Nonlinear Hamiltonian Waves with Constant Frequency
and Surface Waves on Vorticity Discontinuities
JOSEPH BIELLO
University of California at Davis
JOHN K. HUNTER
University of California at Davis
Abstract
Waves with constant, nonzero linearized frequency form an interesting class of
nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model
equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of
multiple scales, we derive a cubically nonlinear, quasi-linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in
two-dimensional inviscid, incompressible fluid flows. Thus, the Burgers-Hilbert
equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that
the asymptotic equation can also be derived by means of a near-identity transformation. We derive a semiclassical approximation of the asymptotic equation
and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers-Hilbert and asymptotic
equations are in excellent agreement in the appropriate regime. In particular, the
lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is
given by the cubically nonlinear timescale predicted by the asymptotic equation.
© 2009 Wiley Periodicals, Inc.
1 Introduction
The most familiar class of nondispersive waves consists of hyperbolic waves
whose linearized phase speed is independent of their wave number [12]. A less familiar class of nondispersive waves consists of waves whose linearized frequency
is nonzero and independent of their wave number. We call these waves constantfrequency waves. A weakly nonlinear, nondispersive hyperbolic wave has a spatial
profile that propagates and slowly distorts as a result of quadratically nonlinear,
three-wave interactions. The wave profile typically satisfies an inviscid Burgers
equation. By contrast, a weakly nonlinear constant-frequency wave has a spatial
profile that oscillates in time and slowly distorts as a result of cubically nonlinear,
Communications on Pure and Applied Mathematics, Vol. LXIII, 0303–0336 (2010)
© 2009 Wiley Periodicals, Inc.
304
J. BIELLO AND J. K. HUNTER
four-wave interactions. In this paper, we introduce a model equation for nonlinear constant-frequency waves and derive an asymptotic equation for the weakly
nonlinear evolution of the wave profile. We show that the same asymptotic equation describes constant-frequency surface waves on a vorticity discontinuity in a
two-dimensional, inviscid, incompressible fluid flow.
The model equation for unidirectional, constant-frequency waves is
1 2
u
D HŒu:
(1.1)
ut C
2
x
In (1.1) and below, H denotes the spatial Hilbert transform, which is defined by
HŒe i kx D i.sgn k/e i kx ;
where sgn is the usual sign function defined in (A.1). We summarize our notation
in Appendix A. Equation (1.1) is an inviscid Burgers equation for u.x; t / with a
lower-order, nonlocal, linear, oscillatory term. We refer to (1.1) as the inviscid
Burgers-Hilbert equation or, for brevity, the Burgers-Hilbert equation. Marsden
and Weinstein [10] wrote down (1.1) as a quadratic approximation for the motion
of the boundary of a vortex patch but did not analyze it.
Positive wave number solutions of the linearized equation u t D HŒu have frequency equal to 1, while negative wave number solutions have frequency equal to
1. Thus, (1.1) describes a single, real-valued nonlinear wave whose linearized
frequency is independent of the magnitude of the wave number. The frequency of
a real wave is an odd function of the wave number, so it necessarily depends on the
sign of the wave number.
Equation (1.1) may be written as
Z 1
ıH
1 3
1
D 0;
H.u/ D
uj@x j u C u dx;
(1.2)
u t C @x
ıu
2
6
where @x denotes the partial derivative with respect to x and j@x j D H@x is the
operator with symbol jkj. Thus, the Burgers-Hilbert equation is Hamiltonian with
respect to the constant Hamiltonian operator @x .
Weakly nonlinear solutions of (1.1) have the form
u.x; t / (1.3)
.x; t /e i t C
.x; t /e i t ;
where .x; t / is a small, complex-valued amplitude function that varies slowly in
time and contains only positive wave number components, meaning that
1
P D .I C iH/:
2
Here P is the projection onto positive spatial wave number components. Using the
method of multiple scales, we show that (1.3) is an asymptotic solution of (1.1) if
.x; t / satisfies the following cubically nonlinear, nonlocal equation:
P
(1.4)
(1.5)
t
D ;
D P@x Πj@x jn nj@x j ;
n D j j2 :
NONLINEAR WAVES WITH CONSTANT FREQUENCY
305
This equation preserves the constraint (1.4). From (1.4), we may alternatively write
j@x j D i @x in (1.5), but there is no similar identity for j@x jn since n contains
both positive and negative wave number components. Equation (1.5) may be written in equivalent spectral and real forms, given in (5.6) and (5.9), respectively.
Equation (1.5) has the complex Hamiltonian form
ıH
(1.6)
D 0;
t C @x
ı where
(1.7)
H
;
Z D
i
4
.
x
x/ 1
2
j@x jŒ
dx:
We also derive (1.5) from (1.1) by a symplectic near-identity transformation that
eliminates the cubic terms from the Hamiltonian (1.2), which are nonresonant,
giving the resonant fourth-order terms in the Hamiltonian (1.7).
An interesting physical example of constant-frequency waves arises in twodimensional incompressible, inviscid fluid flows. A planar discontinuity in vorticity that separates two linear shear flows is linearly stable. This stability contrasts
with the Kelvin-Helmholtz instability of a vortex sheet, across which the tangential
velocity is discontinuous. A vorticity discontinuity supports surface waves that decay exponentially in space away from the discontinuity and oscillate in time with a
constant frequency that is proportional to the jump in vorticity.
Remarkably, when weakly nonlinear effects are taken into account, the complex
amplitude of the displacement of the vorticity discontinuity satisfies an asymptotic
equation of exactly the same form as (1.5). Thus, the Burgers-Hilbert equation,
with a suitably renormalized nonlinear coefficient given in (B.4) below, captures
the resonant cubically nonlinear dynamics of the vorticity discontinuity. This apparent coincidence is explained, in part, by dimensional analysis, which shows that
(1.1) is an appropriate model equation for constant-frequency surface waves.
A planar vorticity discontinuity provides a local approximation of a smooth vortex patch near its boundary, so the asymptotic equation (1.5) is relevant to the
dynamics of vortex patches [3, 9, 11]. A spectral form of an equation related to
(1.5) was derived by Dritschel [5] for the motion of a circular vortex patch.
Numerical computations show that smooth solutions of (1.5) typically develop
singularities in finite time, in which the derivative x blows up. This singularity formation corresponds to the breakdown of smooth solutions of the inviscid Burgers-Hilbert equation (1.1). There is excellent numerical agreement between solutions of the asymptotic equation and small-amplitude solutions of the
Burgers-Hilbert equation. Moreover, the singularity formation time of the asymptotic equation gives an accurate approximation of the singularity formation time of
the Burgers-Hilbert equation for small initial data.
It follows that small, smooth solutions of the inviscid Burgers-Hilbert equation
have a longer, cubically nonlinear lifespan than the quadratically nonlinear lifespan
306
J. BIELLO AND J. K. HUNTER
of smooth solutions of the inviscid Burgers equation. This phenomenon is illustrated in Figure 8.4. The extension in the lifespan of smooth solutions is a result
of the quadratic Burgers nonlinearity averaging out over a period of the temporal
oscillations induced by the Hilbert transform. If part of the wave is compressed
during one phase of the oscillation, then it is rarefied during the other phase. Thus,
the nonlinear steepening of the wave profile in one phase is canceled by its expansion in the other.
The only explicit solution of (1.5) that we know of is the harmonic, spatially
periodic traveling wave
(1.8)
.x; t / D Ae i kxi !t ;
A 2 C, k > 0;
where ! satisfies the nonlinear dispersion relation
(1.9)
! D jAj2 k 2 :
We show that this solution is linearly and modulationally stable.
We conclude the introduction by outlining the contents of the paper. In Section 2, we discuss constant-frequency waves in the context of the general theory of
nonlinear waves. In Section 3, we carry out a dimensional analysis of Hamiltonian
wave motions that shows that the Burgers-Hilbert equation (1.1) is an appropriate
model equation for unidirectional, constant-frequency surface waves. In Section 4,
we use the method of multiple scales to derive the asymptotic equation (1.5) from
(1.1). We outline the derivation of (1.5) for surface waves on a vorticity discontinuity in Appendix B. In Section 5, we describe the Hamiltonian structure of (1.5)
and write the equation in equivalent spectral and real forms. In Section 6, we derive (1.5) from (1.1) by the use of a near-identity transformation. In Section 7, we
show that harmonic solutions are linearly stable and derive a semiclassical approximation of (1.5). Finally, in Section 8, we present and compare some numerical
solutions of (1.1) and (1.5).
2 Constant-Frequency Waves
We begin by recalling some well-known facts about dispersive and nondispersive hyperbolic waves [12, 13]. For simplicity, we consider a unidirectional, realvalued wave motion with a single mode that propagates freely in space. Abusing
notation, we write the linearized dispersion relation between the frequency ! and
the wave number k of a harmonic wave as ! D !.k/, where ! W R ! R is an odd
function.
The wave motion is dispersive if ! 00 ¤ 0, where the prime denotes a derivative
with respect to k. According to the linearized theory, a small-amplitude dispersive
wave spreads out into a locally harmonic wave train. Over longer times, weakly
nonlinear effects come into play, and the complex amplitude of the wave typically
satisfies a cubically nonlinear Schrödinger equation.
Nondispersive hyperbolic waves have a dispersion relation of the form ! D c0 k,
where c0 is a constant wave speed. According to the linearized theory, a wave
NONLINEAR WAVES WITH CONSTANT FREQUENCY
307
has an arbitrary spatial wave profile that propagates at constant velocity without
distortion. Over longer times, weakly nonlinear effects distort the wave profile,
and the profile of a bulk wave typically satisfies the quadratically nonlinear inviscid
Burgers equation [2, 7].
The degree of nonlinearity in a weakly nonlinear equation is a consequence of
the resonances allowed by the linearized dispersion relation. The quadratically
nonlinear effects on nondispersive hyperbolic waves arise from three-wave resonances among harmonics of the form
!1 D !2 C !3 ;
k1 D k2 C k3 :
Since ! D c0 k, this resonance condition is satisfied for any kj such that k1 D
k2 C k3 . As a result, the quadratically nonlinear self-interaction of an initially
harmonic wave with frequency and wave number .!; k/ generates higher harmonics .n!; nk/ for every integer n. We assume throughout this discussion that all
relevant nonlinear interaction coefficients are nonzero.
By contrast, three-wave resonances of dispersive waves do not occur (except in
nongeneric cases, such as second-harmonic resonance, which we do not consider).
However, four-wave resonances of the form
! D ! C ! !;
k D k C k k;
always occur. These resonances, and nearby ones, lead to cubically nonlinear effects on harmonics in a narrow frequency and wave number band about .!; k/, but
they do not lead to the resonant generation of higher harmonics .!.nk/; nk/ for
jnj ¤ 1.
Constant-frequency waves have a linearized dispersion relation of the form
(2.1)
! D !0 sgn k
where !0 is a nonzero constant. The dispersion relation ! D !0 sgn k Cc0 k can be
reduced to (2.1) by means of a Galilean transformation. Constant-frequency waves
are nondispersive, since ! 00 D 0 for k ¤ 0. According to the linearized theory,
a constant-frequency wave has an arbitrary spatial profile that oscillates with frequency !0 between the profile and its Hilbert transform. Three-wave resonances
do not occur since !0 ¤ !0 C !0 , but four-wave resonances occur among any
spatial harmonics with positive wave numbers fk1 ; k2 ; k3 ; k4 g such that
(2.2)
k1 D k2 C k3 k4 ;
kj > 0:
The corresponding condition for the frequencies is satisfied automatically. Thus,
cubically nonlinear interactions among different spatial harmonics generate new
spatial harmonics, leading to a wide wave number spectrum but a narrow frequency spectrum. For example, in the case of spatially periodic waves with zero
mean, we get a spectrum that consists of frequency and wave numbers .!0 ; nk/
for all nonzero integers n. The complex wave-amplitude function that describes
the spatial profile of the wave typically satisfies a cubically nonlinear asymptotic
equation, such as (1.5). The nonlinear dynamics of constant-frequency waves is
308
J. BIELLO AND J. K. HUNTER
therefore fundamentally different from that of either dispersive waves or nondispersive hyperbolic waves.
3 Dimensional Analysis
Constant-frequency waves arise in systems that are invariant under spatial scalings .x; t / 7! .x; t /. This invariance implies that the only space-time parameters
on which the wave motion depends have the dimension of time; the motion cannot
depend, for example, on parameters with the dimension of length, velocity, or acceleration. By contrast, nondispersive hyperbolic waves arise in systems that are
invariant under space-time scalings .x; t / 7! .x; t /, which implies that the only
space-time parameters on which the wave motion depends have the dimension of
velocity.
In this section, we carry out a dimensional analysis of general Hamiltonian
constant-frequency waves that explains why (1.1) provides an appropriate model
equation for surface waves, such as waves on a vorticity discontinuity. A similar
dimensional analysis [2] shows that the inviscid Burgers is an appropriate model
equation for nondispersive hyperbolic bulk waves, such as nonlinear sound waves.
3.1 Hamiltonian Waves
We consider a translation-invariant Hamiltonian wave motion [13] that depends
upon two dimensional parameters: a frequency !0 and a density 0 . We use M , L,
and T to denote the dimensions of mass, length, and time, respectively, and denote
the dimensions of a quantity X by ŒX . Then Œ!0 D 1=T and Œ0 D M=Ln ,
where n is the number of space dimensions.
We denote canonically conjugate complex wave amplitudes by fa.k/; a .k/g
and suppose that they are parametrized by a wave number vector k 2 Rd . Thus,
d D n for bulk waves and d D n 1 for waves that propagate along a surface of
codimension 1. The energy of the wave motion is given by a real-valued Hamiltonian functional H .a; a / of the wave amplitudes.
Assuming that the Hamiltonian is a smooth functional of the wave amplitudes,
we Taylor-expand it in powers of fa; a g to get
H.a; a / D
(3.1)
Z
!0 a.k/a .k/d k
Z
C ı.k1 k2 k3 /V .k1 ; k2 ; k3 /a .k1 /a.k2 /a.k3 /d k1 d k2 d k3
Z
C ı.k1 C k2 k3 /V .k1 ; k2 ; k3 /a.k1 /a .k2 /a .k3 /d k1 d k2 d k3
C O.jaj4 /:
Here !0 is a constant frequency, and V .k1 ; k2 ; k3 / 2 C is an interaction coefficient
for the interaction k1 D k2 C k3 , which satisfies V .k1 ; k2 ; k3 / D V .k1 ; k3 ; k2 /.
NONLINEAR WAVES WITH CONSTANT FREQUENCY
309
For simplicity, we suppose that the expansion of the Hamiltonian does not contain
creation and annihilation terms proportional to
a .k1 /a .k2 /a .k3 / and
a.k1 /a.k2 /a.k3 /;
respectively. These terms are nonresonant and can be removed by a near identity
transformation to the orders we consider here.
Since the Hamiltonian is an energy, it has dimension ŒH D ML2 =T 2 , which
implies that
1=2
1=2
M
M
Œa D
L1Cd=2 ; ŒV D
L3.1Cd=2/Cn :
T
T
It follows that
V .k1 ; k2 ; k3 / D .0 !0 /1=2 W .k1 ; k2 ; k3 /;
where ŒW D L with
nd C2
:
D
2
In particular, we have D 1 for bulk waves and D 32 for surface waves. The
inverse wave number provides the only length scale for a constant-frequency wave
motion, so W is a homogeneous function of degree , meaning that
W .k1 ; k2 ; k3 / D W .k1 ; k2 ; k3 / for all > 0.
Next, we specialize to unidirectional waves and consider some examples of bulk
and surface wave equations that are consistent with this dimensional argument.
3.2 Unidirectional Waves
A unidirectional wave with a single, real-valued mode may be described by
complex-canonical variables parametrized by a positive wave number
fa.k/; a .k/ W 0 < k < 1g:
We consider a cubic, constant-frequency Hamiltonian without creation or annihilation terms,
Z 1
H.a; a / D
!0 a./a ./d 0
Z 1Z 1
(3.2)
C
V . C ; ; /a . C /a./a./d d
0
0
Z 1Z 1
V . C ; ; /a. C /a ./a ./d d;
C
0
0
where the interaction coefficient V .k1 ; k2 ; k3 / is defined on positive wave numbers
kj such that k1 D k2 C k3 and is symmetric in .k2 ; k3 /. The complex canonical
form of Hamilton’s equation is
(3.3)
i a t .k; t / D
ıH
ıa .k; t /
for k > 0:
310
J. BIELLO AND J. K. HUNTER
Using (3.2) in (3.3), we get the following equation for a.k; t /:
Z k
i a t .k; t / D !0 a.k; t/ C
V .k; k ; /a.k ; t /a.; t /d 0
Z 1
C2
V .k C ; k; /a.k C ; t /a .; t /d for k > 0:
0
Defining a.k; t/ D a .k; t /, we may write this equation in convolution form as
Z 1
i.sgn k/a t .k; t / D !0 a.k; t / C
V .k; k ; /a.k ; t /a.; t /d 1
for 1 < k < 1;
where we extend V .k1 ; k2 ; k3 / to a complex-valued function defined for all kj 2 R
such that k1 D k2 Ck3 . As shown in [2], this function is symmetric in .k1 ; k2 ; k3 /
and satisfies V .k1 ; k2 ; k3 / D V .k1 ; k2 ; k3 /.
The three-wave interaction coefficient of a constant-frequency bulk wave is a
symmetric, homogeneous function of degree 1. A simple choice for it is
V .k1 ; k2 ; k3 / D .0 !0 /1=2 jk1 k2 k3 j1=3 :
The corresponding Hamiltonian equation is
i.sgn k/a t .k; t / D !0 a.k; t/
Z 1
C .0 !0 /1=2 jkj1=3
jk j1=3 jj1=3 a.k ; t /a.; t /d :
1
Introducing a noncanonical variable
Z
u.x; t / D .0 !0 /
1=2
1
1
jkj1=3 a.k; t /e i kx d k;
we get
1
Hj@x j2=3 .u2 / D !0 HŒu:
2
This equation, with a nonlocal nonlinearity, therefore provides a model Hamiltonian equation for constant-frequency bulk waves.
The three-wave interaction coefficient of a constant-frequency surface wave is a
symmetric, homogeneous function of degree 32 . A simple choice for it is
ut C
V .k1 ; k2 ; k3 / D .0 !0 /1=2 jk1 k2 k3 j1=2 :
The corresponding Hamiltonian equation is
i.sgn k/a t .k; t / D !0 a.k; t/
Z 1
C .0 !0 /1=2 jkj1=2
jk j1=2 jj1=2 a.k ; t /a.; t /d :
1
Introducing the noncanonical variable
u.x; t / D .0 !0 /
1=2
Z
1
1
jkj1=2 a.k; t /e i kx d k;
NONLINEAR WAVES WITH CONSTANT FREQUENCY
311
we get the Burgers-Hilbert equation,
1 2
u
(3.4)
ut C
D !0 HŒu:
2
x
We may normalize !0 D 1 in (3.4) by the scaling t 7! !0 t , u 7! !01 u, and the
Hamiltonian form of the resulting equation is given in (1.2).
We remark that a simultaneous time reversal and space reflection leaves the linearized dispersion relation (2.1) invariant, but a separate time reversal or space
reflection transforms !0 7! !0 . Similarly, the transformation t 7! t , x 7! x,
u 7! u leaves (3.4) invariant, but a separate time reversal t 7! t , u 7! u or
space reflection x 7! x, u 7! u transforms !0 7! !0 . Thus, by making a
spatial reflection if necessary, we can ensure that !0 is positive. It also follows
that if a system is invariant under both time reversal and space reflection, then
constant-frequency modes must come in pairs, with the linearized dispersion relation ! 2 D !02 .
For nondispersive hyperbolic waves depending on a wave speed c0 , dimensional
analysis [2] shows that the bulk wave interaction coefficients are homogeneous of
degree 32 , leading to an inviscid Burgers equation
1 2
u px D 0:
u t C c0 ux C
2
This analysis explains why the local inviscid Burgers nonlinearity is appropriate
for surface waves depending on a frequency !0 , and bulk waves depending on a
speed c0 .
4 The Burgers-Hilbert Equation
In this section, we use the method of multiple scales to show that weakly nonlinear solutions of (1.1) satisfy the asymptotic equation (1.5). Before doing so,
we shall briefly compare (1.1) with some other nonlinear, nonlocal wave equations
that have been studied previously.
The Benjamin-Ono equation,
1 2
u
C HŒuxx D 0;
ut C
2
x
contains a nonlocal, linear dispersive term, but, unlike the term in (1.1), it is higher
order and prevents the formation of shocks. The inviscid Burgers equation with a
lower-order source term consisting of a spatial convolution with an odd, integrable
function f W R ! R,
1 2
(4.1)
ut C
u
D f u;
2
x
provides a useful model for nonlinear waves with a dispersion relation of the form
! D i fO.k/ (see [7, sec. 3.4.1], [8], and [12, sec. 13.14], for example). One
difference between (4.1) and (1.1) is that convolution with an integrable function
312
J. BIELLO AND J. K. HUNTER
is a bounded linear operator on L1 , whereas the Hilbert transform is a singular
integral operator that is unbounded on L1 . A more significant difference for the
problems considered here is that (4.1) is dispersive, whereas (1.1) is nondispersive.
As a result, small-amplitude solutions of (1.1) and (4.1) have qualitatively different
behaviors.
The nonlinear, nonlocal equation
u t D uHŒu;
introduced in [4] as a one-dimensional model for vortex stretching, includes a
Hilbert transform in the nonlinearity, as does the equation
1 2
C HŒuHŒuxx u
ut D
2
xx
introduced in [6] as a model for nonlinear Rayleigh waves and derived in [1] for
surface waves on a tangential discontinuity in magnetohydrodynamics.
4.1 Linearized Equation
The linearization of (1.1) is
u t D HŒu:
(4.2)
For definiteness, we consider solutions u W R R ! R; with appropriate modifications, similar results apply to spatially periodic solutions u W T R ! R.
The dispersion relation of (4.2) is ! D sgn k, and the general solution is
Z 1
O .k/e i kxi t C O .k/e i kxCi t d k;
u.x; t / D
0
where O W .0; 1/ ! C is arbitrary. Equivalently, we have
u.x; t / D
where
W R ! C is given by
.x/e i t C
Z
.x/ D
1
.x/e i t
O .k/e i kx d k:
0
It follows that P D , where P is the projection onto positive wave numbers
defined in (1.4). Writing
f .x/ C ig.x/
2
where f and g are real-valued functions, we find that g D HŒf , and
.x/ D
u.x; t / D f .x/ cos t C HŒf .x/ sin t:
Thus the spatial profile of the solution of (4.2) oscillates in time between f and
HŒf , where f is an arbitrary function.
NONLINEAR WAVES WITH CONSTANT FREQUENCY
313
Another way to understand this solution is to write v D HŒu and take the Hilbert
transform of (4.2). Using the fact that H2 D I, we find that
u t D v;
v t D u:
Thus, the linearized wave motion consists of simple harmonic oscillators at each
point of space. Oscillations at different points are coupled together only because
their velocity is the spatial Hilbert transform of their displacement. Although the
phase velocity !=k D 1=jkj of the waves is nonzero, the group velocity ! 0 is zero,
so the waves do not transport energy.
4.2 Weakly Nonlinear Waves
Next we consider weakly nonlinear solutions of (1.1). Using the method of
multiple scales, we look for an asymptotic solution of the form
(4.3)
u.x; tI "/ D "u1 .x; t; "2 t / C "2 u2 .x; t; "2 t / C "3 u3 .x; t; "2 t / C O."4 /:
We use (4.3) in (1.1), expand the result in power series with respect to ", and equate
coefficients of ", "2 , and "3 . We find that u1 .x; t; /, u2 .x; t; /, and u3 .x; t; /
satisfy
(4.4)
(4.5)
(4.6)
u1t D HŒu1 ;
1 2
u2t C
u
D HŒu2 ;
2 1 x
C u1 C .u1 u2 /x D HŒu3 :
u3t
The solution of (4.4) for u1 is
(4.7)
u1 .x; t; / D F .x; /e i t C F .x; /e i t ;
where F . ; / W R ! C satisfies
PŒF D F:
(4.8)
We use (4.7) in (4.5) and solve the resulting equation for u2 to get
(4.9)
(4.10)
u2 .x; t; / D G.x; /e 2i t C M.x; / C G .x; /e 2i t ;
1 2
GD
; M D HŒFF x :
iF
2
x
In solving for G, we use (4.8), which implies that PŒF 2 D F 2 and HŒF 2 D
iF 2 .
We obtain an equation for F from the requirement that (4.6) is solvable for u3 .
To state this requirement, consider the following equation for u.x; t /,
(4.11)
u t D HŒu C B.x/e i nt ;
where n 2 Z and B W R ! C.
314
J. BIELLO AND J. K. HUNTER
P ROPOSITION 4.1 If n2 ¤ 1, then (4.11) is uniquely solvable for every B 2
L2 .R/. If n D 1, then (4.11) is solvable for B 2 L2 .R/ if and only if
(4.12)
PŒB D 0;
and if n D 1, then (4.11) is solvable if and only if
(4.13)
QŒB D 0;
where P and Q are defined in (A.4).
P ROOF : A solution of (4.11) has the form
u.x; t / D A.x/e i nt
where A satisfies
(4.14)
i nA C HŒA D B:
Taking the Hilbert transform of this equation, and using the fact that H2 D I, we
get
(4.15)
A i nHŒA D HŒB:
If n2 ¤ 1, the unique solution of (4.14)–(4.15) is
AD
i nB HŒB
:
n2 1
If n D 1, the system (4.14)–(4.15) is solvable if and only if HŒB D iB. From
(1.4), this means that B satisfies (4.12). In that case, the solution is
AD
1
iB C C
2
where C.x/ is an arbitrary function such that PŒC D C . The solution for n D 1
is similar.
We use (4.7)–(4.10) in (4.6), compute the coefficient of the nonhomogeneous
term proportional to e i t , and impose the solvability condition (4.12) that its projection is 0. The solvability condition (4.13) follows by complex conjugation. After
simplifying the result, we find that (4.6) is solvable for u3 if and only if F .x; /
satisfies
(4.16)
F D P F HŒFF x C iFF Fx x :
Equation (4.16) with F D
and D t is equivalent to (1.5).
NONLINEAR WAVES WITH CONSTANT FREQUENCY
315
5 Properties of the Asymptotic Equation
In this section, we describe the formal Hamiltonian structure of the asymptotic
equation (1.5), list the symmetries that we know of, and write it in equivalent spectral and real forms.
We denote by ; W R ! C complex-conjugate functions that satisfy the
constraints
(5.1)
PΠD ;
QŒ
D
;
where P and Q are the projections onto positive and negative wave number components, respectively, defined in (A.4). The functional derivative of a functional
H. ; / with respect to is given by
ıH
D PŒh;
ı where h W R ! C is any function such that
ˇ
ˇ
d
ˇ
H. ;
C /ˇ
d
Z
D
h dx
D0
for all W R ! C with QΠD . One may verify the Hamiltonian form of
(1.5) directly by computing the functional derivative of H given in (1.7) and using
the result in (1.6).
5.1 Symmetries
The Hamiltonian (1.7) has the following symmetries:
(1) time translation invariance, .x; t / 7! .x; t C /, generated by the
Hamiltonian H;
(2) space translation invariance, .x; t / 7! .x ; t /, generated by the momentum
Z
dxI
PD
(3) phase translation invariance,
tion
Z
SD
.x; t / 7! e i .x; t /, generated by the ac
j@x j1
dxI
(4) amplitude translation invariance, .x; t / 7! .x; t / C 0 for any
generated for functions defined for x 2 R by 0 M , where
Z
M D x dx:
0
2 C,
The first three symmetries have a clear physical origin. The origin of the fourth
symmetry is more mysterious. Its existence depends on a cancellation between the
terms i n.
x / and nj@x jn in the Hamiltonian (1.7), and it would not
x hold if the coefficients of those terms were in a different ratio.
316
J. BIELLO AND J. K. HUNTER
The generating functionals of these symmetries are conserved quantities for
(1.5). In addition, the Poisson bracket associated with @x ,
Z
ıG
ıF
dx;
(5.2)
fF; Gg D @x
ıu
ıu
has the Casimir functional
Z
CD
dx:
Thus if x 2 R, the functionals H, P, S, M, and C are conserved for smooth solutions of (1.5) that decay sufficiently rapidly at infinity (as may be verified directly
from the equation). If x 2 T , the functionals H, P, S, and C are conserved. We
do not know, however, of an analogue of M for spatially periodic solutions.
5.2 Spectral Form
Let O .k; t / denote the spatial Fourier transform of .x; t /. The projection condition (1.4) implies that O .k; t / D 0 for k 0, so
Z 1
Z 1
O .k; t /e i kx dx;
O .k; t /e i kx dx:
.x; t / D
.x; t / D
0
0
Using these expressions in the spatial expression of the Hamiltonian (1.7) and simplifying the result, we find that the spectral expression of the Hamiltonian is
Z
O
O
H. ; / D ı.k1 C k2 k3 k4 /ƒ.k1 ; k2 ; k3 ; k4 /
(5.3)
O .k1 / O .k2 / O .k3 / O .k4 /d k1 d k2 d k3 d k4 ;
where ı denotes the delta function, the integrals are taken over kj > 0, and
(5.4)
ƒ.k1 ; k2 ; k3 ; k4 / D 2 min fk1 ; k2 ; k3 ; k4 g:
Writing (1.6) in terms of O , we get
O t D i k ıH :
2
ı O (5.5)
Use of (5.3) in (5.5) gives the spectral form of (1.5) for O .k; t /,
(5.6)
O t .k; t / C i k
Z
ı.k C k2 k3 k4 /ƒ.k; k2 ; k3 ; k4 /
O .k2 ; t/ O .k3 ; t/ O .k4 ; t /d k2 d k3 d k4 D 0
for k > 0;
where ƒ is defined in (5.4). This equation describes the evolution of a nonlinear wave due to four-wave resonant interactions (2.2) with the interaction coefficient ƒ. One can also verify the result directly by taking the Fourier transform
of (1.5).
NONLINEAR WAVES WITH CONSTANT FREQUENCY
5.3 Real Form
To write the complex equation (1.5) for
(5.7)
It follows that
v.x; t / D
317
in an equivalent real form, we define
.x; t /e i t C
.x; t /e i t :
is given in terms of v by
1
.x; t / D fv.x; t / C iHŒv.x; t /ge i t :
2
The function v corresponds to the leading-order approximation for u in the expansion (4.3) and to the leading-order approximation for the displacement of a
vorticity discontinuity in (B.1).
We differentiate (5.7) with respect to t , then use (1.5) and (5.8) to express the
result in terms of spatial derivatives of v. After some algebra, the details of which
we omit, we find that v.x; t / satisfies the equation
1
1
1
(5.9)
v t C @x j@x jŒv 3 vj@x jŒv 2 C v 2 j@x jŒv D HŒv:
6
2
2
(5.8)
Equation (5.9) has the Hamiltonian form
ıK
D 0;
v t C @x
ıv
(5.10)
Z 1
1 3
1 2
1
2
K.v/ D
vj@x j Œv C v j@x jŒv v j@x jŒv dx:
2
6
8
6 Near-Identity Transformation
In this section, we derive the real form (5.9) of the asymptotic equation from the
Burgers-Hilbert equation (1.1) by a symplectic near-identity transformation that
removes the quadratically nonlinear terms from (1.1), which are nonresonant, and
gives the cubically nonlinear terms in (5.9).
We introduce an amplitude parameter " and write the Hamiltonian in (1.2) for
the Burgers-Hilbert equation (1.5) as
H D H2 C "H3 ;
where Hj is homogeneous of degree j in u. Explicitly,
Z
Z
1
1 3
1
uj@x j Œudx; H3 .u/ D
u dx:
(6.1)
H2 .u/ D
2
6
We define a symplectic, near-identity transformation v 7! u."/ by
ıF
D 0; u.0/ D v;
u" C @x
ıu
where the generating functional F has the form
F.u/ D F3 .u/ C "F4 .u/;
318
J. BIELLO AND J. K. HUNTER
with Fj homogeneous of degree j in u. Then, since H" D fH; Fg; where f ; g
denotes the Poisson bracket (5.2), a Taylor expansion with respect to " gives
H.u."//
(6.2)
D H2 .v/ C ".H3 C fH2 ; F3 g/.v/
1
2
C " fH2 ; F4 g C fH3 ; F3 g C ffH2 ; F3 g; F3 g .v/ C O."3 /:
2
To eliminate the cubically nonlinear terms from H.u/, we choose
Z
1
F3 .v/ D HŒv3 dx:
6
Using (A.2) and the fact that the Hilbert transform is a skew-adjoint isometry
on L2 , we compute that
fH2 ; F3 g D H3 :
Thus, the cubic terms in (6.2) cancel, and we find that
(6.3)
(6.4)
(6.5)
H.u."// D H2 .v/ C "2 K4 .v/ C O."3 /;
e 4 .v/;
K4 .v/ D fH2 ; F4 g.v/ C H
Z 1
2 3
e 4 .v/ D 1
H
v j@x jŒv HŒv2 C v 2 j@x jŒv 2 dx:
12
3
2
The choice of F4 in the near-identity transformation and the form of the fourthdegree terms in the Hamiltonian is not uniquely determined. We choose
Z 1
1
2
2 2
vx HŒvHŒv HŒv .v /x dx;
(6.6)
F4 .v/ D
6
8
which leads to the asymptotic Hamiltonian. Using (6.1), (6.6), and (A.2)–(A.3),
we compute that
Z 1
4 3
2
2
2
2
(6.7) fH2 ; F4 g.v/ D
HŒv j@x jŒv v j@x jŒv C v j@x jŒv dx:
12
3
The use of (6.5) and (6.7) in (6.4) then gives, after some simplification, that
Z 1
1 2
1 3
2
(6.8)
K4 .v/ D
v j@x jŒv v j@x jŒv dx:
8
1 6
Neglecting terms of higher order than the quartic terms, setting the amplitude parameter " equal to 1, and using (6.1) and (6.8) in (6.3), we see that the near-identity
transformation v 7! u generated by F transforms the Hamiltonian equations (1.2)
into (5.10).
Up to quadratic terms, generated by F3 , the near-identity transformation is
given by
1
u D v j@x j.HŒv2 / C :
2
NONLINEAR WAVES WITH CONSTANT FREQUENCY
319
One may verify directly using (A.2) that this transformation eliminates the quadratically nonlinear terms from (1.1).
7 Stability of Periodic Waves
The harmonic-wave solution (1.8)–(1.9) is a constant-frequency analogue of the
Stokes wave solution for the NLS equation. A significant difference, however, is
that the dependence of the frequency on the wave amplitude A is multiplicative
rather than additive.
A single positive wave number harmonic is an exact solution of (1.5) because
four-wave resonant interactions of the form k C k k do not generate any new
spatial harmonics. On the other hand, resonant interactions that involve more than
one spatial harmonic generate infinitely many new harmonics (for example, 3k D
2k C 2k k, 4k D 3k C 2k k, and so on).
7.1 Linearized Stability
To study the linearized stability of the solution (1.8)–(1.9), we normalize A D
k D ! D 1, without loss of generality, and write
.x; t / D e i.xt / Œ1 C .x; t /:
(7.1)
Using this expression in (1.5) and linearizing the result with respect to , we get
t i D e ix PŒe ix .j@x jŒm m C i x /x
where m D C . We write this equation as
t i D @Qx PQ j@x jΠC j@x jΠC i x 2
where
@Qx D e ix @x e ix D @x C i I; PQ D e ix Pe ix :
O t/
The Fourier transform O .k; t / is supported in 0 < k < 1, so from (7.1) .k;
Q
is supported in 1 < k < 1. It follows that PΠD , and therefore
t i D @Qx PQ j@x jΠC .@x C i /.j@x jΠC i x 2/:
To compute the action of PQ on , we decompose as
.x; t / D .x; t/ C .x; t /
where
Z
.x; t/ D
Then
1
1
O t/e i kx d k;
.k;
Z
1
.x; t / D
O t /e i kx d k:
.k;
1
@Qx PQ j@x jΠD .@x C i I/ j@x jΠand
(7.2) t C 3x i j@x jŒ C i D j@x jŒx C i xx C .@x C i /.j@x jŒ /:
320
J. BIELLO AND J. K. HUNTER
Projecting (7.2) onto Fourier components with 1 < k < 1 and using the fact
that j@x jΠD ix , we get
t C 2x C i D 0:
Thus perturbations with wave number k > 1 are independent of the other modes.
They are stable, with velocity 2 and frequency 1.
Projecting (7.2) onto Fourier components with 1 < k < 1, we get
(7.3)
t C 3x i j@x jŒ C i D j@x jŒx C i xx C .@x C i /.j@x jŒ /:
Thus, if 0 < k < 1, the k and k modes are coupled through their interaction with
the unperturbed wave.
To solve (7.3), we write
.x; t/ D .x; t / C .x; t /
where
Z
1
.x; t / D
O t/e i kx d k;
.k;
Z
1
.x; t / D
0
O .k/e i kx d k:
0
Then HΠD i, HΠD i , and
j@x jΠD ix C i x :
Using this equation in (7.3), projecting the result onto Fourier components with
0 < k < 1 and 1 < k < 0, respectively, and simplifying the result, we get
t C 2x C i C i .@2x C 1/ D 0;
t C 4x i C 2ixx C i .@x i /2 D 0:
We look for Fourier solutions of this system of the form
.x; t / D M e i kxi !t ;
.x/ D Ne i kxi !t ;
where 0 < k < 1. Then
! C 2k C 1
1 k2
M
D 0:
N
.1 k/2
! 1 C 4k 2k 2
Setting the determinant of the matrix in this equation equal to 0 and introducing
the phase speed c D !=k, we get
c 2 C 2.k 3/c C 6 2k k 2 D 0:
The solutions are real, with
(7.4)
q
c D .3 k/ ˙ 1 C 2.1 k/2 :
It follows that the harmonic solutions (1.8)–(1.9) of (1.5) are linearly stable to small
perturbations.
We remark that in this analysis, unlike the analogous analysis of dispersive
waves using the NLS equation, the perturbations are not assumed to be of long
NONLINEAR WAVES WITH CONSTANT FREQUENCY
321
wavelength relative to the wavelength of the carrier wave. Their frequency is,
however, close to the frequency of the carrier wave.
7.2 Semiclassical Approximation
To derive a semiclassical approximation for (1.5), it is convenient to proceed
informally. We write
(7.5)
.x; t / D a.x; t /e iS.x;t /
where a and S are real-valued functions. Using (7.5) in (1.5), we get
(7.6)
i aS t C a t D e iS P faj@x jŒa2 a3 Sx C i a2 ax ge iS x :
We suppose that phase S varies much more rapidly than a, so that the spectrum of
is concentrated near Sx . Assuming that Sx > 0, we have for any slowly varying
function b that
PŒbe iS be iS :
We may therefore approximate (7.6) by
i aS t C a t D iSx faj@x jŒa2 a3 Sx C i a2 ax g
C faj@x jŒa2 a3 Sx C i a2 ax gx :
Expanding derivatives and equating real and imaginary parts in this equation, we
get
S t C a2 Sx2 D Sx j@x jŒa2 C aaxx C 2ax2 ;
a t C 4Sx a2 ax C a3 Sxx D .aj@x jŒa2 /x :
Introducing n D a2 and k D Sx , we may write these equations as
1
n2x
2
;
k t C .k n/x D kj@x jŒn C nxx C
2
4n x
(7.7)
n t C .2k n2 /x D 2nj@x jŒnx C nx j@x jŒn:
The terms on the right-hand side of (7.7) are small in the semiclassical limit.
The leading-order semiclassical equations are therefore
(7.8)
k t C .k 2 n/x D 0;
n t C .2k n2 /x D 0:
These equations form a hyperbolic system for .k; n/,
k
k
2k n k 2
(7.9)
C
D 0:
n t
2n2 4k n n x
The eigenvalues and eigenvectors R of the matrix in (7.9) are given by
k
D k n; R D
;
. 2/n
322
J. BIELLO AND J. K. HUNTER
where
(7.10)
D3˙
p
3:
The hyperbolicity of the semiclassical equations means that periodic wave trains
are modulationally stable and is consistent with
p the linearized stability of periodic
waves. The characteristic
velocities
D
3
˙
3 of (7.8) at k n D 1 agree with the
p
velocities in c D 3˙ 3 at k D 0 in (7.4), both of which limits describe linearized,
long-wave perturbations of the solution .x; t / D e ixi t .
Riemann invariants ˆ of (7.9) are given by
ˆ.k; n/ D k n. 2/=2 :
It follows that f.k; n/ W k > 0; n > 0g is an invariant region for smooth solutions
of (7.9), consistent with the assumption made in deriving the system that k; n > 0.
The characteristics of (7.9) are genuinely nonlinear for k n ¤ 0, with
p
(7.11)
r R D ˇk n; ˇ D 9 ˙ 5 3:
Thus, in general, semiclassical solutions steepen until the higher-order dispersive
terms on the right-hand side of (7.7) become important.
The dominant long-wave dispersive terms in (7.7) are the ones proportional to
j@x jŒnx D HŒuxx . Omitting the explicit introduction of a small parameter, we
find that small-amplitude, long-wave perturbations of k, n about k D 1, n D 1 are
given by
k.x; t/ D 1 C u.x; t / C ;
n.x; t / D 1 C . 2/u.x; t / C ;
where u.x; t / satisfies a Benjamin-Ono equation
(7.12)
u t C ux C ˇuux D ˛HŒuxx :
Here is the linearized wave speed (7.10), ˇ is the genuine nonlinearity coefficient
(7.11), and
p
2 3
˛ D1˙
3
is a linear dispersive coefficient. This value agrees with the long-wave expansion
of the linearized phase velocity (7.4), which is c D ˛jkj C O.k 2 / as k ! 0.
The soliton solutions of (7.12) are
a
4˛
u.x; t / D
ˇ .x ct /2 C a2
where a is a large, positive parameter and
˛
:
a
For the .C/-branch, with ˛ > 0, the solitons travel faster than periodic long waves,
while for the ./-branch, with ˛ < 0, the solitons travel slower than periodic long
waves. The solitons have the same phase velocity as a periodic short wave with
cDC
NONLINEAR WAVES WITH CONSTANT FREQUENCY
323
p
wave number close to D 4 ˙ 2 3. This suggests that solitary waves corresponding to the BO-solitons may not exist in the full asymptotic equation due to
the radiation of short waves, but we will not pursue that question further here.
8 Numerical Solutions
We used a pseudospectral method to numerically integrate both the BurgersHilbert equation (1.1) and the asymptotic equation (1.5). We computed the spatial
derivatives and the Hilbert transform spectrally and carried out the time integration
by means of a fourth-order Runge-Kutta scheme. The numerical solutions show
that small-amplitude solutions of the Burgers-Hilbert equation are well described
by the corresponding solutions of the asymptotic equation. They also show that
solutions of the asymptotic equation steepen and form a singularity.
Using 212 points for x 2 Œ0; 2
, we could integrate the equations quickly in
MATLAB up to the formation time of the singularity, which will be discussed
below. For both equations, energy was conserved to within a relative error of less
than 107 for the duration of the integration, and for the asymptotic equation, the
momentum and action were conserved to within a relative error of less than 108 .
We performed several integrations of the Burgers-Hilbert equation with the same
initial profile scaled to different amplitudes:
1
2
(8.1)
u.x; 0/ D A cos x C cos Œ2.x C 2
/
2
where A is a real constant. This data has maximum negative slope
D jmin ux .x; 0/j
where D cA with c 1:225.
A linear timescale Th for (1.1) is given by the period Th D 2
of solutions
of the linearized equation (4.2). A nonlinear timescale Tb is given by the shock
formation time Tb D 1 for the inviscid Burgers equation. Thus,
Th
Tb
is an appropriate parameter for measuring the effect of nonlinearity on solutions
of the Burgers-Hilbert equation. If is small, which is the regime in which the
asymptotic equation applies, then singularities form over the course of many oscillations, while if is large, they form quickly relative to the period of a single
oscillation.
For smaller-amplitude data, with 2
1:2, the solutions evolve in a qualitatively similar fashion. Over shorter times, they approximately oscillate with period 2
between a profile and its Hilbert transform. Over longer times, the profile
deforms and steepens until a singularity in the derivative ux develops.
Figure 8.1 shows a numerical solution of (1.1) for the initial data (8.1) with
A D 102 and 1:2 102 . In the top frame of Figure 8.1, the solid line
is the initial data and the dashed line is the solution at t D 2 . The dashed line
2
D
324
J. BIELLO AND J. K. HUNTER
F IGURE 8.1. A numerical solution of the Burgers-Hilbert equation (1.1)
for the initial condition (8.1) with A D 0:01. (a) The solid line shows
u.x; 0/. The dashed line shows the solution u.x; 2 /, one quarter period
after the initial condition. (b) The solution for the initial condition in (a)
just prior to the formation of a singularity. The solid and dashed line
show u.x; Ts / for Ts D 2
n for n D 104 and n D 104 C 14 , respectively.
is almost equal to the Hilbert transform of the solid line. In the bottom frame of
Figure 8.1, the solid line shows the solution u.x; t / at t D 2
n with n D 104 , just
prior to the development of the singularity, and the dashed line shows the solution
at t D 2
n C 2 . The dashed curve is again almost equal to the Hilbert transform
of the solid curve.
Several snapshots in time of u.x; t / are shown in Figure 8.2, where the times of
each slice are integer multiples of the linearized oscillation period. The singularity
develops ahead of a peak in u. The solution steepens in a similar fashion to solutions of the Burgers equation, but the Burgers-Hilbert equation has an additional
fast oscillation that has been factored out of the plot.
As the singularity forms, the solution of the Burgers-Hilbert equation develops
a small-scale structure, which is shown in Figure 8.3. We determined the time Ts
and location of singularity formation numerically. The solid lines show the solution
near the location of the singularity at times Ts 2
, Ts , and Ts C 2
. The dashed
lines show the solution at times Ts 1:5
, Ts C0:5
, and Ts C2:5
. In both phases,
the wave profile forms a small dip ahead of its steepest part, followed by a corner,
or “knee,” after which the profile becomes relatively flat. The integration can be
continued to times slightly beyond the development of the singularity, presumably
NONLINEAR WAVES WITH CONSTANT FREQUENCY
325
F IGURE 8.2. Snapshots of u.x; t / for the solution in Figure 8.1. Time
increases to the foreground from the initial condition to the development
of the singularity, and each time slice is taken at an integer multiple of
2
in order to factor out the fast oscillation. The singularity develops as
a steepening ahead of the maximum of ux .x; 0/.
because de-aliasing in the pseudospectral method introduces a small amount of
numerical dissipation and dispersion.
For larger-amplitude data, with 2
1:7, the qualitative nature of the solution is dramatically different. The characteristic time of the nonlinear term,
Tb D 1 < 3:7, is then significantly less than the period 2
of the linear term. The
solutions do not oscillate between one profile and its Hilbert transform; instead,
they develop shocks in a similar way to solutions of the Burgers equation. In this
regime, the asymptotic equation no longer provides a good approximation for the
evolution of u.
The transition between these two qualitatively different regimes is remarkably
rapid. For example, when 2
D 1:154, the singularity forms at time Ts 77:9
in the twelfth oscillation of period 2
; but when 2
D 1:731, the singularity
forms at Ts 3:94 in the first oscillation. Thus, once the data is small enough
that a singularity does not form in the first few oscillations, the effect of the nonlinearity becomes much weaker as a result of the alternation between compression
and expansion in each oscillation, leading to a greatly increased lifespan of smooth
solutions.
326
J. BIELLO AND J. K. HUNTER
F IGURE 8.3. Magnification of u.x; t / near the singularity for the solution in Figure 8.1. The solid curves show the solution one period before,
at, and after the time of singularity formation. The dashed curves show
the solution a quarter period before, a quarter period after, and one and a
quarter periods after the time of singularity formation; they are approximately the Hilbert transforms of the solid curves.
Figure 8.4 shows a log-log plot of the numerically computed singularity formation time Ts against 2
for several integrations. The equation
(8.2)
2 Ts D 2:37
provides an excellent fit to the numerical values for 2
1. The scaling Ts k 2 as ! 0 agrees with the cubically nonlinear scaling used to derive the
asymptotic equation (1.5) from equation (1.1), and the value k 2:37 is obtained
from the numerical integration of the asymptotic equation described below.
For large values of , we expect that (1.1) should behave like the inviscid Burgers
equation and that
Ts 1
as ! 1:
This line is also plotted on Figure 8.4; the agreement with the numerical values is
excellent for 2
2.
In the transition regime, 1:2 < 2
< 1:7, the singularity formation time Ts
appears to increase in a series of steps as decreases. These results suggest that
Ts W .0; 1/ ! R is a decreasing, lower-semicontinuous function of with jump
discontinuities at a decreasing sequence fn W n 2 Ng of values of , where n
is the largest value of for which the singularity forms in the nth oscillation. We
NONLINEAR WAVES WITH CONSTANT FREQUENCY
327
F IGURE 8.4. Logarithm of the singularity formation time Ts for the
Burgers-Hilbert equation versus the logarithm of 2
for the numerical
experiments (blue dots) along with the prediction from the asymptotic
equation, Ts D 2:37 2 (steeper line) and the prediction from the Burgers equation Ts D 1 (shallower line).
further expect that n ! 0 as n ! 1 and
Ts .nC1 / Ts .n / 2
:
Next we describe a numerical solution of the asymptotic equation. We integrated
(1.5) for .x; t / with initial data
1 2i.xC2 2 /
:
e
2
The corresponding asymptotic solution of the Burgers-Hilbert equation (1.1) with
initial data (8.1) is
1 2
1
u.x; t / A x; A t e i t C c:c: as A ! 0.
2
4
(8.3)
.x; 0/ D e ix C
The real and imaginary parts of .x; t / are shown in Figure 8.5 just prior to the
development of the singularity at t D s where s 0:395. Since u oscillates
between the real and imaginary parts of
on the fast time scale, the curves in
Figure 8.5 should be compared to the corresponding curves in Figure 8.1(b). While
=. / shows a sharper feature right at the point of the singularity, the two figures
otherwise agree extremely well with each other.
328
J. BIELLO AND J. K. HUNTER
F IGURE 8.5. The solution of the asymptotic equation (1.5) with initial
data (8.3) at the onset of the singularity. The solid line is <. /; the
dashed line is =. /. Compare with Figure 8.1(b).
The asymptotic theory predicts that the singularity formation time Ts for the
Burgers-Hilbert equation has the asymptotic behavior
Ts 4s A2
Setting A D
c
as A ! 0:
and using the numerically computed value for s , we get
Ts 2:37 2
as ! 0;
which is the relation used to fit the Burgers-Hilbert data in equation (8.2). The
mean of 2 Ts for the five values of from the Burgers-Hilbert data whose Ts lie
near the asymptotic fit is given by
2 Ts D 2:47:
This differs by 4% from the coefficient predicted by the asymptotic equation; moreover, the difference is less for the smaller values of . Thus, the solution of the
asymptotic equation is in excellent quantitative agreement with the solutions of the
Burgers-Hilbert equation.
Appendix A: Notation
In this appendix, we define and summarize notation that is used throughout the
paper. For definiteness, we consider square-integrable functions defined on R.
Similar considerations apply to periodic functions, in which case we project constant Fourier modes to 0.
NONLINEAR WAVES WITH CONSTANT FREQUENCY
329
We denote the Fourier transform of f 2 L2 .R/ by fO 2 L2 .R/, where
Z 1
f .x/ D
fO.k/e i kx d k:
The Hilbert transform H W
L2 .R/
1
1
! L2 .R/
is defined by
HŒf .k/ D i.sgn k/fO.k/;
where
8
ˆ
<C1 if k > 0,
sgn k D 0
if k D 0,
:̂
1 if k < 0.
(A.1)
Equivalently,
1
HŒf D p: v:
f:
x
The Hilbert transform is a skew-adjoint isometry on L2 .R/, and H2 D I.
If u W R ! R is a real-valued L2 -function, then u C i HŒu is the boundary
value on the real axis of a function that is holomorphic in the upper half-plane with
uniformly bounded L2 -norms on lines with constant positive imaginary part. If F
and G are holomorphic functions whose boundary values have real parts v and w,
respectively, then a consideration of the holomorphic functions F G and F 3 shows
that, under suitable assumptions on v and w,
(A.2)
H vw HŒvHŒw D vHŒw C wHŒv;
1
1 3
2
3
2
(A.3)
v HŒv HŒv D H v vHŒv :
3
3
For example, it is sufficient that v; w 2 Lp for p > 2 in (A.2), and v 2 Lp for
p > 3 in (A.3).
We define orthogonal self-adjoint projections
P; Q W L2 .R/ ! L2 .R/
onto positive and negative wave number components by
1
1
(A.4)
P D .I C iH/; Q D .I i H/:
2
2
Then
PŒf D QŒf ; QŒf D PŒf ;
where the star denotes the complex conjugate, and
Z 1
Z 1
PŒf .x/ g.x/dx D
f .x/ QŒg.x/dx:
1
1
We denote the differentiation operator by @x , and define
j@x j D H@x ;
2
j@x jŒf .k/ D jkjfO.k/:
330
J. BIELLO AND J. K. HUNTER
Appendix B: Surface Waves on a Vorticity Discontinuity
In this appendix, we study surface waves on an interface between two halfspaces with constant vorticities in a two-dimensional, inviscid, incompressible
fluid shear flow. We suppose that the unperturbed interface is located at y D 0, and
denote the vorticities in y > 0 and y < 0 by ˛C and ˛ , respectively, where
˛C ¤ ˛ . We denote the location of the perturbed interface by y D .x; tI "/.
First, we summarize the asymptotic solution. The location of the vorticity discontinuity is given by
(B.1) .x; tI "/ D "fF .x; "2 t /e i !0 t C F .x; "2 t /e i !0 t g C O."2 / as " ! 0;
where F .x; / satisfies
(B.2)
F D 0 P@x ŒF j@x jŒFF C iFF Fx ;
PŒF D F:
The frequency parameters !0 and 0 in (B.1)–(B.2) are given by
˛C ˛
!0 D
;
2
(B.3)
2
2
C ˛
˛C
:
0 D
˛C ˛
Equation (B.2) with F D and 0 D t is equivalent to (1.5).
After accounting for the coefficients, we find that (B.2) is identical to the asymptotic equation derived in Section 4 for the Burgers-Hilbert equation
(B.4)
t C
1
ˇ0 2
2
D !0 HŒ;
x
ˇ02 D
2
2
C ˛
˛C
:
2
Either choice of sign for ˇ0 in (B.4) leads to the same asymptotic equation (B.2).
This is because the transformation 7! corresponds to a half-period phase shift
in the linearized oscillations and does not affect the averaged, long-time dynamics.
Since (B.4) leads to the same asymptotic equation as the one derived from the
primitive fluid equations, it provides an effective equation for the motion of a planar vorticity discontinuity with slope of the order " over times of the order !01 "2 .
The coefficient ˇ0 in (B.4) is not equal to the coefficient ˛0 of the quadratic nonlinearity in the equations of motion for a vorticity discontinuity, which, from (B.15)
below, is given by
(B.5)
˛0 D
˛C C ˛
:
2
Instead, ˇ0 describes the combined effect of both quadratic and cubic nonlinearities. For example, if the shear flow is symmetric and ˛C D ˛ , then ˇ0 D ˛C ,
even though ˛0 D 0; while
p if the flow in the lower half-space is irrotational and
˛ D 0, then ˇ0 D ˛C = 2 and ˛0 D ˛C =2.
NONLINEAR WAVES WITH CONSTANT FREQUENCY
331
B.1: Formulation
Consider a velocity perturbation .u; v/ of an unperturbed shear flow .˛y; 0/,
where .x; y/ are spatial coordinates and ˛ is a constant shear rate. The twodimensional incompressible Euler equations for .u; v/ and the pressure p are
u t C .˛y C u/ux C v.˛ C uy / C px D 0;
v t C .˛y C u/vx C vvy C py D 0;
ux C vy D 0:
Since the vorticity is advected by the perturbed flow, and the unperturbed vorticity
is constant (equal to ˛), it is consistent to assume that the flow perturbations are
irrotational.
We therefore introduce a velocity potential .x; y; t / such that
v D y ;
u D x ;
and a stream function
.x; y; t / such that
uD
y;
vD
x:
Then the incompressibility condition implies that
D 0;
(B.6)
and an integration of the momentum equations gives
1
/ C jrj2 C p D 0:
2
Next, we consider perturbations of a shear flow given by .˛C y; 0/ in y > 0 and
.˛ y; 0/ in y < 0, where the shear rates ˛˙ are distinct constants. The unperturbed
flow has a jump in vorticity across y D 0. We assume that the flow perturbations
are irrotational and that the interface is a graph. We write the perturbed location of
the interface where the vorticity jumps as
t C ˛.yx (B.7)
y D .x; t /:
We use the notation
(B.8)
(
˛D
˛C
˛
if y > .x; t /,
if y < .x; t /,
with similar notation for other quantities that jump across the interface.
The kinematic boundary condition states that the interface moves with the fluid,
meaning that
(B.9)
t C .˛ C x /x y D 0 on y D .x; t /˙ :
The dynamic boundary condition states that the pressure is continuous across the
interface, meaning that
(B.10)
Œp D 0;
332
J. BIELLO AND J. K. HUNTER
where
Œf .x; t / D f .x; .x; t /C ; t / f .x; .x; t / ; t /
denotes the jump in a quantity f .x; y; t / across y D .x; t /. Using (B.7) in (B.10),
we get
1
2
(B.11)
t C ˛.yx / C jrj D 0:
2
Finally, we require that the flow perturbation and the pressure decay to zero
away from the interface. This condition, together with (B.6), gives
(B.12)
D 0
in y > .x; t /, y < .x; t /;
.x; y; t / ! 0
as y ! ˙1:
The full set of equations consists of (B.9), (B.11), and (B.12), where
harmonic conjugate of such that ! 0 as y ! ˙1.
is the
B.2: Linearized Equations
Linearizing (B.9), (B.11), and (B.12) around the unperturbed solution D 0,
D 0, we get
D 0
in y > 0 and y < 0;
t y D 0
on y D 0˙ ;
Œ t ˛ D 0;
.x; y; t / ! 0
as jyj ! 1;
where Πnow denotes a jump across y D 0. Taking the jump of the kinematic
boundary condition t y D 0 across y D 0, we get
Œy D 0:
A Fourier solution of D 0 that decays as jyj ! 1 has the form
(
AOC .k/e i kxjkjyi !t in y > 0;
.x; y; t / D
AO .k/e i kxCjkjyi !t in y < 0;
where k 2 R. We write this as
i kx jkjyi !t
O
.x; y; t / D A.k/e
;
where AO D AO˙ , and
(
D
C1
1
if y > 0;
if y < 0:
The corresponding stream function and interface displacement are
i kxjkjyi !t
O
;
.x; y; t / D B.k/e
.x; t / D FO .k/e i kxi !t ;
NONLINEAR WAVES WITH CONSTANT FREQUENCY
333
where
k O
FO .k/ D B.k/:
!
O we find that
Using these solutions in the jump conditions and eliminating A,
O D 0; ŒB
O D 0:
Œf! sgn.k/ ˛gB
O
O
A.k/
D i sgn.k/B.k/;
(B.13)
It follows that BO is continuous across y D 0, and
! D !0 sgn.k/
(B.14)
where the frequency !0 is given in (B.3).
Superposing Fourier solutions, we get the linearized solution with general spatial dependence,
.x; y; t / D A.x; y/e i !0 t C A .x; y/e i !0 t
.x; y; t / D B.x; y/e i !0 t C B .x; y/e i !0 t ;
.x; t/ D F .x/e i !0 t C F .x/e i !0 t :
Here the complex amplitudes A, B, and F consist of positive wave numbers,
Z 1
i kx ky
O
A.x; y/ D
d k;
A.k/e
B.x; y/ D
Z0 1
Z
0
F .x/ D
1
i kx ky
O
d k;
B.k/e
FO .k/e i kx d k;
0
and, from (B.13),
i
Bx .x; 0/:
!0
We remark that the linearized tangential fluid velocity U on the interface is
the sum of the x-velocity components of the unperturbed shear flow and the flow
perturbation, so
ˇ
U D ˛ C x ˇyD0 :
A.x; y/ D i sgn.y/B.x; y/;
F .x/ D
We compute from the linearized solution that U is continuous across the interface,
as it must be, and is given by
(B.15)
U.x; t / D ˛0 .x; t /
where ˛0 is defined in (B.5). Thus, since .x; t / has zero mean with respect to t ,
the time-averaged translation of the interface in the x-direction is 0 according to
the linearized theory. In the symmetric case ˛0 D 0, the linearized x-velocity is
identically 0. In the general case, the advection of the interface in one direction
for positive displacements is canceled by its advection in the opposite direction for
negative displacements. There is, however, a nonlinear Stokes drift of the interface
that is second-order in the amplitude.
334
J. BIELLO AND J. K. HUNTER
For use in the asymptotic expansion, we state a solvability condition for the
nonhomogeneous linearized problem,
D 0
in y > 0 and y < 0;
Œy D f .x/e i n!0 t
(B.16)
Œ t ˛ D g.x/e
on y D 0;
i n!0 t
.x; y; t / ! 0
on y D 0;
as jyj ! 1:
P ROPOSITION B.1 If n2 ¤ 1, then (B.16) is solvable for for any functions
f; g 2 L2 .R/. If n D 1, then (B.16) is solvable if and only if
PŒ˛0 f C g D 0;
(B.17)
where ˛0 is defined in (B.5) and P is defined in (1.4).
P ROOF : Write .x; y; t / D ˆ.x; y/e i n!0 t , take the Fourier transform of
(B.16) with respect to x, and solve the resulting equations. The details are omitted.
B.3: Weakly Nonlinear Solution
We look for an asymptotic solution of (B.9), (B.11), and (B.12), depending on a
small parameter ", of the form
" .x; y; t / D "1 .x; y; "2 t; t / C "2 2 .x; y; "2 t; t / C "3 3 .x; y; "2 t; t / C ;
"
.x; y; t / D "
1 .x; y; "
2
t; t / C "2
2 .x; y; "
2
t; t / C "3
3 .x; y; "
2
t; t / C ;
" .x; t / D "1 .x; "2 t; t/ C "2 2 .x; "2 t; t / C "3 3 .x; "2 t; t / C :
The velocity potentials n and stream functions
n D 0;
ny
D nx ;
n
satisfy
nx
D ny :
Expansion and simplification of the boundary conditions yields the following perturbation equations: at the order ",
1t 1y D 0;
Œ1t ˛
1
D 0I
at the order "2 ,
2t 2y C ˛1 1x C 1x 1x 1 1yy D 0;
1
2
2t ˛ 2 C 1 1ty C jr1 j D 0I
2
and, at the order "3 ,
3t 3y C 1 C ˛.1 2x C 2 1x / C 1x 2x C 2x 1x
1
C 1 1x 1xy 1 2yy 2 1yy 21 1yyy D 0;
2
NONLINEAR WAVES WITH CONSTANT FREQUENCY
3t ˛
3
335
1 2
1tyy
2 1
C r1 r2 C 1 r1 r1y D 0:
C 1 C 2 1ty C 1 2ty C
1 2
˛
2 1
1xx
Here all boundary conditions and jumps are evaluated at y D 0.
The solution of the first-order equations is
1 .x; y; ; t / D A.x; y; /e i !0 t C A .x; y; /e i !0 t ;
1 .x; y; ; t /
D B.x; y; /e i !0 t C B .x; y; /e i !0 t ;
1 .x; ; t / D F .x; /e i !0 t C F .x; /e i !0 t ;
where B.x; y; / is continuous across y D 0 with PŒB.x; 0; / D 0, and
A.x; y; / D i B.x; y; /;
F .x; / D i
Bx .x; 0; /:
!0
Using these expressions in the second-order equations and solving the result, we
find that
2 .x; y; ; t / D C.x; y; /e 2i !0 t C M.x; y; / C C .x; y; /e 2i !0 t ;
2 .x; y; ; t /
D D.x; y; /e 2i !0 t C N.x; y; / C D .x; y; /e 2i !0 t ;
2 .x; ; t / D G.x; /e 2i !0 t C G .x; y; /e 2i !0 t ;
where
˛
B 2 .x; y; /;
2!02 x
ˇ
N.x; y; t/ D 2 Bx .x; y; /Bx .x; y; /
!0
i ˛0
fB 2 .x; 0; /gx ;
G.x; / D
2!03 x
D.x; y; / D with
(
ˇD
˛
˛C
if y > 0,
if y < 0,
C D i D;
Mx D Ny ;
My D Nx :
We use these expressions in the third-order equations, collect terms proportional
to e i !0 t , and impose the solvability condition (B.17). After writing this condition
in terms of F and simplifying the result, we find that F .x; / satisfies (B.2).
Acknowledgment. JB was partially supported by National Science Foundation
Grant DMS-0604947. JKH was partially supported by National Science Foundation Grant DMS-0607355.
336
J. BIELLO AND J. K. HUNTER
Bibliography
[1] Alì, G.; Hunter, J. K. Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics. Quart. Appl. Math. 61 (2003), no. 3, 451–474.
[2] Alì, G.; Hunter, J. K.; Parker, D. Hamiltonian equations for scale invariant waves. Stud. Appl.
Math. 108 (2002), no. 3, 305–321.
[3] Chemin, J.-Y. Two-dimensional Euler system and the vortex patches problem. Handbook of
mathematical fluid dynamics. Vol. III, 83–160. North-Holland, Amsterdam, 2004.
[4] Constantin, P.; Lax, P.; Majda, A. J. A simple one-dimensional model for the three-dimensional
vorticity equation. Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724.
[5] Dritschel, D. G. The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid
Mech. 194 (1988), 511–547.
[6] Hamilton, M. F.; Il0 insky, Y. A.; Zabolotskaya, E. A. Evolution equations for nonlinear Rayleigh
waves. J. Acoust. Soc. Amer. 97 (1995), no. 2, 891–897.
[7] Hunter, J. K. Asymptotic equations for nonlinear hyperbolic waves. Surveys in applied mathematics. Vol. 2, 167–276. Plenum, New York, 1995.
[8] Liu, H. Wave breaking in a class of nonlocal dispersive wave equations. J. Nonlinear Math.
Phys. 13 (2006), no. 3, 441–466.
[9] Majda, A. J.; Bertozzi, A. L. Vorticity and incompressible flow. Cambridge Texts in Applied
Mathematics, 27. Cambridge University Press, Cambridge, 2002.
[10] Marsden, J.; Weinstein, A. Coadjoint orbits, vortices, and Clebsch variables for incompressible
fluids. Phys. D 7 (1983), no. 1-3, 305–323.
[11] Saffman, P. G. Vortex dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, New York, 1992.
[12] Whitham, G. B. Linear and nonlinear waves. Wiley-Interscience, New York–London–Sydney,
1974.
[13] Zakharov, V. E.; Falkovich, G.; L0 vov, V. S. Kolmogorov spectra of turbulence. I. Wave turbulence. Springer, Berlin–New York, 1992.
J OSEPH B IELLO
University of California at Davis
Department of Mathematics
One Shields Avenue
Davis, CA 95616
E-mail: biello@
math.ucdavis.edu
Received January 2009.
J OHN K. H UNTER
University of California at Davis
Department of Mathematics
One Shields Avenue
Davis, CA 95616
E-mail: [email protected]