M x M matrices defined by A”,.” II
=
Pf + P.i’ ’
with f!, j = 1, . . . , M and v,i = pjx + ipjt; the complex numbers pj = aj - ibj, aj > 0, determine the amplitude and speed of the jth soliton; the N-component complex vectors rj (j = 1,. , . , M) determine the polarizations and envelope phases of the solitons; (ii) @ is the column vector Qce) = eqr, C = 1, . . . , M; (iii) Y, is the row vector Pk(i) = -rjk’, j = 1,. . . , M. The one-soliton solution is obtained with M = 1 in the determinants ( 13). This yields 4= &
aiei’m71 sech(Re7,
+ S),
where Revi
= six - 2qb,t,
Irnvi
= -blx
+ (a: - bt)t,
S=log!$ I
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M.J. Ablowitz et al. /Physics Letters A 253 (1999) 287-304
and, without loss of generality of the form ( 11) - where
we assume at > 0. Hence, the one-soliton
solution
is a reduction
solution - i.e.
and the scalar function 4 is the one-soliton solution of scalar NLS - i.e. (8) - with a = al, b = bl and 8 = -6 (without loss of generality, the parameter I$ in (8) is absorbed into the polarization vector). Expressions for solutions with two or more solitons are obtained simply by the expansion of the determinants. However, the soliton interaction can be analyzed more easily by computing the long-time limits of ( 13) before expanding the determinants. 3.2. Interaction of the solitons Now we describe the interaction of solitons in VNLS. As in the scalar case, the IST [ 31 (or analysis of the formulae obtained by Hirota’s method) show that solitons interact elastically. By elastic, it is meant that the solitons retain their shape and speed after interaction as shown by comparing the forward and backward long-time limits. For the vector system, care is needed: the nature of the phase shift for vector solitons is somewhat different than the scalar case. In addition to a shift in the center of the peak (as in the scalar case), vector solitons also undergo a change in polarization upon collision. That is, in the forward and backward long-time limit, the solution asymptotically approaches a linear superposition of the individual solitons:
qwCq,f, where qj* = cfijf speed. domparisbn
as t+fco,
and, for each j, df and ~$7 are one-soliton solutions of NLS with the same amplitude of the forward (‘i) and backward (-) long-time limits shows that
and
c,: # ci’, but (14) A closed formula for this shift in the polarization can be calculated by considering the eigenfunction of the associated scattering problem [ 31. We refer to the squared modulus of a component of the polarization vector as the intensity of that component of the polarization. That is, if c,:‘) is the &h component of Cj, the polarization vector of the soliton j, then the intensity of the of the Cth component of soliton j is lc~“‘/*. The relation (14) implies that the total intensity of each soliton is preserved, specifically the sum of the intensities is equal to one. However, the distribution of intensity between the components of the polarization of an individual soliton will not, in general be equal in the forward and backward long-time limits due to interaction with other solitons: subject to the constraint of Eq. (14), in general,
I@- 12# ,q)+12. This change in the distribution of intensity is a distinctive feature of the vector system. There is no corresponding phenomenon in the soliton interactions of the scalar equation (in both the scalar and vector equations, the location of the peak of an individual soliton is shifted by the soliton interaction).
M.J. Ablowitz et al. /Physics
Manakov [3] also observed the following every pair of solitons j and k, either
Letters A 253 (X999) 287-304
special case for the interaction
of the solitons
293
of VNLS: If, for
(15a)
Ic,; . CJ = 1,
(15b)
where - denotes the dot product, then
Ic,, - CT1= 1,
(16)
for all j. As a consequence of ( 16), ]cjf’- /* = Ic,iE)+]* for all j, 1. Physically, this means that, in any polarization basis, the distribution of intensity among the components of polarization for an individual soliton is not changed by the soliton interaction. 3.3. Soliton interaction
in non-integrable
vector NLS
The nature of solitary-wave interaction in the generalized system (3a)-(3b) contrasts strongly with the soliton interaction in the integrable case. First, we note that when B # 1 the symmetry transformation (10) no longer holds and therefore general reduction solutions of the form ( 11) do not exist. However, reduction solutions do exist for the particular polarizations
where 0 < $1, (62 < 27r. In particular, there are solitary-waves of the form ( 11) where c is one of the three allowed polarizations and 4 is the one-soliton solution of NLS. We contrast the interaction of these solitary waves with the elastic collisions described in the previous section. When two solitary waves of the same polarization collide, the vector nature of the system does not play a role. In this degenerate case, the system (3a)-( 3b) reduces to the scalar NLS (1). As a consequence, the solitary waves interact elastically. These elastic solitary-wave interactions occur only because the coupling between polarizations plays no role. The interaction of solitary waves of different polarizations is, however, strongly inelastic. These phenomena can be investigated by numerical simulation (cf. Ref. [ 161). When solitary waves of different polarizations collide, a number of results are possible. In general, the speed and height of the solitary waves are not preserved in the collision. For some parameter values, the solitary waves reflect one another while, for different parameter values, the waves pass through each other but emerge with different speeds and amplitudes. Additionally, in some simulations, the soliton interaction can lead to large and rapidly oscillating radiative tails.
4. Asymmetric
discrete vector NLS
The preceding derivations of IDNLS and VNLS from NLS suggest two methods to obtain an operator pair which has a discrete form of VNLS as its compatibility condition: (i) discretize the operator pair for vector NLS in a manner analogous to that used to obtain the operator pair of IDNLS or (ii) appropriately substitute vectors into the operator pair for IDNLS. These approaches yield the linear operator pair
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M.J. Ablowitz et al. /Physics
T
iq,rL_, - i(1/2h2)(z - I/z)~ZN i(l/h)((l/z)rT - zri-i)
=
”
where IN is the N x N identity this pair. Under the reduction 4” =
Letters A 253 (1999) 287-304
-i(llh)((llz)q, - zq,-,) +i( 1/2h2)(z - ]/z)~
-irzq,_,
matrix. The asymmetric
system (6a)-(6b)
(17b)
’
is the compatibility
condition
r, = dg,‘,
CQ”,
of
(18)
where c and d are constant vectors such that c - d = 1 and 4” is a scalar function of n and t, the asymmetric system (6a)-(6b) reduces to IDNLS (4). Solutions of this form ( 18) are referred to as reduction solutions. Hence, every solution of IDNLS generates a family of reduction solutions of the asymmetric system. We are particularly interested in reduction solutions where d = -c* because, in this case, r, = -q,* and (6a)-(6b) reduces to a single equation. 4.1. General formula for multi-soliton solutions Multi-soliton solutions of the asymmetric system can be derived, without the use of the IST machinery, Hirota’s method. First, make the independent-variable transformations
by
r,=-, & fn
q”=F, R
where g, and 2, are vectors with N components
and fn is a scalar. Then, solutions
of the system of bilinear
equations (19a)
fnil fit-1 - f,’ = 2&T ih2D,f,
- 2f,g,
- g, = f+&+l
-ih2D,f,
- & = f+$,+,
+ f,+lg,-,
- 2f&,
- H&
+ fn-$,,+l
(19b)
- fi&,,
(19c)
fnH,+l
= h2k,g;+,
- g,+,g;L
(19d)
f,,&-,
= h2@,_&
-#,&L
( 19e)
where the N x N matrices H, and Z?,, are auxiliary variables, are solutions of the asymmetric system (6a)-( 6b). The M-soliton solutions for these bilinear equations (19a)-( 19e) are given by the following determinants:
gLk’=
“; z 0
/ H”..” ”
=
A,
Z
@n-l
-’
B
0
pt
0
0
Y,j
0
B Y,
@n1
’
0
where Z is the A4 x M identity matrix, and
@n 0 0
,
gck) = n A,, -I
1” &+,
Z B 0 0
A, -Z
Z B
&
0 0
0
!@‘k, 0
?PF ‘Pi 0 0
(20a)
0
0 0
(2Ob)
M.J. Ablowitz
et al. /Physics
295
Letters A 253 (I 999) 287-304
(i) A,, and B are M x M matrices defined by A”..”
h
=
n
e wr+P,*) _ *
@Lj) _ - eh(p;+:,) _ , rh
ew”+v;,,(
with vj,n
pjn}l
=
+
$2 - e”“’ -
e-““’
)t;
(21)
the complex numbers pj = a,j - ibj, aj > 0, determine the amplitude and speed of the jth soliton; the Ncomponent complex vectors yj and yj (j = 1,. . . , M) determine the polarizations and envelope phases of the solitons; (ii) cP”, 6, are, respectively, the M-component column and row vectors, -(j) _ Qn _ e 7.1.11 ;
@W = e-““L eS’JJ II (iii)
P,,
@k are, respectively,
the M-component
row and column
vectors,
q.(P) = e-rlpf7;k].
p;.j) = _y,;k],
k
4.2. The one-soliton solution From the above formulae
Yl qn =
(gy,)l/2
r, = -
e
(20a)-(
(M = 1) solution
is
-ihbl sinh( a] h) e’.p‘,‘Isech( a],n + 6, ) ,
(224
h
-i/tb, sinh(ah) (+&1/2
20b), the one-soliton
e_ipI,,,
h
e
(22b)
sect-da,,, + 6, ),
where
a],, = Rev],,
= qhn - 2
PI,,,
= -hhn
= Imvl,,
+
sinh( a] h) sin( b] h) t
h2
(234
’
2( 1 - cosh(a,h) cos(b,h)) h2
(23b)
and
s, = log
+{y;y]}
- log C+-
Note that to get a solution (22a)-(22b) with 6, real we restrict our attention to the case where y:y, is real and positive. The one-soliton solution (22a)-(22b) is a reduction solution - i.e. of the form ( 18) - where
and the scalar function 4” is given by Eq. (9) - the one-soliton 0 = 4,. In this solution, d = -c* if, and only if, 7, = y;.
solution of IDNLS - where a =
UI ,
b = 6, and
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287-304
4.3. Two-soliton interaction To show that, for A4 > 1, the determinants the long-time limits. In these limits, solitons M = 2 and
(20a)-(20b) indeed give a multi-soliton solution, we consider moving at different speeds are separated. For concreteness, let
sinh( at h) sin( bt h) > sinh( azh) sin( b2h) ,
(24)
for the given at, bl, ~2, b2. The condition (24) ensures that each soliton travels with a different speed (the analysis below holds, in general for M solitons as long as each has a different speed). When the solitons travel at different speeds, we determine the asymptotic form of an individual soliton by taking long-time limits in a coordinate frame moving with that soliton. The limits n N 2sinh(u,ih)
are long-time
sin(b,ih) t , a,ih3
t-+Itcm,
limits in a coordinate
Rerlt.n = const The substitution
and
of (26)
* 4, -+ 4l.n
and
(25)
frame moving with soliton j. In the limit (25) with j = 1,
Revz,,, + &o,
as t -+ fca
into Eqs. (20a)-(20b),
(26)
where M = 2, yields
as t -+ *co,
r, -+ r:,,
(27)
which is the asymptotic form of soliton 1 in the forward (+) and backward (-) long-time limits. The long-time limits for soliton 2 are similar. The coordinate frame of soliton 2 is obtained by the limit (25) with j = 2 which yields Re v ,,, + loo
and
Re 772,,,= const,
as t -+ *co.
Note the change in relative sign between ~1,” and t in the coordinate in the long-time limit. To obtain +
4, *
and
42,”
r,
-+
* r2,,,
(28) frame of soliton 2 - the slower soliton -
as t -i *oo,
(29)
which is the asymptotic form of soliton 2 in the forward (+) and backward (28) into (20a)-( 20b), where M = 2. Combining (27) and (29) gives 4n N q$
+ 4:”
and
r,, -
if,
+ r.f,,
where
Cl 41,n* A-
=
r<,=4-G .
’
long-time
limits, substitute
as t -+ ice.
The effect of interaction on the solitons is determined by comparing q,Tnand We give the formulae for the asymptotic forms of the solitons below. First, we consider soliton 1. In the backward long-time limit, 41.n
(-)
r&
with qln and
rin
for j = 1,2.
M.J. Ablowitz et al. /Physics
and i,,
is a one-soliton
e=-s,
solution
291
Letters A 253 (1999) 287-304
of IDNLS (the 4 in Eq. (9) ) with a = ai, b = bl and
=-log~{~~;fy,}+log
(31) F+
As in the one-soliton
case, we require that 7;~~
is real and positive. The forward long-time
for soliton
1 is
where 1
CT = - (e-b1 g2 - P*)(c+2(&>rl
- P*@Y,)Y2)*
X
d: = _’ X
e-hPi (g2 - PI
and (i{n is a one-soliton 0 = 4;
((72@Y2H,
solution
= - log X + log
-
m;v,
H2)’
of IDNLS with a = al, b = bl and
(+2 +
log{&),
also
and we require that ri, yj for j = 1,2 are such that X and yly2 are given by h
u’ =
,h(p,+p;)
h _
1
u2 =
’
e/1(/,2+/J;;)
are real and positive. The constants
go, ~2, p
h _
1
’
p = eM/h+P;)
_
1 .
Hence, the expression ((~1~2 - 1~1~) is real and positive. Note that, although cl f CT, d, # d:, and 6, # ST, the parameters ai and 61 are the same in both long-time limits. Thus, soliton 1 has the same form in both long-time limits but undergoes a phase shift due to interaction with soliton 2. The asymptotic limits of soliton 2 are similar to those for soliton 1 except that the relative change in sign in (28) reverses the calculations, that is, q2.n
- *= c2 42.n
and
rTn = d; g-;,
where c; has the form (33a) and d, has the form (33b) with the indices 1 and 2 exchanged. The scalar function C& is a one-soliton solution of IDNLS with a = ~2, b = rzz and 19= -8, where 8, is equal to ST with the indices 1 and 2 exchanged. Similarly, f _ +-+ 42,” - c2 q2.n
and
r 2f,, = d,fB&
where cl has the form (30a)
and dr has the form (30b)
with the indices
one-soliton solution of IDNLS with a = ~2, b = n2 and 0 = -4: and 2 exchanged. In order to construct a solution such that
1 and 2 exchanged
where 6; is equal to 8,
and Si,
is a
with the indices
1
298
M.J. Ablnwitz
for j = I,2 it is necessary
and sufficient
et al. /Physics
Letters A 253 (1999)
287-304
that
7, =rT*
t 3W Jy32
+
~,-/luly~2’)2
(1 _e-2/ra,)y3/j2)*
For this choice, x and $Y,i then
(1 _ e-2/la,)yy)*y;2) ~e-halyy(2
+
ly;2)12
for j = 1,2 are real and positive as we required.
(34b)
However, if (34a) -( 34b) hold
rjtn + -9.1;9
(35)
for j = 1,2. In fact, llr& - (-qIJ)Il and
7, = r;
Yz=yG.
(36)
Then, if (36) holds, x and $‘yi r1.n = -4c,:t
= O(h). An alternative is to set
ri,
- for j = 1,2 - are real and positive and
= -4;;
but
IIG - (-!$;:)II = O(h)= lb;, - (-42,n*)II. In contrast, recall that the solutions of the scalar IDNLS converge to solutions of the scalar NLS with 0(/z’) convergence. This unavoidable asymmetry of the system (6a)-(6b) makes it less desirable as a discrete approximation of VNLS. Hence, we turn our attention to the symmetric system.
5. Symmetric
discrete vector NLS
The symmetric system (5) is the natural vector generalization of IDNLS (4). We now describe some important symmetries of the symmetric system. In analogy with the scalar case (4)) the symmetric system can be thought of as the reduction of the system i&7,=
$(4+t
-i$r,
= &(rR-t
-29”+fq,+t) -2r,
obtained by letting r, = -ql, Under the reduction 4,, =
*
cqn,
-t-r,+t)
-Cqn(qn-t
+qn+t)’
(37a)
+r,+t),
(37b)
- qXr,(r,-t
which is the symmetry
that is broken by the asymmetric
system.
(38)
where llcll = 1, the symmetric system reduces to the scalar IDNLS (4). Furthermore, if qn satisfies the symmetric system, then so does Uq, where ZJ is a unitary matrix (this symmetry is a discrete form of Eq. ( I1 )). Therefore, the symmetric discretization retains the reductions and symmetries of the asymmetric system and has the additional symmetry r, = -q;T. Despite the above-mentioned symmetries, there is, to date, no known associated linear operator pair for the symmetric system. Without such a pair, the system (37a)-( 37b) cannot be solved by IST. The question remains, however, whether the symmetric system is integrable. In the absence of IST, the existence of multisoliton solutions provides strong circumstantial evidence of integrability.
M.J. Ablowitz et al. /Physics
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Letters A 253 (1999) 287-304
In order to determine whether there are solitary waves which interact elastically - i.e. solitons, we first identify a solitary-wave solution of the symmetric system: a solution of the form (38) where the scalar &, is a one-soliton solution of IDNLS (as in Eq. (9) > is a solitary-wave solution of the symmetric system. Note that such a solution can have any polarization c subject only to the constraint that l/c11*= 1. In order to determine whether these vector solitary waves interact exactly, we use both direct and numerical methods. 5.1. Soliton solutions by Hirota’s method Soliton solutions of the symmetric system can be found by Hirota’s method. For concreteness, the case where q,, has two components (N = 2). Under the independent variable transformation,
we consider
qn&, .fil where g is a vector and $
is a scalar, solutions
of the bilinear equations
ih2Dtfn-g, = fn-lg,,+, - 2.fng, + fn+lg,,_,, fn+~frv-I - f,’ = h211g,/12~ are solutions of the symmetric system. The following solution of the bilinear
equations
yields a two-soliton
f = 1 + e’7’.‘,+“;!,+ ea..+s:,,, + IB, l*e1)l.,,+9;,,+112.,1+?1;
solution:
(JOa)
,,,(
(40b) g
(2)
h (eb2 _ e--kt$)et12.~~( 1 + B2e’?l.,‘+‘l;,,,),
(4Oc)
where li,n and p,i are as in Eq. (21) and (&I B, =
_ ehy2) (&PI + ekPJ )
(&(Pl+Pz) + l)(e/~(/~l+P;) _ 1)’
(&I B2 = -
(&~(Pl+Pz)
_ &Pl ) (e/l/J; + ehp2) +
l)(&(P;+/B)
_
1)
In order to see that the above solution indeed gives a two-soliton solution, consider the long-time limits. We calculate the long-time limits of the Hirota form of the solution (40a)-(40~) by the same approach as was used for the asymmetric case. In the forward (+) and backward (-) long-time limits, the solution (40a)-(40c) asymptotically approaches the form
4, -
q:n +q,fn,
The long-time
limits for soliton
sinh(ah) h sinh( ai h) h
eiP1,,
1 are SeWal,,),
e’PI.risech( (Yr,n+ log (B, I),
where $1 =hbl,
4: = hbl +argB,,
(ala) (41b)
300
M.J. Ablowitz
and LY~,~,Pi,n are as in (23a)-(23b). a one-soliton
solution
of IDNLS
et al. /Physics
Letters A 253 (1999) 287-304
In both limits, this is a solution
(as in (9)).
of the form qtn = ~t(il,~ where $,
Note that the polarization,
c:,
is
is such that lc{‘)* )* = 1 (and,
therefore, jci*)*1* = 0 since llc~j/* = 1). The long-time limits for soliton 2 are sinh( azh)
_ 42,”
e’p2.*s
=
+ 42,,, -
sech(a2,n + log IBi I),
t42d
11
sinh( azh) h
e’p2.9s sech( (YZ,~),
t42b)
where 4;
= hb:! + arg B2,
4;
= hb2,
and CY~,~,/?z,~ are as in (23a)-(23b) with ~22and b2 replacing ai and bl respectively. In both the forward and backward long-time limit, this is a solution of the form CAL& where lci2)*12 = 1 (and, therefore, ]@)*I2 = 0 since llc~]/* = 1). The solution (4 1a) -( 4 1b) , (42a) -( 42b) is not the most general two-soliton
Ic;.cJ=Ic;.c2fl=o.
interaction
because (43)
The two-soliton solution (41a)-( 4 1b), (42a)-(42b) can be multiplied by a unitary matrix to obtain a twosoliton solution with any cl, CT such that [CT SC; I = 0 (under any such transformation, the condition ICI’*cl I = 0 will still hold). Thus, these two-soliton solutions are a constrained class of two-soliton solutions where (43) is the constraint. More generally, Hirota’s method can be used to derive solutions of the symmetric system with more than two solitons. These solutions can be represented as combinations of Pfaffians (these formulae are quite technical; they are presented in a separate publication [ 171). The M-soliton solutions derived in this manner are constrained such that either
Ic,; ‘CJ = 1,
(44a)
lc.7 - CJ = 0,
(44b)
or
for all .j, k = 1,. . . , M. Moreover, the above satisfy
Ic, *CT1= 1,
(45)
M. These multi-soliton solutions are reminiscent of the special case of soliton interactions in forallj= l,..., the PDE discussed at the end of Section 3: conditions (44a)-(44b) and Eq. (45) for the discrete symmetric system are the counterparts of (15a)-( 15b) and Eq. (16) in the PDE. To date, there is no known analytic formula for a general multi-soliton solution of the symmetric system such that 0 < 1c.r SC
301
M.J. Ablmitz et al. /Physics Letters A 253 (1999) 287-304
t
-
t
-2.46
= 6.54
9.82.
9.82
7
4.91
_____*_fi__
a:
4.g1s.A__o,-
-8.
Fig. I. Two-soliton interaction for the symmetric system. The filled boxes are ]qh”1 and the open boxes are \q)1*‘1, Increasing time is read down column-wise. Soliton 1 is on the left and soliton 2 is on the right at I = -8.46. They are reversed at t = 6.54. The c; = (0.60.0.80). at soliton parameters are: al = I, a2 = 3. bl = 0.1, b2 = -0.1. The polarization vectors are: CT = (1.0). at t = 6.54. This is a typical two-soliton interaction where t = -8.46, and CT = (0.33ei.2”“,0.95ei.12W), c: = (O.Me- i0”2R , 0 54~‘“.~‘“) 0<~c;.c;~=0.6<1,~c;~c~~=0.204<1and~c;~c~~=0.993<1.
discrete symmetric system for these general multi-soliton interactions, we investigated waves by numerical simulation. In the simulations, the initial conditions were taken to be of the form
the collision
of solitary
(46) where q& = cj& and L&&is of the form (9) and the peaks are well-separated. Then, the symmetric system was integrated in time by an adaptive Runge-Kutta-Merson routine (from the NAG library) until the peaks were again well-separated (see Fig. 1 for an example). The separation of peaks in the initial and final conditions makes these conditions comparable to, respectively, the backward and forward long-time limits. The solitarywave interactions simulated in this manner comprised initial conditions in which 0 < Ic,; - CL 1 < 1. Visually, in Fig. 1 (and in other simulations) the solitary waves appear to interact without any radiation. We confirmed this finding by quantitative comparison of the solitary waves at the final time and exact solitary waves with the same height and speed parameters as the initial data. The error at the final time was defined separately for each solitary wave by (47)
M.J. Abtowitz et al. /Physics Letters A 253 (1999) 287-304
302
Table 1 Two-soliton interaction for the symmetric system (5). The soliton amplitude parameters are (11 = I and (12 = I or 02 = 3 as is given in both parts of the table. For these values of Cl), the soliton width is comparable to the grid size and the solution is not “close” to the continuum limit. The soliton speed parameters, 61 = 0.1 and b2 = -0.1, are such that the solitons move slowly relative to one another thereby increasing the strength of the interaction. The polarizations before interaction are c; = ( 1,O) and c; as is given in the table. The polarizations after interaction are c: and c:. Aj where j = 1.2, is the difference, as given by Eq. (47), between the numerical solution at the final time and an exact solitary wave. The values Ic,: . cj+l < I indicate that the polarization vectors are shifted by the soliton interaction, The small errors, Aj, show that the solitary waves interact nearly elastically with the measured deviation accounted for by error in the numerical time integration.
=;
ICY. c:i
Ic; *c:I
W, AI
1% ,o 42
=
I
(0.0,l.O) (0.2, .98) (0.4, .92) (0.6,0.8) (0.8.0.6) (1.0,O.O)
1.000 0.575 0.492 0.624 0.806
1.000 0.575 0.492 0.624 0.806
-7.57 -7.54 -1.52 -7.58 -7.56
I .ooo
I.000
-7.51
-7.57 -7.54 -7.52 -758 -7.56 -7.57
‘12 =
3
(0.0, 1.0) (0.2, .98) (0.4, .92) (0.6.0.8) (0.8,0.6) (1.0,O.O)
I .ooo 0.900 0.617 0.204 0.396 1.000
I .ooo 0.999 0.996 0.993 0.994 1.000
-8.92 -1.75 -7.41 -1.27 -7.37 -7.61
-6.21 -6.12 -6.06 -6.07 -6.12 -6.25
Q
where: - d,, is the numerical data at the final time; - q,l,, is a solitary wave with the same amplitude and speed parameters - u,i and b,i - as the initial data, q,Ln; - A,j = sinh( u,ih) /h is the amplitude of the exact solitary wave with amplitude parameter aj; - L?j is the set of points containing the main contribution of the jth solitary wave - i.e. cj = {n : [jq:[i > c} l small compared to maxj,r,z Al. The resulting errors were small (see Table 1 for an example). Furthermore, when the user-supplied error bound in the adaptive integration scheme was decreased, the errors, A,, decreased proportionally. Therefore, the differences between the final wave forms obtained by numerical simulation and exact solitary waves are accounted for by errors in the time integration. These results strongly suggest that the solitary waves interact elastically - i.e. the solitary waves are solitons. In the numerical simulations described in Table 1, and in other experiments, we considered initial data such that 0 < 1~; - c; 1 < 1, the general case of the soliton interaction for which there is no known analytical solution. In the PDE (2)) such conditions result in the shift of the polarizations of the individual vector solitons - i.e. Icy - C,i +I < 1, j = 1,2. The simulations described in Table 1, consistent with other numerical experiments, show the same distinctive vector soliton behavior for the discrete symmetric system. The mechanism of the more general elastic soliton interactions observed in the symmetric system remains to be explained analytically. In contrast with the above, simulations of the more general discrete system (7a)-(7b) with B + 1 yield results which are qualitatively the same as those of its continuum limit (3a)-( 3b) - i.e. the solitary waves collide inelastically and the nature of the interaction depends on the initial solitary-wave parameters. For example, when B = 0.5 in (7a)-(7b), we consider initial conditions of the type (46), where the amplitude parameters are al = 1, u2 = 2 and the polarizations are cl = ( 1 0 ), c; = ( 0 1 ). If the initial speed parameters are b, = 0.1, bZ = -0.1, then the solitary waves reflect one another (see Fig. 2). However, if bl = 1.O and bz = - 1.O, then the solitary waves instead pass through one another and there is a visible polarization shift in both waves. We note that these phenomena persisted at several levels of discretization. These results suggest
for
M.J. Ablowia
er al. /Physics
3.58.
3.58.
1.79.
1.79.
-10.
0. t = -22.1
20.
303
Letters A 253 (1999) 287-304
-20.
0. t = 17.
20.
Fig. 2. Two-soliton interaction for the system (7a)-(7b) with B = 0.5. The filled boxes are lq,$“j and the open boxes are jqi’)/. Increasing time is read down column-wise. Soliton I is on the left and soliton 2 is on the right at r = -48.2. The initial soliton parameters are: (I, = I. n:! = 2, hi = 0.1, b2 = -0.1. The initial polarization vectors are: ct = (I, 0). c2 = (0, 1). In this example, the solitary waves reflect off one another. This interaction is inelastic in the sense that the speeds of the solitons are changed by the interaction.
that the discretization (7a)-(7b) is a valuable numerical scheme to study the PDE system (3a)-( 3b). Finally, we remark that the complex solitary-wave interactions observed with the system (7a)-(7b) demonstrate that elastic soliton interactions depend critically on the coefficients of the equation - the restriction B = 1. This is the same criterion for integrability in the case of the PDE (3a)-( 3b). The fact that inelastic soliton interactions appear to be generic in the case B # 1 strongly suggests that the elastic solitary waves of the symmetric system (5) are indeed true solitons. This provides strong evidence that the symmetric system is an integrable discretization of vector NLS.
Acknowledgement This effort was sponsored in part by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number F49620-97-l-0017, by the Office of Naval Research, USN, under grant number NOOO14-94-1-0915 and the National Science Foundation under grant number DMS-9703850. One of the authors (YO) is grateful to Dr. S. Tsujimoto for valuable discussions and also acknowledges the financial support of the Japan Ministry of Education through the Foreign Study Program.
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Letters A 253 (1999) 287-304
References I Ii M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, Vol. 4 (SIAM, Philadelphia, PA, 198 1) 121 V.E. Zakharov, A.B. Shabat, Sov. Phys. JETP 34 (1972) 62. [ 3 1 S.V. Manakov, Sov. Phys. JETP 38 (1974) 248. [ 4 I C.R. Menyuk, IEEE J. Quantum Electron. 23 (1987) 174. IS] SC. Evangelides, L.F. Mollenauer, J.P. Gordon, N.S. Bergano, J. Lightwave Technol. 10 ( 1992) 28. 16 I CR. Menyuk, IEEE J. Quantum Electron. 25 (1989) 2674. 17 1 V.E. Zakharov, E.I. Schulman, Physica D 4 ( 1982) 270. 181 M.J. Ablowitz, J.F. Ladik, Stud. Appl. Math. 55 (1976) 213. 191 Ch. Claude, Yu.S. Kishvar, 0. Kluth, K.H. Spatscheck, Phys. Rev. B 47 ( 1993) 14228. IlO] J.C. Eilbeck, P.S. Lombdahl, A.C. Scott, Physica D 16 (1985) 318. [ 1I I A.R. Its, A.G. Izergin, V.E. Korepin, N.A. Slavnov, Phys. Rev. Lett. 70 (1993) 1704. [ 121 A.A. Vakhnenko, Yu.B. Gaididei, Theor. Math. Phys. 68 (1987) 873. 113) M.J. Ablowitz, P.A. Clarkson, Solitons Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, Vol. 149 (Cambridge Univ. Press, Cambridge, 1991). 1141 Y.B. Suris, Phys. Lett. A 189 (1994) 281. 115I R. Radakrishnan, M. Lakshmanan, J. Hietarinta, Phys. Rev. E 56 ( 1997) 2213. I 16 I C. Sophocleus. D.F. Parker, Opt. Commun. 112 ( 1994) 214. 1I7 I Y. Ohta, Pfatlian solutions for coupled discrete nonlinear Schrodinger equation, in: Proc. of Brussels Meeting II: Integrability and Chaos in Discrete Systems, Brussels, 2-6 July 1997, Chaos, Solitons and Fractals, to appear.
PHYSICS LETTERS A
ELSEVIER
Physics Letters A 253 ( 1999) 287-304
On discretizations of the vector nonlinear SchrGdinger equation Mark J. Ablowitz”, Yasuhiro Ohtab, A. David Trubatch a a Department of Applied Mathematics, Universify ofColorado-Boulder, Boulder. CO 80309, USA h Department ofApplied Mathematics, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, H&ash&Hiroshima 739-8527. Japan Received 27 April 1998; revised manuscript
received 3 November 1998; accepted for publication Communicated
14 January
1999
by A.P. Fordy
Abstract Two discretizations of the vector nonlinear Schrodinger (NLS) equation are studied. One of these discretizations, referred to as the symmetric system, is a natural vector extension of the scalar integrable discrete NLS equation. The other discretization, referred to as the asymmetric system, has an associated linear scattering pair. General formulae for soliton solutions of the asymmetric system are presented. Formulae for a constrained class of solutions of the symmetric system may be obtained. Numerical studies support the hypothesis that the symmetric system has general soliton solutions. @ 1999 Elsevier Science B.V. PAC.‘+ 03.40.Kf;
46IO.+z;
42.81.D~;
42.81.G
1. Introduction In recent years there has been wide interest in the study of solitons and integrable systems. Researchers have found that not only are continuous systems (i.e. PDEs) integrable via the inverse scattering transform (IST) but also that interesting classes of discrete systems (semi-discrete as well as partial difference equation) are integrable (cf. Ref. [l] for an early review). In this Letter, we discuss soliton solutions and the integrable nature of certain discrete systems associated with the vector extensions of the nonlinear Schrodinger (NLS) equation: iq, = qXx + 2&12.
(1)
NLS is a centrally important and physically significant nonlinear equation which possesses solitons and is integrable via IST [ 21. Furthermore, NLS arises in many areas of physics, such as the evolution of small amplitude slowly varying wave packets in: deep water, nonlinear optics and plasma physics (see e.g. Ref. [ I] ) . In 1974, Manakov [ 31 showed that the vector NLS (VNLS), iq, = 4xX + 211ql12q~ where q is an N-component integrated via IST (actually,
(2) vector and )I . 1) denotes the vector norm, also possessed solitons and could be in Ref. [3] only the case N = 2 was studied in detail; however the extension to
0375-9601/99/$ - see front matter @ 1999 Elsevier Science B.V. All rights reserved. PII SO375-9601(99)00048-1
288
M.J. Ablowitz et al. /Physics
Letters A 253 (1999) 287-304
the Nth order vector system is straightforward). The second order (N = 2) VNLS equation is relevant in the study of electromagnetic waves in optical media in which the electric field has two nontrivial components. In optical fibers, the components of q in Eq. (2) correspond to components of the electric field transverse to the direction of wave propagation. These components of the transverse field compose a basis of the polarization states. In optical fibers, the equations governing the field are more generally non-integrable variations of vector NLS [ 451, although there are circumstances in which Eq. (2) is the appropriate model [ 61. A related model with experimental significance is . (‘) = qg ‘41
+ 2(lq”‘[2
‘4t(*) = qL2,’+ 2(Blq(‘)j2
+ Blq(*y)q(‘),
(3a)
+ )q’*y)q’*‘,
(3b)
which is equivalent to (2) when B = 1 [ 61. In contrast to the Manakov equation (2), the system (3a)-( 3b) is not integrable when B # 1 (cf. Ref. [7] ) and cannot be solved by IST. This system further illustrates that integrability depends specifically on the form of the equation - i.e. the integrability of (3a)-(3b) depends on the precise value of B. In 1976, Ablowitz and Ladik [ 81 found that the following discrete system is integrable via IST: i$q,
= i(qn-t
-2q,
+qrl+t)
+ lqn12(qn+i ++I).
(4)
The continuum (h + 0) limit of this discrete system (4) is NLS ( 1) hence this system can be referred to as integrable discrete NLS (IDNLS). The discrete system itself is also useful in physical applications (see e.g. Refs. [9-121). IDNLS provides an excellent numerical scheme to solve NLS ( 1) (cf. Ref. [ 131) . Computations demonstrate that, by simply replacing the nonlinear term in (4) with 21q,(*q,, one obtains a poor numerical scheme. In contrast, the preservation of the integrable structure in IDNLS (4) plays a key role in its utility as a numerical scheme. This points out that care must be taken in the choice of discretization of the nonlinear terms for the purpose of numerical simulations. It is natural to look for useful discretizations of VNLS (2). However, the special character of IDNLS as compared to other discretizations of NLS, suggests that different discretizations of VNLS may have very different dynamics, Therefore, the choice of discretization merits close analysis. In this Letter, we discuss two discretizations of VNLS which we refer to as the symmetric and asymmetric discretizations. These discretizations are as follows: Symmetric discretization: i$q,
= ;(4,-t
- 2q, + qn+t) +
where q, is an N-component Asymmetric discretization: d iz4” -i$,,
1 = -((4+i h* = $(r,_l
where, as before, qn Both systems (5), take r = -q*, where (6a)-(6b), however,
l14nl12(4n-, + %+I))
(5)
vector.
- 2% + 4n+l) - ezndz, - 2r, + r,+l)
- (Cq,)r,-l
- a?,)%,,~ - (C+,q,)rn9
(6a) (6b)
is an N-component vector as is r,. The superscript T denotes the transpose. (6a)-(6b) reduce to VNLS (2) in the continuum limit (h -+ 0) where in (6a)-(6b) we * denotes the complex conjugate, after taking the continuum limit. The asymmetric system does not admit the symmetry r, = -4; and hence remains “asymmetric” for h finite.
M.J. Ablowitz et al. /Physics
Letters A 253 (1999) 287-304
289
These systems have a number of interesting properties. Both systems (5), (6a) -( 6b) reduce to the integrable discrete scalar case (4), just as VNLS (2) reduces to NLS ( 1) . The asymmetric system (6a) -( 6b) is associated with a linear operator pair (see Eqs. (17a)-( 17b)). Although there is no known linear operator pair for the symmetric system (5), under the reduction qn = eiO’u,, where U, is independent of t, this equation reduces to an N-dimensional difference equation which is known to be integrable (cf. Ref. [ 141). The results in Ref. [ 141 suggest that the linear operator for the symmetric system (5) may be closely related to the matrix generalizations discussed below (cf. ( 17a) ) . Both the symmetric (5) and asymmetric (6a)-(6b) systems possess a class of soliton solutions. The asymmetric system has a class of soliton solutions which correspond to all the soliton solutions of the continuum limit NLS (2). However, the exact multi-soliton solutions found so far for the symmetric system (5) only reduce to a subset of those associated with Eq. (2). Since the known exact soliton solutions of Eq. (5) are only a subset of those known in the continuum limit (2)) we numerically examined the interactions of solitary waves associated with the symmetric system which lie outside this class. The numerical evidence indicates that, in the symmetric system, the solitary waves interact elastically and are therefore true solitons. However, as we discuss below, when (5) is modified, solitary-wave interactions are not elastic. These findings strongly suggest that the symmetric system (5) is indeed integrable. The generalization of (5) where the nonlinear term in the jth component, (C,“=, lq,$k) I*) (St>, + q$ ), is replaced by (cf=, of the form
Bj,klqLk)(*) (q;!,
+ ql;‘:{) is an interesting
discrete system. For example,
+q::),)+(lq~1’12+Blq~2)12)(q~l),
+q;:‘,),
+ q:2+)1)+ (Blq:“l’+
+q;$.
lq:*)j*)(q~,
the N = 2 system
(7b)
where B is a parameter, is a discretization of (3a)-( 3b). We concretely contrast the two-component symmetric system (i.e. (5) with N = 2) with (7a)-( 7b). In numerical simulations of the system (7a)-(7b) with B # 1, described below, solitary waves do not, in general, interact elastically. Rather, the solitary-wave interactions are qualitatively analogous to those of the continuum limit - i.e. the PDEs (3a)-( 3b). These results suggest that: (i) the discretization (7a)-( 7b) retains the qualitative properties of the PDE and (ii) the elastic soliton interactions observed in the symmetric system (5) depend on the precise structure of that system (e.g. B,i,k= 1) . We conclude that the systems (5)) (7a) -( 7b) - and generalizations - are extremely useful as numerical schemes for the study of their continuous analogs (2)) (3a)-( 3b) - and generalizations.
2. NLS and integrable
discrete NLS
We briefly review the relationship between NLS and IDNLS in order to illustrate the way that solitons are solutions preserved by an integrable discretization. NLS ( 1) can be obtained as compatibility condition of a linear operator pair (e.g. a Lax pair). The linear pair can be used to solve ( 1) by IST. The one-soliton solution of NLS is q(x, t) = ae -i(bx-(b’-J)+d)
sech(ax _ abr _ 0) = ei+gx,
t),
(8)
where a, b, 0 and C$ are arbitrary parameters. Either IST or Hirota’s method can be used to obtain the multisoliton solutions. The existence of an associated linear operator pair and IST are hallmarks of integrability. Therefore, in the effort to construct an integrable discretization of NLS, it is natural to look for a discrete version of the operator
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Letters A 253 (1999)
287-304
pair and the compatibility condition. In fact, the integrable discretization of NLS (4) was obtained in exactly this manner [S]. The extent to which the integrable structure of NLS is preserved under the integrable discretization is significant: The IST for IDNLS is analogous to the theory for NLS. In particular, the one-soliton solution of IDNLS (4) given by qn(t)
= Ae-i(bkn-w+)
sech( ahn - ut - 0) = e’“&(t)
,
(9)
with A = sinh(ah) h
’
6J=
2( 1 - cosh(ah) h2
cos(bh))
u = 2sinh(ah)
sin(bh) h2
’
converges (as h -+ 0) to the one-soliton solution of NLS (8) with the same four free parameters. More generally, solutions of IDNLS converge to the solutions of NLS with an error of O( h2). For both NLS and IDNLS, generic rapidly-decaying initial data resolves itself into some number solitons plus radiation that vanishes as t -+ 00 (in the sup norm). Even though the solitons interact nonlinearly upon collision, it is well-known that they retain their shape and speed after the collision. This can be determined by comparing the forward (t + +oo) and backward (t --+ -oa) long-time limits of the solutions. In these long-time limits, solitons traveling at different speeds are well-separated. Although this collision is elastic, the position and overall complex phase of the individual solitons are shifted between the forward and backward long-time limits. This elastic interaction of solitons up to a shift in phase (position and overall complex phase) is typical of ( 1 + I)-dimensional integrable systems.
3. Vector NLS Now we consider Vector NLS (2) and illustrate those properties which we seek to preserve under discretization. In particular we give the Hirota formula for the general M-soliton solution and describe soliton interaction in the vector system. VNLS can be obtained by substituting the vector q for q in NLS (1) and replacing complex conjugation with the Hermitian conjugate. Similarly, the linear operator pair for VNLS can be obtained by appropriately making the same substitution of scalars by vectors in the linear operator pair for NLS (cf. Ref. [ 31) . When considered as a model for the propagation of electromagnetic waves in optical fibers, the components of VNLS play the role of a basis for the polarization vector. In the derivation of VNLS by Manakov [3] the choice of basis is arbitrary. Therefore, the vector system (2) ought to be, and is, invariant under a change of basis for the polarization. Mathematically, a change of basis is obtained by multiplying the independent variable q by a unitary matrix, U. Therefore, the freedom in the choice of basis is reflected in the fact that the vector system (2) is invariant under the transformation
q+uq. This symmetry is an important feature in distinguishing among discrete versions of vector NLS (notably, the versions of discrete vector NLS discussed in this Letter (5)) (6a)-( 6b) retain this symmetry). Under the reduction q = cli,
(10) both
(11)
where c is a constant, N-component vector such that ]lc112= 1 and 6 is a scalar function of x and t, VNLS (2) reduces to NLS ( 1). This is a manifestation of the fact that the vector system is a generalization of NLS obtained by allowing the polarization to be non-constant. When the solution of VNLS has a constant
M.J. Ablowirzet al./Physics LerfersA 253 (1999) 287-304
291
polarization, NLS is recovered. As a consequence, any solution of scalar NLS has a corresponding family of solutions of VNLS. We call a solution of the form ( 11) a reduction solution. Because the vector c is arbitrary (up to the constraint that it is of unit length), a reduction solution has correspondingly more parameters than the solution of scalar NLS to which it reduces. We refer to the vector c in Eq. ( 11) as the polarization of the reduction solution. In particular we show below that the one-soliton solution of vector NLS is the reduction solution (i.e. of the form ( 11) where 9 is the one-soliton solution of NLS). 3.1. Soliton solutions The soliton solutions of vector NLS can be found by IST [3] or by Hirota’s method. The Hirota formula for the general M-soliton solution suggests the formulae for the soliton solutions associated with the discrete systems (5) and (6a)-(6b). The derivation of the general multi-soliton solution by Hirota’s method has not been previously reported in the literature. (The Hirota formula for the two-soliton solution is given in Ref. [ 151 and the derivation by IST is described in Ref. [3].) Here, we give the general Hirota formula, for both (i) completeness and (ii) as a motivation for the soliton solutions of the discrete vector systems. In order to obtain the Hirota formula, we rewrite VNLS (2) in bilinear form. With the independent variable transformation
where g is a vector and iD,f
.g=
Dif
f is a scalar, solutions of the bilinear equations
-g,
are solutions of VNLS. In particular, the M-soliton
D:f
- f = llgl12~
solution of VNLS is given by the determinant
formulae:
(13) where Z is the M x M identity matrix and (i) A, and B are M x M matrices defined by A”,.” II
=
Pf + P.i’ ’
with f!, j = 1, . . . , M and v,i = pjx + ipjt; the complex numbers pj = aj - ibj, aj > 0, determine the amplitude and speed of the jth soliton; the N-component complex vectors rj (j = 1,. , . , M) determine the polarizations and envelope phases of the solitons; (ii) @ is the column vector Qce) = eqr, C = 1, . . . , M; (iii) Y, is the row vector Pk(i) = -rjk’, j = 1,. . . , M. The one-soliton solution is obtained with M = 1 in the determinants ( 13). This yields 4= &
aiei’m71 sech(Re7,
+ S),
where Revi
= six - 2qb,t,
Irnvi
= -blx
+ (a: - bt)t,
S=log!$ I
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M.J. Ablowitz et al. /Physics Letters A 253 (1999) 287-304
and, without loss of generality of the form ( 11) - where
we assume at > 0. Hence, the one-soliton
solution
is a reduction
solution - i.e.
and the scalar function 4 is the one-soliton solution of scalar NLS - i.e. (8) - with a = al, b = bl and 8 = -6 (without loss of generality, the parameter I$ in (8) is absorbed into the polarization vector). Expressions for solutions with two or more solitons are obtained simply by the expansion of the determinants. However, the soliton interaction can be analyzed more easily by computing the long-time limits of ( 13) before expanding the determinants. 3.2. Interaction of the solitons Now we describe the interaction of solitons in VNLS. As in the scalar case, the IST [ 31 (or analysis of the formulae obtained by Hirota’s method) show that solitons interact elastically. By elastic, it is meant that the solitons retain their shape and speed after interaction as shown by comparing the forward and backward long-time limits. For the vector system, care is needed: the nature of the phase shift for vector solitons is somewhat different than the scalar case. In addition to a shift in the center of the peak (as in the scalar case), vector solitons also undergo a change in polarization upon collision. That is, in the forward and backward long-time limit, the solution asymptotically approaches a linear superposition of the individual solitons:
qwCq,f, where qj* = cfijf speed. domparisbn
as t+fco,
and, for each j, df and ~$7 are one-soliton solutions of NLS with the same amplitude of the forward (‘i) and backward (-) long-time limits shows that
and
c,: # ci’, but (14) A closed formula for this shift in the polarization can be calculated by considering the eigenfunction of the associated scattering problem [ 31. We refer to the squared modulus of a component of the polarization vector as the intensity of that component of the polarization. That is, if c,:‘) is the &h component of Cj, the polarization vector of the soliton j, then the intensity of the of the Cth component of soliton j is lc~“‘/*. The relation (14) implies that the total intensity of each soliton is preserved, specifically the sum of the intensities is equal to one. However, the distribution of intensity between the components of the polarization of an individual soliton will not, in general be equal in the forward and backward long-time limits due to interaction with other solitons: subject to the constraint of Eq. (14), in general,
I@- 12# ,q)+12. This change in the distribution of intensity is a distinctive feature of the vector system. There is no corresponding phenomenon in the soliton interactions of the scalar equation (in both the scalar and vector equations, the location of the peak of an individual soliton is shifted by the soliton interaction).
M.J. Ablowitz et al. /Physics
Manakov [3] also observed the following every pair of solitons j and k, either
Letters A 253 (X999) 287-304
special case for the interaction
of the solitons
293
of VNLS: If, for
(15a)
Ic,; . CJ = 1,
(15b)
where - denotes the dot product, then
Ic,, - CT1= 1,
(16)
for all j. As a consequence of ( 16), ]cjf’- /* = Ic,iE)+]* for all j, 1. Physically, this means that, in any polarization basis, the distribution of intensity among the components of polarization for an individual soliton is not changed by the soliton interaction. 3.3. Soliton interaction
in non-integrable
vector NLS
The nature of solitary-wave interaction in the generalized system (3a)-(3b) contrasts strongly with the soliton interaction in the integrable case. First, we note that when B # 1 the symmetry transformation (10) no longer holds and therefore general reduction solutions of the form ( 11) do not exist. However, reduction solutions do exist for the particular polarizations
where 0 < $1, (62 < 27r. In particular, there are solitary-waves of the form ( 11) where c is one of the three allowed polarizations and 4 is the one-soliton solution of NLS. We contrast the interaction of these solitary waves with the elastic collisions described in the previous section. When two solitary waves of the same polarization collide, the vector nature of the system does not play a role. In this degenerate case, the system (3a)-( 3b) reduces to the scalar NLS (1). As a consequence, the solitary waves interact elastically. These elastic solitary-wave interactions occur only because the coupling between polarizations plays no role. The interaction of solitary waves of different polarizations is, however, strongly inelastic. These phenomena can be investigated by numerical simulation (cf. Ref. [ 161). When solitary waves of different polarizations collide, a number of results are possible. In general, the speed and height of the solitary waves are not preserved in the collision. For some parameter values, the solitary waves reflect one another while, for different parameter values, the waves pass through each other but emerge with different speeds and amplitudes. Additionally, in some simulations, the soliton interaction can lead to large and rapidly oscillating radiative tails.
4. Asymmetric
discrete vector NLS
The preceding derivations of IDNLS and VNLS from NLS suggest two methods to obtain an operator pair which has a discrete form of VNLS as its compatibility condition: (i) discretize the operator pair for vector NLS in a manner analogous to that used to obtain the operator pair of IDNLS or (ii) appropriately substitute vectors into the operator pair for IDNLS. These approaches yield the linear operator pair
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M.J. Ablowitz et al. /Physics
T
iq,rL_, - i(1/2h2)(z - I/z)~ZN i(l/h)((l/z)rT - zri-i)
=
”
where IN is the N x N identity this pair. Under the reduction 4” =
Letters A 253 (1999) 287-304
-i(llh)((llz)q, - zq,-,) +i( 1/2h2)(z - ]/z)~
-irzq,_,
matrix. The asymmetric
system (6a)-(6b)
(17b)
’
is the compatibility
condition
r, = dg,‘,
CQ”,
of
(18)
where c and d are constant vectors such that c - d = 1 and 4” is a scalar function of n and t, the asymmetric system (6a)-(6b) reduces to IDNLS (4). Solutions of this form ( 18) are referred to as reduction solutions. Hence, every solution of IDNLS generates a family of reduction solutions of the asymmetric system. We are particularly interested in reduction solutions where d = -c* because, in this case, r, = -q,* and (6a)-(6b) reduces to a single equation. 4.1. General formula for multi-soliton solutions Multi-soliton solutions of the asymmetric system can be derived, without the use of the IST machinery, Hirota’s method. First, make the independent-variable transformations
by
r,=-, & fn
q”=F, R
where g, and 2, are vectors with N components
and fn is a scalar. Then, solutions
of the system of bilinear
equations (19a)
fnil fit-1 - f,’ = 2&T ih2D,f,
- 2f,g,
- g, = f+&+l
-ih2D,f,
- & = f+$,+,
+ f,+lg,-,
- 2f&,
- H&
+ fn-$,,+l
(19b)
- fi&,,
(19c)
fnH,+l
= h2k,g;+,
- g,+,g;L
(19d)
f,,&-,
= h2@,_&
-#,&L
( 19e)
where the N x N matrices H, and Z?,, are auxiliary variables, are solutions of the asymmetric system (6a)-( 6b). The M-soliton solutions for these bilinear equations (19a)-( 19e) are given by the following determinants:
gLk’=
“; z 0
/ H”..” ”
=
A,
Z
@n-l
-’
B
0
pt
0
0
Y,j
0
B Y,
@n1
’
0
where Z is the A4 x M identity matrix, and
@n 0 0
,
gck) = n A,, -I
1” &+,
Z B 0 0
A, -Z
Z B
&
0 0
0
!@‘k, 0
?PF ‘Pi 0 0
(20a)
0
0 0
(2Ob)
M.J. Ablowitz
et al. /Physics
295
Letters A 253 (I 999) 287-304
(i) A,, and B are M x M matrices defined by A”..”
h
=
n
e wr+P,*) _ *
@Lj) _ - eh(p;+:,) _ , rh
ew”+v;,,(
with vj,n
pjn}l
=
+
$2 - e”“’ -
e-““’
)t;
(21)
the complex numbers pj = a,j - ibj, aj > 0, determine the amplitude and speed of the jth soliton; the Ncomponent complex vectors yj and yj (j = 1,. . . , M) determine the polarizations and envelope phases of the solitons; (ii) cP”, 6, are, respectively, the M-component column and row vectors, -(j) _ Qn _ e 7.1.11 ;
@W = e-““L eS’JJ II (iii)
P,,
@k are, respectively,
the M-component
row and column
vectors,
q.(P) = e-rlpf7;k].
p;.j) = _y,;k],
k
4.2. The one-soliton solution From the above formulae
Yl qn =
(gy,)l/2
r, = -
e
(20a)-(
(M = 1) solution
is
-ihbl sinh( a] h) e’.p‘,‘Isech( a],n + 6, ) ,
(224
h
-i/tb, sinh(ah) (+&1/2
20b), the one-soliton
e_ipI,,,
h
e
(22b)
sect-da,,, + 6, ),
where
a],, = Rev],,
= qhn - 2
PI,,,
= -hhn
= Imvl,,
+
sinh( a] h) sin( b] h) t
h2
(234
’
2( 1 - cosh(a,h) cos(b,h)) h2
(23b)
and
s, = log
+{y;y]}
- log C+-
Note that to get a solution (22a)-(22b) with 6, real we restrict our attention to the case where y:y, is real and positive. The one-soliton solution (22a)-(22b) is a reduction solution - i.e. of the form ( 18) - where
and the scalar function 4” is given by Eq. (9) - the one-soliton 0 = 4,. In this solution, d = -c* if, and only if, 7, = y;.
solution of IDNLS - where a =
UI ,
b = 6, and
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Letters A 253 (1999)
287-304
4.3. Two-soliton interaction To show that, for A4 > 1, the determinants the long-time limits. In these limits, solitons M = 2 and
(20a)-(20b) indeed give a multi-soliton solution, we consider moving at different speeds are separated. For concreteness, let
sinh( at h) sin( bt h) > sinh( azh) sin( b2h) ,
(24)
for the given at, bl, ~2, b2. The condition (24) ensures that each soliton travels with a different speed (the analysis below holds, in general for M solitons as long as each has a different speed). When the solitons travel at different speeds, we determine the asymptotic form of an individual soliton by taking long-time limits in a coordinate frame moving with that soliton. The limits n N 2sinh(u,ih)
are long-time
sin(b,ih) t , a,ih3
t-+Itcm,
limits in a coordinate
Rerlt.n = const The substitution
and
of (26)
* 4, -+ 4l.n
and
(25)
frame moving with soliton j. In the limit (25) with j = 1,
Revz,,, + &o,
as t -+ fca
into Eqs. (20a)-(20b),
(26)
where M = 2, yields
as t -+ *co,
r, -+ r:,,
(27)
which is the asymptotic form of soliton 1 in the forward (+) and backward (-) long-time limits. The long-time limits for soliton 2 are similar. The coordinate frame of soliton 2 is obtained by the limit (25) with j = 2 which yields Re v ,,, + loo
and
Re 772,,,= const,
as t -+ *co.
Note the change in relative sign between ~1,” and t in the coordinate in the long-time limit. To obtain +
4, *
and
42,”
r,
-+
* r2,,,
(28) frame of soliton 2 - the slower soliton -
as t -i *oo,
(29)
which is the asymptotic form of soliton 2 in the forward (+) and backward (28) into (20a)-( 20b), where M = 2. Combining (27) and (29) gives 4n N q$
+ 4:”
and
r,, -
if,
+ r.f,,
where
Cl 41,n* A-
=
r<,=4-G .
’
long-time
limits, substitute
as t -+ ice.
The effect of interaction on the solitons is determined by comparing q,Tnand We give the formulae for the asymptotic forms of the solitons below. First, we consider soliton 1. In the backward long-time limit, 41.n
(-)
r&
with qln and
rin
for j = 1,2.
M.J. Ablowitz et al. /Physics
and i,,
is a one-soliton
e=-s,
solution
291
Letters A 253 (1999) 287-304
of IDNLS (the 4 in Eq. (9) ) with a = ai, b = bl and
=-log~{~~;fy,}+log
(31) F+
As in the one-soliton
case, we require that 7;~~
is real and positive. The forward long-time
for soliton
1 is
where 1
CT = - (e-b1 g2 - P*)(c+2(&>rl
- P*@Y,)Y2)*
X
d: = _’ X
e-hPi (g2 - PI
and (i{n is a one-soliton 0 = 4;
((72@Y2H,
solution
= - log X + log
-
m;v,
H2)’
of IDNLS with a = al, b = bl and
(+2 +
log{&),
also
and we require that ri, yj for j = 1,2 are such that X and yly2 are given by h
u’ =
,h(p,+p;)
h _
1
u2 =
’
e/1(/,2+/J;;)
are real and positive. The constants
go, ~2, p
h _
1
’
p = eM/h+P;)
_
1 .
Hence, the expression ((~1~2 - 1~1~) is real and positive. Note that, although cl f CT, d, # d:, and 6, # ST, the parameters ai and 61 are the same in both long-time limits. Thus, soliton 1 has the same form in both long-time limits but undergoes a phase shift due to interaction with soliton 2. The asymptotic limits of soliton 2 are similar to those for soliton 1 except that the relative change in sign in (28) reverses the calculations, that is, q2.n
- *= c2 42.n
and
rTn = d; g-;,
where c; has the form (33a) and d, has the form (33b) with the indices 1 and 2 exchanged. The scalar function C& is a one-soliton solution of IDNLS with a = ~2, b = rzz and 19= -8, where 8, is equal to ST with the indices 1 and 2 exchanged. Similarly, f _ +-+ 42,” - c2 q2.n
and
r 2f,, = d,fB&
where cl has the form (30a)
and dr has the form (30b)
with the indices
one-soliton solution of IDNLS with a = ~2, b = n2 and 0 = -4: and 2 exchanged. In order to construct a solution such that
1 and 2 exchanged
where 6; is equal to 8,
and Si,
is a
with the indices
1
298
M.J. Ablnwitz
for j = I,2 it is necessary
and sufficient
et al. /Physics
Letters A 253 (1999)
287-304
that
7, =rT*
t 3W Jy32
+
~,-/luly~2’)2
(1 _e-2/ra,)y3/j2)*
For this choice, x and $Y,i then
(1 _ e-2/la,)yy)*y;2) ~e-halyy(2
+
ly;2)12
for j = 1,2 are real and positive as we required.
(34b)
However, if (34a) -( 34b) hold
rjtn + -9.1;9
(35)
for j = 1,2. In fact, llr& - (-qIJ)Il and
7, = r;
Yz=yG.
(36)
Then, if (36) holds, x and $‘yi r1.n = -4c,:t
= O(h). An alternative is to set
ri,
- for j = 1,2 - are real and positive and
= -4;;
but
IIG - (-!$;:)II = O(h)= lb;, - (-42,n*)II. In contrast, recall that the solutions of the scalar IDNLS converge to solutions of the scalar NLS with 0(/z’) convergence. This unavoidable asymmetry of the system (6a)-(6b) makes it less desirable as a discrete approximation of VNLS. Hence, we turn our attention to the symmetric system.
5. Symmetric
discrete vector NLS
The symmetric system (5) is the natural vector generalization of IDNLS (4). We now describe some important symmetries of the symmetric system. In analogy with the scalar case (4)) the symmetric system can be thought of as the reduction of the system i&7,=
$(4+t
-i$r,
= &(rR-t
-29”+fq,+t) -2r,
obtained by letting r, = -ql, Under the reduction 4,, =
*
cqn,
-t-r,+t)
-Cqn(qn-t
+qn+t)’
(37a)
+r,+t),
(37b)
- qXr,(r,-t
which is the symmetry
that is broken by the asymmetric
system.
(38)
where llcll = 1, the symmetric system reduces to the scalar IDNLS (4). Furthermore, if qn satisfies the symmetric system, then so does Uq, where ZJ is a unitary matrix (this symmetry is a discrete form of Eq. ( I1 )). Therefore, the symmetric discretization retains the reductions and symmetries of the asymmetric system and has the additional symmetry r, = -q;T. Despite the above-mentioned symmetries, there is, to date, no known associated linear operator pair for the symmetric system. Without such a pair, the system (37a)-( 37b) cannot be solved by IST. The question remains, however, whether the symmetric system is integrable. In the absence of IST, the existence of multisoliton solutions provides strong circumstantial evidence of integrability.
M.J. Ablowitz et al. /Physics
299
Letters A 253 (1999) 287-304
In order to determine whether there are solitary waves which interact elastically - i.e. solitons, we first identify a solitary-wave solution of the symmetric system: a solution of the form (38) where the scalar &, is a one-soliton solution of IDNLS (as in Eq. (9) > is a solitary-wave solution of the symmetric system. Note that such a solution can have any polarization c subject only to the constraint that l/c11*= 1. In order to determine whether these vector solitary waves interact exactly, we use both direct and numerical methods. 5.1. Soliton solutions by Hirota’s method Soliton solutions of the symmetric system can be found by Hirota’s method. For concreteness, the case where q,, has two components (N = 2). Under the independent variable transformation,
we consider
qn&, .fil where g is a vector and $
is a scalar, solutions
of the bilinear equations
ih2Dtfn-g, = fn-lg,,+, - 2.fng, + fn+lg,,_,, fn+~frv-I - f,’ = h211g,/12~ are solutions of the symmetric system. The following solution of the bilinear
equations
yields a two-soliton
f = 1 + e’7’.‘,+“;!,+ ea..+s:,,, + IB, l*e1)l.,,+9;,,+112.,1+?1;
solution:
(JOa)
,,,(
(40b) g
(2)
h (eb2 _ e--kt$)et12.~~( 1 + B2e’?l.,‘+‘l;,,,),
(4Oc)
where li,n and p,i are as in Eq. (21) and (&I B, =
_ ehy2) (&PI + ekPJ )
(&(Pl+Pz) + l)(e/~(/~l+P;) _ 1)’
(&I B2 = -
(&~(Pl+Pz)
_ &Pl ) (e/l/J; + ehp2) +
l)(&(P;+/B)
_
1)
In order to see that the above solution indeed gives a two-soliton solution, consider the long-time limits. We calculate the long-time limits of the Hirota form of the solution (40a)-(40~) by the same approach as was used for the asymmetric case. In the forward (+) and backward (-) long-time limits, the solution (40a)-(40c) asymptotically approaches the form
4, -
q:n +q,fn,
The long-time
limits for soliton
sinh(ah) h sinh( ai h) h
eiP1,,
1 are SeWal,,),
e’PI.risech( (Yr,n+ log (B, I),
where $1 =hbl,
4: = hbl +argB,,
(ala) (41b)
300
M.J. Ablowitz
and LY~,~,Pi,n are as in (23a)-(23b). a one-soliton
solution
of IDNLS
et al. /Physics
Letters A 253 (1999) 287-304
In both limits, this is a solution
(as in (9)).
of the form qtn = ~t(il,~ where $,
Note that the polarization,
c:,
is
is such that lc{‘)* )* = 1 (and,
therefore, jci*)*1* = 0 since llc~j/* = 1). The long-time limits for soliton 2 are sinh( azh)
_ 42,”
e’p2.*s
=
+ 42,,, -
sech(a2,n + log IBi I),
t42d
11
sinh( azh) h
e’p2.9s sech( (YZ,~),
t42b)
where 4;
= hb:! + arg B2,
4;
= hb2,
and CY~,~,/?z,~ are as in (23a)-(23b) with ~22and b2 replacing ai and bl respectively. In both the forward and backward long-time limit, this is a solution of the form CAL& where lci2)*12 = 1 (and, therefore, ]@)*I2 = 0 since llc~]/* = 1). The solution (4 1a) -( 4 1b) , (42a) -( 42b) is not the most general two-soliton
Ic;.cJ=Ic;.c2fl=o.
interaction
because (43)
The two-soliton solution (41a)-( 4 1b), (42a)-(42b) can be multiplied by a unitary matrix to obtain a twosoliton solution with any cl, CT such that [CT SC; I = 0 (under any such transformation, the condition ICI’*cl I = 0 will still hold). Thus, these two-soliton solutions are a constrained class of two-soliton solutions where (43) is the constraint. More generally, Hirota’s method can be used to derive solutions of the symmetric system with more than two solitons. These solutions can be represented as combinations of Pfaffians (these formulae are quite technical; they are presented in a separate publication [ 171). The M-soliton solutions derived in this manner are constrained such that either
Ic,; ‘CJ = 1,
(44a)
lc.7 - CJ = 0,
(44b)
or
for all .j, k = 1,. . . , M. Moreover, the above satisfy
Ic, *CT1= 1,
(45)
M. These multi-soliton solutions are reminiscent of the special case of soliton interactions in forallj= l,..., the PDE discussed at the end of Section 3: conditions (44a)-(44b) and Eq. (45) for the discrete symmetric system are the counterparts of (15a)-( 15b) and Eq. (16) in the PDE. To date, there is no known analytic formula for a general multi-soliton solution of the symmetric system such that 0 < 1c.r SC
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M.J. Ablmitz et al. /Physics Letters A 253 (1999) 287-304
t
-
t
-2.46
= 6.54
9.82.
9.82
7
4.91
_____*_fi__
a:
4.g1s.A__o,-
-8.
Fig. I. Two-soliton interaction for the symmetric system. The filled boxes are ]qh”1 and the open boxes are \q)1*‘1, Increasing time is read down column-wise. Soliton 1 is on the left and soliton 2 is on the right at I = -8.46. They are reversed at t = 6.54. The c; = (0.60.0.80). at soliton parameters are: al = I, a2 = 3. bl = 0.1, b2 = -0.1. The polarization vectors are: CT = (1.0). at t = 6.54. This is a typical two-soliton interaction where t = -8.46, and CT = (0.33ei.2”“,0.95ei.12W), c: = (O.Me- i0”2R , 0 54~‘“.~‘“) 0<~c;.c;~=0.6<1,~c;~c~~=0.204<1and~c;~c~~=0.993<1.
discrete symmetric system for these general multi-soliton interactions, we investigated waves by numerical simulation. In the simulations, the initial conditions were taken to be of the form
the collision
of solitary
(46) where q& = cj& and L&&is of the form (9) and the peaks are well-separated. Then, the symmetric system was integrated in time by an adaptive Runge-Kutta-Merson routine (from the NAG library) until the peaks were again well-separated (see Fig. 1 for an example). The separation of peaks in the initial and final conditions makes these conditions comparable to, respectively, the backward and forward long-time limits. The solitarywave interactions simulated in this manner comprised initial conditions in which 0 < Ic,; - CL 1 < 1. Visually, in Fig. 1 (and in other simulations) the solitary waves appear to interact without any radiation. We confirmed this finding by quantitative comparison of the solitary waves at the final time and exact solitary waves with the same height and speed parameters as the initial data. The error at the final time was defined separately for each solitary wave by (47)
M.J. Abtowitz et al. /Physics Letters A 253 (1999) 287-304
302
Table 1 Two-soliton interaction for the symmetric system (5). The soliton amplitude parameters are (11 = I and (12 = I or 02 = 3 as is given in both parts of the table. For these values of Cl), the soliton width is comparable to the grid size and the solution is not “close” to the continuum limit. The soliton speed parameters, 61 = 0.1 and b2 = -0.1, are such that the solitons move slowly relative to one another thereby increasing the strength of the interaction. The polarizations before interaction are c; = ( 1,O) and c; as is given in the table. The polarizations after interaction are c: and c:. Aj where j = 1.2, is the difference, as given by Eq. (47), between the numerical solution at the final time and an exact solitary wave. The values Ic,: . cj+l < I indicate that the polarization vectors are shifted by the soliton interaction, The small errors, Aj, show that the solitary waves interact nearly elastically with the measured deviation accounted for by error in the numerical time integration.
=;
ICY. c:i
Ic; *c:I
W, AI
1% ,o 42
=
I
(0.0,l.O) (0.2, .98) (0.4, .92) (0.6,0.8) (0.8.0.6) (1.0,O.O)
1.000 0.575 0.492 0.624 0.806
1.000 0.575 0.492 0.624 0.806
-7.57 -7.54 -1.52 -7.58 -7.56
I .ooo
I.000
-7.51
-7.57 -7.54 -7.52 -758 -7.56 -7.57
‘12 =
3
(0.0, 1.0) (0.2, .98) (0.4, .92) (0.6.0.8) (0.8,0.6) (1.0,O.O)
I .ooo 0.900 0.617 0.204 0.396 1.000
I .ooo 0.999 0.996 0.993 0.994 1.000
-8.92 -1.75 -7.41 -1.27 -7.37 -7.61
-6.21 -6.12 -6.06 -6.07 -6.12 -6.25
Q
where: - d,, is the numerical data at the final time; - q,l,, is a solitary wave with the same amplitude and speed parameters - u,i and b,i - as the initial data, q,Ln; - A,j = sinh( u,ih) /h is the amplitude of the exact solitary wave with amplitude parameter aj; - L?j is the set of points containing the main contribution of the jth solitary wave - i.e. cj = {n : [jq:[i > c} l small compared to maxj,r,z Al. The resulting errors were small (see Table 1 for an example). Furthermore, when the user-supplied error bound in the adaptive integration scheme was decreased, the errors, A,, decreased proportionally. Therefore, the differences between the final wave forms obtained by numerical simulation and exact solitary waves are accounted for by errors in the time integration. These results strongly suggest that the solitary waves interact elastically - i.e. the solitary waves are solitons. In the numerical simulations described in Table 1, and in other experiments, we considered initial data such that 0 < 1~; - c; 1 < 1, the general case of the soliton interaction for which there is no known analytical solution. In the PDE (2)) such conditions result in the shift of the polarizations of the individual vector solitons - i.e. Icy - C,i +I < 1, j = 1,2. The simulations described in Table 1, consistent with other numerical experiments, show the same distinctive vector soliton behavior for the discrete symmetric system. The mechanism of the more general elastic soliton interactions observed in the symmetric system remains to be explained analytically. In contrast with the above, simulations of the more general discrete system (7a)-(7b) with B + 1 yield results which are qualitatively the same as those of its continuum limit (3a)-( 3b) - i.e. the solitary waves collide inelastically and the nature of the interaction depends on the initial solitary-wave parameters. For example, when B = 0.5 in (7a)-(7b), we consider initial conditions of the type (46), where the amplitude parameters are al = 1, u2 = 2 and the polarizations are cl = ( 1 0 ), c; = ( 0 1 ). If the initial speed parameters are b, = 0.1, bZ = -0.1, then the solitary waves reflect one another (see Fig. 2). However, if bl = 1.O and bz = - 1.O, then the solitary waves instead pass through one another and there is a visible polarization shift in both waves. We note that these phenomena persisted at several levels of discretization. These results suggest
for
M.J. Ablowia
er al. /Physics
3.58.
3.58.
1.79.
1.79.
-10.
0. t = -22.1
20.
303
Letters A 253 (1999) 287-304
-20.
0. t = 17.
20.
Fig. 2. Two-soliton interaction for the system (7a)-(7b) with B = 0.5. The filled boxes are lq,$“j and the open boxes are jqi’)/. Increasing time is read down column-wise. Soliton I is on the left and soliton 2 is on the right at r = -48.2. The initial soliton parameters are: (I, = I. n:! = 2, hi = 0.1, b2 = -0.1. The initial polarization vectors are: ct = (I, 0). c2 = (0, 1). In this example, the solitary waves reflect off one another. This interaction is inelastic in the sense that the speeds of the solitons are changed by the interaction.
that the discretization (7a)-(7b) is a valuable numerical scheme to study the PDE system (3a)-( 3b). Finally, we remark that the complex solitary-wave interactions observed with the system (7a)-(7b) demonstrate that elastic soliton interactions depend critically on the coefficients of the equation - the restriction B = 1. This is the same criterion for integrability in the case of the PDE (3a)-( 3b). The fact that inelastic soliton interactions appear to be generic in the case B # 1 strongly suggests that the elastic solitary waves of the symmetric system (5) are indeed true solitons. This provides strong evidence that the symmetric system is an integrable discretization of vector NLS.
Acknowledgement This effort was sponsored in part by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under grant number F49620-97-l-0017, by the Office of Naval Research, USN, under grant number NOOO14-94-1-0915 and the National Science Foundation under grant number DMS-9703850. One of the authors (YO) is grateful to Dr. S. Tsujimoto for valuable discussions and also acknowledges the financial support of the Japan Ministry of Education through the Foreign Study Program.
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Letters A 253 (1999) 287-304
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