Variational approach to the modulational instability.
ABSTRACT We study the modulational stability of the nonlinear Schrödinger equation using a time-dependent variational approach. Within this framework, we derive ordinary differential equations (ODE's) for the time evolution of the amplitude and phase of modulational perturbations. Analyzing the ensuing ODE's, we rederive the classical modulational instability criterion. The case (relevant to applications in optics and Bose-Einstein condensation) where the coefficients of the equation are time dependent, is also examined.
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ABSTRACT: When atoms in a gas are cooled to extremely low temperatures, they will-under the appropriate conditions-condense into a single quantum-mechanical state known as a Bose-Einstein condensate. In such systems, quantum-mechanical behaviour is evident on a macroscopic scale. Here we explore the dynamics of how a Bose-Einstein condensate collapses and subsequently explodes when the balance of forces governing its size and shape is suddenly altered. A condensate's equilibrium size and shape is strongly affected by the interatomic interactions. Our ability to induce a collapse by switching the interactions from repulsive to attractive by tuning an externally applied magnetic field yields detailed information on the violent collapse process. We observe anisotropic atom bursts that explode from the condensate, atoms leaving the condensate in undetected forms, spikes appearing in the condensate wavefunction and oscillating remnant condensates that survive the collapse. All these processes have curious dependences on time, on the strength of the interaction and on the number of condensate atoms. Although the system would seem to be simple and well characterized, our measurements reveal many phenomena that challenge theoretical models.Nature 08/2001; 412(6844):295-9. · 38.60 Impact Factor
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ABSTRACT: We consider the quantum correlations between fields generated through cross-phase-modulational instability in an optical fiber. Two examples of this instability are addressed, one relating to the linear-polarization eigenstates in a high-birefringence fiber, and the other to two-color propagation in a fiber. The degrees of second-order coherence of the quantum noise sidebands generated in this chi(3) process are calculated, and the violation of a classical Cauchy-Schwarz inequality and intensity-difference squeezing is reported for light transmitted by a passive filter. Around 50% intensity-difference squeezing is possible in the case of the two-color instability. This illustrates the quantum-statistical nature of the sideband correlations and how passive processing may in some circumstances help accentuate quantum correlations.Physical Review A 09/1991; 44(3):2113-2123. · 3.04 Impact Factor
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ABSTRACT: We study the nonlinear evolution in optical fibers of a modulationally unstable continuous wave. We find that a good description of the propagation can be analytically obtained by a simple truncated three-wave model. The dynamic viewpoint may give an important physical insight in different processes, such as frequency conversion of modulated signals, all-optical generation of temporal codes, and the onset of spatiotemporal chaos.Optics Letters 07/1991; 16(13):986-8. · 3.39 Impact Factor
arXiv:cond-mat/0404601v1 [cond-mat.soft] 25 Apr 2004
Variational approach to the modulational instability
Z. Rapti1, P.G. Kevrekidis1, A. Smerzi2,3and A.R. Bishop3
1Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA
2Istituto Nazionale di Fisica per la Materia BEC-CRS and Dipartimento di Fisica, Universita’ di Trento, I-38050 Povo, Italy
3Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(February 2, 2008)
We study the modulational stability of the nonlinear Schr¨ odinger equation (NLS) using a time-
dependent variational approach. Within this framework, we derive ordinary differential equations
(ODEs) for the time evolution of the amplitude and phase of modulational perturbations. Analyzing
the ensuing ODEs, we re-derive the classical modulational instability criterion. The case (relevant
to applications in optics and Bose-Einstein condensation) where the coefficients of the equation are
time-dependent, is also examined.
Modulational instability (MI) is a general feature of discrete as well as continuum nonlinear wave equations. This
instability shows that in such settings, a specific range of wavenumbers of plane wave profiles of the form u(x,t) ∼
exp(i(kx − ωt)) becomes unstable to modulations. The latter effect leads to an exponential growth of the unstable
modes and eventually to delocalization (upon excitation of such wavenumbers) in momentum space. That is, in turn,
equivalent to localization in position space, and hence the formation of localized, coherent solitary wave structures
The realizations of this instability span a diverse set of disciplines ranging from fluid dynamics  (where it is
usually referred to as the Benjamin-Feir instability) and nonlinear optics  to plasma physics . One of the earliest
contexts in which its significance was appreciated was the linear stability analysis of deep water waves. It was much
later recognized that the conditions for MI would be significantly modified for discrete settings relevant to, for instance,
the local denaturation of DNA  or coupled arrays of optical waveguides . In the latter case, the relevant model
is the discrete nonlinear Schr¨ odinger equation (DNLS), and its MI conditions were discussed in . Most recently,
the MI has been recognized as responsible for dephasing and localization phenomena in the context of Bose-Einstein
condensates (BEC) in the presence of an optical lattice [8–11].
In this brief report, we present an alternative approach to the modulational stability of plane waves in the context
of the nonlinear Schr¨ odinger equation
iψt= −ψxx− U|ψ|2ψ,
where ψ is a complex field, the subscripts denote partial derivatives with respect to the corresponding variable and
U is a constant prefactor (the strength of the nonlinearity). We examine MI using a time dependent variational
approach (TDVA), and study the results in comparison with the standard linear stability (LS) calculations. It should
be mentioned that the use of TDVA for the study of solitons at the classical  and even at the quantum 
level is not novel. What distinguishes our study from these earlier ones is the use of the MI-motivated ansatz in the
TDVA (see below). We also note in passing that MI and solitons from a quantum mechanical point of view have
been considered in a number of references including (but not limited to) . Furthermore, a similar in spirit, 3-mode
approximation was systematically developed in the works of . There are however a number of differences between
the latter and the present approach such as e.g., our use of the variational formulation of the problem (instead of the
application of the 3-mode ansatz in the dynamical equation in the context of ), as well as the fact that we are
perturbing around an exact plane wave solution, while in the case of , the plane wave is an additional mode in the
In the linear stability framework (see e.g., [7,9] and references therein for relevant details), the stability of the plane
waves has been examined. The latter are of the form
ψ(x,t) = ψ0exp[i(kx − ωt)],
and constitute exact solutions of the nonlinear Schr¨ odinger equation with a dispersion relation
ω = k2− Uψ2
Then, the MI is examined in the LS framework using the linearization
u(x,t) = (ψ0+ ǫc)exp[i((kx − ωt) + ǫd(x,t))](4)
and analyzing the O(ǫ) terms as
c(x,t) = c0exp(iβ(x,t)),d(x,t) = d0exp(iβ(x,t)).
Using β(x,t) = qx − Ωt, the dispersion relation connecting the wavenumber q and frequency Ω of the perturbation
(see e.g., )
(−Ω + 2kq)2= q2(q2− 2Uψ2
is obtained. This implies that the instability region for Eq. (1) appears for perturbation wavenumbers q2< 2Uψ2
and in particular only for focusing nonlinearities (to which we restrict this study).
We now attempt to identify the interval of unstable wavenumbers by means of the TDVA. In particular, we start
from the Lagrangian L
t) − |ψx|2+U
and consider a modulation of the plane wave of the form
ψ = [ψ0+ a(t)exp(iφa(t))exp(iqx) + b(t)exp(iφb(t))exp(−iqx)]exp(i(kx − ωt)).
However, instead of considering the modulation directly at the level of the equation, the variation of our approach is
that we use the modulational ansatz in the Lagrangian. This constitutes the basic novel ingredient of this variational-
type approach to the modulational instability.
Here we consider an annular (1-dimensional) geometry, which imposes periodic boundary conditions on the wave-
function ψ(x) and integration limits 0 ≤ x < 2π in Eq. (7). This results in the quantization of the wavenumbers
k,q = 0,±1,±2,.... However, it is clear that our results can be easily generalized to the case of an infinite, open
system and to higher dimensions.
After substitution of Eq. (8) into Eq. (7), we obtain the variational Lagrangian
L = π[−2(a2˙φa+ b2˙φb) + 2(Uψ2
+U(a4+ b4+ 4a2b2)].
0− q2)(a2+ b2) − Uψ4
0− 4qk(a2− b2) + 4Uψ2
It is clear from this Lagrangian that the pair φa(t),φb(t) can be interpreted as the generalized coordinates of the
system, while A(t) = 2a2(t),B(t) = 2b2(t) are the corresponding momenta. In particular, the pairs A(t),φa(t) and
B(t),φb(t) are canonically conjugate with respect to the effective Hamiltonian
0− q2)(A + B) − 2qk(A − B) + 2Uψ2
√AB cos(φa+ φb) +U
4(A2+ B2+ 4AB),
which is an exact integral of motion on the subspace spanned by Eq. (8).
The Lagrangian equations of motion are:
∂a⇒ a˙φa= C1a + C2bcos(φa+ φb) + Ua(a2+ 2b2)
∂b⇒ b˙φb= C3b + C2acos(φa+ φb) + Ub(b2+ 2a2)
⇒ ˙ a = C2bsin(φa+ φb)(12)
⇒˙b = C2asin(φa+ φb),
where C1= Uψ2
If we now keep all terms to O(a) in Eqs. (11)-(14) [which is consistent with an approximation linear in a], we obtain
0− q2− 2qk, C2= Uψ2
0, and C3= Uψ2
0− q2+ 2qk are constant prefactors.
a = b
˙ a = C2asin(φ)
˙φ = (C1+ C3) + 2C2cos(φ),
where φ = φa+ φb. The latter equation has the solution
φ(t) = 2arctan
Two different cases arise here, corresponding respectively to whether the instability criterion is satisfied or not.
Namely, when 2Uψ2
0− q2< 0 the solution of Eq. (16) is
while when 2Uψ2
0− q2> 0 the solution of Eq.(16) is
The solutions signal the appearance of the modulational instability when the threshold condition q2= 2Uψ2
crossed (passing from higher to lower perturbation wavenumbers). This is also clearly shown in the time evolution of
a(t) in accordance with Eqs. (19)-(20) also shown in Fig. 1 in the case of q = 2 for
= 1.2 (see the right panel of Fig. 1).
= 0.2 (see the left panel of
Fig. 1) and
FIG. 1. The left panel shows the (stable oscillatory) time evolution of a(t) for the case
= 0.2, in accordance with
Eq. (19). The right panel shows the (unstable, exponentially growing) time evolution of a(t) for the case of
accordance with Eq. (20).
= 1.2, in
An alternative, more intuitive way to appreciate the linear stability result from a dynamical systems viewpoint.
This consists of reducing Eqs. (11)-(14) to a one degree of freedom setting with an effective potential energy landscape
whose (parametric) variation will elucidate the instability. Along these lines, using A(t = 0) = 0 in (without loss of
generality) in Eq. (9) and A(t) = B(t) (from Eqs. (12) and (14)) in Eq. (10), we have:
˜H = π
0− q2)A +3
Eliminitating φ from Eqs. (12) and (21) for A(0) = 0, we obtain the “energy equation” for A
˙A2+ Veff= 0,
where the effective potential Veff(A) is of the form:
Veff(A) = 2q2(q2− 2Uψ2
One can then examine the stability of the effective potential by evaluating its curvature at A = 0. We thus obtain
0) and hence the potential will be convex (and therefore the dynamics will be stable)
eff(A)|A=0= 4q2(q2− 2Uψ2
for q2> 2Uψ2
retrieve the modulational stability criterion. The effective potential is shown for the modulationally stable, unstable
and marginal case in Fig. 2.
0, while it will be concave (and the dynamics unstable) for q2< 2Uψ2
0. Hence in this case also, we
0 0.20.40.6 0.8126.96.36.199.8
FIG. 2. The effective potential of Eq. (23) is shown for as a function of A for U = ψ0 = 1 and three different values of
q: q = 1 (modulationally unstable; solid line), q =
√2 (at the threshold; dashed line) and q = 2 (modulationally stable;
Now we turn to a more interesting case, where the coefficient of the dispersion term, as well as the coefficient of
the nonlinear term in Eq. (1) are temporally modulated, namely we examine the equation
iψt= −D(t)ψxx− U(t)|ψ|2ψ.
Our aim is to derive the modulational stability equation via the TDVA, for general D(t) and U(t). It is interesting
to note that this equation has become of increasing importance in the past decade due to applications both in
optics and also, more recently, in soft condensed-matter physics. In particular, in optics, the case of D(t) periodic
and U(t) constant is of relevance in the context of the so-called dispersion management. The latter is based on
periodic alternation of fibers with opposite signs of the group-velocity dispersion . We note in passing that in this
application time t is, in reality, the propagation distance (i.e., space), while x corresponds to a retarded time variable.
An alternative setting where D(t) is constant, but U(t) can be temporally modulated (via a Feshbach resonance, i.e.,
an external magnetic field; see e.g., ) can be found in Bose-Einstein condensation. In the latter setting, there has
been an explosion of interest recently in time dependent scattering length and its effect on patterns, coherent structures
and collapse thereof (a number of very recent references can be found in [18–21]. While our primary motivation in
considering MI through the TDVA in Eq. (24) principally stems from this recently explored experimental potential
in Bose-Einstein condensates, we should note that this type of problem was investigated earlier in nonlinear optics,
see e.g., .
We consider the perturbation of the form
ψ = ψ1[1 + w(t)cos(qx)](25)
to the plane wave solution ψ1= ei(−k2?t
variant of the ansatz of Eq. (8), with a = b, φa= φband (without loss of generality) ψ0= 1. Following the same
procedure as above, we can obtain the stability equations for w = wr+ iwi:
0U(s)ds+kx). Notice that in Eq. (25), we are using for simplicity a
˙ wr= q2D(t)wi−3
˙ wi= −?q2D(t) − 2U(t)?wr+3
Hence, at the linear level, we can derive the following stability equation
D(t)˙ wr− q2D(t)?q2D(t) − 2U(t)?wr
By determining the windows of stability of the ordinary differential equation of (28), the modulational stability of Eq.
(24) is determined. It is further worth noting that for D(t) constant and U(t) time-periodic, Eq. (28) falls into Eq.
(2) of  and becomes Hill’s equation for which many stability results are known in the mathematical literature .
Furthermore, in the case of U(t) = 1 + 2αcos(ωt), Eq. (28) falls into Eq. (2) of  and the resulting equation is of
the Mathieu type for which explicit stability windows can be computed (for details see  and references therein).
It then becomes naturally an interesting problem in mathematical physics to determine the stability of Eq. (28) for
more general cases (e.g., with both coefficients periodically varying etc.).
In this brief report, we have revisited the modulational instability from a different point of view, namely a vari-
ational one. We have used this dynamical systems’ type approach to derive the Euler-Lagrange equations for the
time-dependent perturbation ansatz parameters and have examined their stability for different wavenumbers of the
perturbation (which affect the constants of the ensuing set of ordinary differential equations). We have retrieved,
in a simple and intuitive way, the criterion for the instability. The technique has also been generalized in cases in
which the coefficients of the dispersion and/or nonlinearity are temporally varying (a case which we have argued to be
relevant to a variety of applications). We have found the corresponding stability condition obtaining a novel ordinary
differential equation, whose special cases correspond to stability/instability criteria established previously. It would
be interesting to extend the considerations of this method (which seems applicable to any setting with an underlying
Lagrangian/Hamiltonian structure) to contexts with explicit spatial dependence of the potential (see e.g., [9,11]).
The support of NSF (DMS-0204585), UMass and the Clay Institute (PGK) is gratefully acknowledged. Work at
Los Alamos is supported by the US DoE.
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