Table Of ContentDiscrete and Continuous Nonlinear Schr¨odinger systems
M.J.Ablowitz, B.Prinari and A.D.Trubatch
Department of Applied Mathematics, University of Colorado,
Campus Box 526, Boulder C0 80309-0526, USA
October 18, 2001
Contents
1 Introduction 3
2 Scalar Nonlinear Schr¨odinger equation 7
2.1 The inverse scattering transform for NLS . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Direct scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Conserved Quantities and Hamiltonian structure . . . . . . . . . . . . . . . . . . . 23
3 Integrable Discrete Nonlinear Schr¨odinger equation 27
3.1 The Inverse Scattering Transform for IDNLS . . . . . . . . . . . . . . . . . . . . . 28
3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Direct scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.4 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Conserved quantities and Hamiltonian structure . . . . . . . . . . . . . . . . . . . 57
4 Vector Nonlinear Schr¨odinger equation 59
4.1 The inverse scattering transform for VNLS . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 Direct scattering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.2 Inverse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.1 One soliton solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2.2 Transmission coefficients for the pure one soliton potential . . . . . . . . . . 75
4.2.3 Vector Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.4 Vector Soliton Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Conserved quantities and Hamiltonian structure . . . . . . . . . . . . . . . . . . . 79
5 Discrete matrix NLS 82
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 The Inverse Scattering Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.1 Direct Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.2 Inverse Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.3 Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3 Vector Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1
5.3.1 One soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.3.2 Transmission coefficients for the pure 1-soliton potential . . . . . . . . . . . 123
5.3.3 Vector soliton interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Conserved Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A Summation by Parts Formula 134
B Transmission of the Jost function through a localized potential 136
C Scattering theory for the discrete Schr¨odinger equation 138
C.1 Direct Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
C.1.1 Existence and analyticity of the Jost functions . . . . . . . . . . . . . . . . 141
C.1.2 Scattering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
C.1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.1.4 Eigenvalues and norming constants . . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2.1 Recovery of the Jost functions . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2.2 Recovery of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2
Chapter 1
Introduction
This paper is devoted to the study of integrable discrete and continuous nonlinear Schr¨odinger
systems. Thepurposeistoputmostoftheknownresults,andsomenewones,intoacomprehensive
and unified framework within the Inverse Scattering Transform approach(IST).
The IST methodusedto solvebothdiscrete andcontinuousnonlinearequationsis wellknown;
thereadercanfindnumerousmonographsdescribingthemathematicalformulation(cf. forinstance
[1], [2]). In this paper we essentially follow the methodology discussed in the monograph [3]. The
analysis involves
1. Direct Problem:
(a) obtaining analytic eigenfunctions using Green’s function methods
(b) finding relevant scattering data
(c) finding the symmetries of the data
2. Inverse Problem:
(a) formulating a generalized Riemann-Hilbert (RH) boundary value problem which takes
into account the analytic properties of the eigenfunctions
(b) obtaining Gel’fand-Levitan-Marchenko integral equations
(c) obtaining reflectionless potentials from the RH problem
3. Time Dependence:
(a) finding the time evolution of the scattering data
(b) obtaining explicit soliton solutions
(c) for vector NLS systems, obtaining explicit formulae for the polarization shift which is
relevant to vector soliton collisions.
In1965ZabuskyandKruskal[4]discoveredthatthesolitarywavesolutiontotheKdVequation
had the property of interacting “elastically” with another such solution and they called these
solutions“solitons”. Shortlyafterward[5],Gardner,Green,KruskalandMiuraproposedamethod
for solving the KdV equation by making use of the ideas of direct and inverse scattering. In
1972,Zakharovand Shabat [6] showed that the method indeed workedalso for another physically
significant nonlinear evolutionequation, namely the nonlinear Schr¨odinger equation (NLS). Using
3
theseideas,in[7]Ablowitz,Kaup,NewellandSegurdevelopedamethodtofindaratherwideclass
of nonlinear evolution equations solvable by this technique which they named Inverse Scattering
Transform (IST), since, in analogy with the Fourier transform for linear equations, it allows one
to exactly solve the initial value problem for a nonlinear evolution equation. Shortly thereafter,
significant progress was made in the understanding of certain discrete problems. In [8] Flaschka
showed how IST could be used to solve the Toda lattice equation and Manakov [9] used a similar
formulation to solve a nonlinear ladder network. Ablowitz and Ladik [10]-[11] provided a broad
formulation of the IST method for discrete problems and found that integrable semi-discrete and
doubly discrete NLS equations were solvable.
The above equations are all 1+1 (one space - one time) dimensional. We note that in the
1980’s significant progress was made in the study of IST in 2+1 dimensions (a review of some of
this work can be found in [3]). The Davey-Stewartson equation, for instance, provides a natural
2+1 dimensional integrable extension of the NLS equation, but its study is outside the scope of
what we will discuss in this paper.
The scalar nonlinear Schr¨odinger equation (NLS)
iq =q ±2|q|2q (1.1)
t xx
is a physically and mathematically significant nonlinear evolution equation. The NLS equation
arises in a generic situation. It describes the evolution of small amplitude, slowly varying wave
packets in nonlinear media [13]. Indeed, it has been derived in such diverse fields as: deep water
waves[15],[14],plasmaphysics[16],nonlinearopticalfibers[17],magnetostaticspinwaves[18]etc.
Mathematically it attains broadsignificancesince it is integrableby IST, admits solitonsolutions,
has an infinite number of conservedquantities etc.
WealsonotethattheformoftheNLSequation(1.1)withaminussigninfrontofthenonlinear
term is sometimes referred to as the “defocusing” case. The defocusing NLS equation does not
admit soliton solutions which vanish at infinity. However, it does admit soliton solutions which
have a nontrivial background intensity (called dark solitons). We will only discuss the IST for
functions decaying sufficiently rapidly at infinity.
The vector nonlinear Schr¨odinger equation (VNLS) arises, physically, under conditions simi-
lar to those described by NLS when there are multiple wavetrains. Also,VNLS models systems
where the field has more that one component. For example, in optical fibers and waveguides,
the propagating electric field has two components transverse to the direction of propagation. The
system
iq(1) =q(1)+2 |q(1)|2+|q(2)|2 q(1) (1.2)
t xx
(cid:16) (cid:17)
iq(2) =q(2)+2 |q(1)|2+|q(2)|2 q(2)
t xx
(cid:16) (cid:17)
as an asymptotic model for the propagationof the electric field in a waveguidewas first examined
byManakov[12]. Subsequently,thissystemwasderivedasakeymodelforlight-wavepropagation
in optical fibers [19]–[21]. In literature this equation is sometimes referred to as the coupled NLS
equation.
Both the VNLS equation (1.2) and its vector generalization
iq =q ±2kqk2q (1.3)
t xx
where q is an N component vector and k·k is the Euclidean norm, are integrable by the IST. In
[12]onlythe caseN =2is studied, butthe extensiontomorecomponents isstraightforward. The
N-component equation can be derived, with some additional conditions, as an asymptotic model
of the interaction of N wavetrains in a weakly nonlinear, conservative medium.
4
In optical fibers and waveguides, depending on the physics of the particular system, the prop-
agation of the electromagnetic waves may be described by variations of equation (1.2). Note that
the VNLS equation is the ideal (exactly integrable) case. For example, a model with physical
significance is
iq(1) = q(1)+2 |q(1)|2+B|q(2)|2 q(1) (1.4)
t xx
iq(2) = q(2)+2(cid:16)B|q(1)|2+|q(2)|2(cid:17)q(2)
t xx
(cid:16) (cid:17)
whichis equivalentto the eq. (1.2)whenB =1. However,basedonthe propertiesofthe equation
(1.4), apparently it is not integrable when B 6=1.
VNLS (1.3) has a natural matrix generalization in the system
iQ = Q −2QRQ (1.5)
t xx
−iR = R −2RQR (1.6)
t xx
where Q and R are N ×M and M ×N matrices, respectively, and the superscript H denotes the
Hermitian (conjugate transpose). When R=∓QH the system (1.5)–(1.6) reduces to the single
matrix equation
iQ =Q ±2QQHQ (1.7)
t xx
which we refer to as matrix NLS or MNLS. VNLS corresponds to the special case when Q is an
N-component row vector and R is an M-component column vector or vice-versa. In particular,
we obtain equation (1.2) when N =2.
Both the NLS and the VNLS equations admit integrable discretizations which, besides being
used as the basis for constructing numerical schemes for the continuous counterparts, also have
physical applications as discrete systems (see e.g. [23]-[26]).
A natural discretization of NLS (1.1) is the following
d 1
i q = (q −2q +q )±|q |2(q +q ) (1.8)
dt n h2 n+1 n n−1 n n+1 n−1
which here is referred to as the integrable discrete NLS (IDNLS). It is a O(h2) finite-difference
approximation of (1.1) which is integrable via the IST and has soliton solutions on the infinite
lattice [10]- [11]. We note that if we change the nonlinear term in (1.8) to 2|q |2q , the equation
n n
in apparently no longer integrable and it has been found that in certain circumstances chaotic
dynamics results [30].
Correspondingly,we will consider the discretization of VNLS given by the following system
d 1
i q = (q −2q +q )−r ·q (q +q ) (1.9)
dt n h2 n+1 n n−1 n n n+1 n−1
d 1
−i r = (r −2r +r )−r ·q (r +r ) (1.10)
dt n h2 n+1 n n−1 n n n+1 n−1
where q and r are N component vectors and · is the inner product. Under the symmetry
n n
reduction r = ∓q∗ (here and in the following ∗ indicates the complex conjugate), the system
n n
(1.9)–(1.10) reduces to the single equation
d 1
i q = (q −2q +q )±kq k2(q +q ) (1.11)
dt n h2 n+1 n n−1 n n+1 n−1
5
which, for q = q(nh) in the limit h → 0, nh = x gives VNLS (1.3). In [29] it was shown that
n
solitarywavesolutions interactelastically,andthat(1.11)admits multisolitonsolutions. Thus the
expectation was that the discrete vector NLS system (1.11) is indeed integrable. An associated
pair of linear operators (Lax pair) for the system (1.9)–(1.10) was constructed in [28]. Actually,
the system is a reduction of a matrix generalization derived in [29]-[31] of the Lax pair pair used
in the IST of IDNLS (1.8). Indeed, the matrix analog of IDNLS (1.8) is given by
d
i Q =Q −2Q +AQ +Q B+Q −Q R Q −Q R Q (1.12)
dτ n n+1 n n n n−1 n+1 n n n n n−1
d
−i R =R −2R +BR +R A+R −R Q R −R Q R (1.13)
n n+1 n n n n−1 n+1 n n n n n−1
dτ
whereQ ,R areN×M andM×N matrices,respectively,AisanN×N diagonalmatrixandB
n n
is an M ×M diagonal matrix. A and B represent a gauge freedom in the definition of integrable
discrete MNLS which will be used in the following.
Note that the system (1.12)–(1.13) does not, in general, admit the reduction
R =∓QH. (1.14)
n n
However, for N =M one can restrict R and Q to be such that
n n
R Q =Q R =α I, (1.15)
n n n n n
where I is the identity N × N matrix and α is a scalar, and make R = ∓QH a consistent
n n n
reduction of the system (1.12)–(1.13) resulting in the single matrix equation
d
i Q =Q −2Q +AQ +Q B+Q −α (Q +Q ). (1.16)
dτ n n+1 n n n n−1 n n+1 n−1
Similarly, the IST follows the same lines as that presented for (1.12)–(1.13) with additional sym-
metryconditionsimposed. Theadditionalsymmetry(1.15)(whichhasnoanaloginthecontinuous
case) has essential consequences for the IST which will be illustrated in detail.
The outline of the paper is the following: in Chap. 2 we review the results for the scalar
NLS equation; in Chap. 3 we consider the integrable discretization of NLS equation, eq. (1.8); in
Chap.4 weconsider VNLS (1.2)andin Chap.5we dealwith the discrete matrixNLS (DMNLS),
i.e. the system (1.12)–(1.13). In all cases we solve the direct problem and find the explicit reg-
ularity conditions on the potentials for which such problem is well-defined, formulate the inverse
scattering problem as a Riemann-Hilbert boundary value problem and derive from it Gel’fand-
Levitan-Marchenko integral equations, find the time evolution and explicit soliton solutions. We
also consider the problem of a multisoliton collision and find an analytic formula for the polar-
ization shift of vector solitons (this last result is new as far as the discrete vector solitons are
concerned).
6
Chapter 2
Scalar Nonlinear Schr¨odinger
equation
2.1 The inverse scattering transform for NLS
The scalar NLS equation (1.1) is well known to be the compatibility condition of the following
linear equations (cf. [1]-[2])
−ik q
v = v (2.1)
x r ik
(cid:18) (cid:19)
and
2ik2+iqr −2kq−iq
x
v = v (2.2)
t −2kr+ir −2ik2−iqr
x !
where v is a 2-component vector, v(x,t)= v(1)(x,t),v(2)(x,t) T and q =q(x,t), r =r(x,t). The
term compatibility condition arises from the fact that the equality of the mixed derivatives, i.e.
(cid:0) (cid:1)
v =v , is equivalent to the statement that q and r satisfy the evolution equations
xt tx
iq =q −2rq2 (2.3)
t xx
−ir =r −2qr2 (2.4)
t xx
if k, the scattering parameter, is independent of x and t. The system (2.3)–(2.4) reduces to the
single PDE (1.1) under the reduction r = ∓q∗. We refer to the equation with the x derivative,
eq. (2.1), as the scattering problem and the equation with the t derivative, eq. (2.2), as the time
dependence.
In this section we develop the inverse scattering transform for the NLS equation, eq. (1.1), on
the infinite line. The solution q of NLS, at a fixed time, is the potential of the scattering problem
(2.1). WeformulatetheISTprocedureforthesomewhatmoregeneralsystem(2.3)–(2.4)andthen
consider the reductions r =∓q∗ as a special case.
The IST can be broken into three parts: (i) the direct problem - constructing x-independent
scatteringdatafromthepotential-(ii)theinverseproblem-reconstructingthepotentialfromthe
scattering data - (iii) time evolution - determining the evolution of the scattering data by making
useofthetime-dependenceoperator(2.2). TheISTprocedureforsolvingtheinitial-valueproblem
proceeds by first constructing the scattering data at t = t from the initial data q(t ),r(t ) - i.e.
i i i
7
step (i) - then computing the evolution of the scattering data from t to t 6= t - step(iii) - and,
i i
finally, recovering q(t),r(t) by solving the inverse problem - step (ii).
Thetreatmentofthedirectproblemgivenherefollows[1],[3]. Theinverseproblemisformulated
as a Riemann-Hilbert boundary value problem, following [3]. The Gelfand-Levitan-Marchenko
integral equations follow from the RH problem.
2.1.1 Direct scattering problem
Jost functions and integral equations
Werefertosolutionsofthescatteringproblem(2.1)aseigenfunctionswithrespecttotheparameter
k. When the potentials q,r → 0 rapidly as x → ±∞, the eigenfunctions are asymptotic to the
solutions of
−ik 0
v = v
x 0 ik
(cid:18) (cid:19)
when |x| is large. Therefore it is natural to introduce the eigenfunctions defined by the following
boundary conditions
1 0
φ(x,k)∼ e−ikx, φ¯(x,k)∼ eikx as x→−∞ (2.5)
0 1
(cid:18) (cid:19) (cid:18) (cid:19)
0 1
ψ(x,k)∼ eikx, ψ¯(x,k)∼ e−ikx as x→+∞. (2.6)
1 0
(cid:18) (cid:19) (cid:18) (cid:19)
In the following analysis, it is convenient to consider functions with constant boundary condi-
tions. Hence, we define the Jost functions as follows
M(x,k)= eikxφ(x,k), M¯(x,k)=e−ikxφ¯(x,k), (2.7)
N(x,k)= e−ikxψ(x,k), N¯(x,k)=eikxψ¯(x,k).
If the scattering problem (2.1) is rewritten as
v =(ikJ+Q)v (2.8)
x
where
−10 0 q
J= , Q= (2.9)
0 1 r 0
(cid:18) (cid:19) (cid:18) (cid:19)
and I denotes the 2×2 identity matrix,then the Jostfunctions M(x,k) and N¯(x,k) are solutions
of the differential equation
χ (x,k)=ik(J+I)χ(x,k)+(Qχ)(x,k) (2.10)
x
while N(x,k) and M¯(x,k) satisfies
χ˜ (x,k)=ik(J−I)χ˜(x,k)+(Qχ˜)(x,k) (2.11)
x
with the constant boundary conditions
1 0
M(x,k)∼ , M¯(x,k)∼ as x→−∞ (2.12)
0 1
(cid:18) (cid:19) (cid:18) (cid:19)
0 1
N(x,k)∼ , N¯(x,k)∼ as x→+∞. (2.13)
1 0
(cid:18) (cid:19) (cid:18) (cid:19)
8
Solutions of the differential equations (2.10)–(2.11) can be represented by means of the following
integral equations
+∞
χ(x,k) =w+ G(x−ξ,k)(Qχ)(ξ,k)dξ
Z−∞
+∞
χ˜(x,k) =w˜+ G˜(x−ξ,k)(Qχ˜)(ξ,k)dξ
Z−∞
or, in component form, for j =1,2
+∞ 2
χ (x,k) =w + G (x−ξ,k)(Qχ) (ξ,k)dξ
j j j` `
Z−∞ `=1
X
+∞ 2
χ˜ (x,k) =w˜ + G˜ (x−ξ,k)(Qχ˜) (ξ,k)dξ
j j j` `
Z−∞ `=1
X
where w =(w ,0)T, w˜ =(0,w˜ )T and the (matrix) Green’s functions G(x,k)=(G (x,k))
1 2 j` j,`=1,2
and G˜(x,k)= G˜ (x,k) satisfy the differential equations
j`
j,`=1,2
(cid:16) (cid:17)
L+G(x,k) =δ(x)I, L−G˜(x,k)=δ(x)I
0 0
L± =I ∂ −ik(J±I).
0 x
TheGreen’sfunctionsarenotunique,and,asweshowbelow,thechoiceoftheGreen’sfunction
and the choice of the inhomogeneous term together determine the Jost function and its analytic
properties.
Using the Fourier transform method, it is easy to find
1 p−1 0
G(x,k) = eipxdp
2πi 0 (p−2k)−1
ZC(cid:18) (cid:19)
1 (p+2k)−1 0
G˜(x,k) = eipxdp
2πiZC˜(cid:18) 0 p−1(cid:19)
where C and C˜ are appropriate contours. It is naturalto consider G (x,k) and G˜ (x,k) defined
± ±
by
1 p−1 0
G (x,k) = eipxdp
± 2πi 0 (p−2k)−1
ZC±(cid:18) (cid:19)
1 (p+2k)−1 0
G˜ (x,k) = eipxdp
± 2πi 0 p−1
ZC±(cid:18) (cid:19)
where C are the contours from −∞ to +∞ which, respectively, pass below and above both the
±
singularities at p=0 and p=2k (see Fig. 2.1).
Therefore
1 0 e−2ikx 0
G (x,k)=±θ(±x) , G˜ (x,k)=∓θ(∓x) (2.14)
± 0e2ikx ± 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
where θ(x) is the Heaviside function (θ(x)=1 if x>0, θ(x)<0 if x<0). The “+” functions are
analytic in the upper half plane of k and the “−” functions are analytic in the lower half plane.
9
