Asymptotic Linear Stability of the Benney-Luke equation in 2D

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by  Tetsu Mizumachi

ASYMPTOTIC LINEAR STABILITY OF THE BENNEY-LUKE
EQUATION IN 2D
TETSU MIZUMACHI ANDYUSUKESHIMABUKURO
Abstract. In this paper, we study transverse linear stability of line solitary waves to the
2-dimensionalBenney-Lukeequationwhicharisesinthestudyofsmallamplitudelongwater
waves in 3D. In the case where the surface tension is weak or negligible, we find a curve of
7
resonant continuous eigenvalues near 0. Time evolution of these resonant continuous eigen-
1
0 modes is described by a 1D damped wave equation in thetransverse variable and it gives a
2 linearapproximationofthelocalphaseshiftsofmodulatinglinesolitarywaves. Inexponen-
tially weighted space whose weight function increases in the direction of the motion of the
n
a linesolitarywave,theotherpartofsolutionstothelinearizedequationdecaysexponentially
J as t→∞.

2
1

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P
A
.
h
Contents
t
a
m
1. Introduction 2
[ 2. Statement of the result 4

1 3. Resonant modes of the linearized operator 7
v
3.1. Spectral stability in the KP-II scaling regime 7
0
9 3.2. Resonant modes 10
3 4. Properties of the free operator 18
0
3 L
5. Spectral stability for small line solitary waves 25
0
. 5.1. Spectral stability for high frequencies in y 25
1
0 5.2. Spectral stability for low frequencies in y 28
7
5.3. Proof of Theorem 2.4 34
1
: 6. Proof of Theorem 2.1 34
v
i 7. Proof of Corollary 2.2 37
X
8. Proof of Theorem 2.3 38
r
a Appendix A. Miscellaneous estimates of operator norms 40
Acknowledgment 43
References 43
2010 Mathematics Subject Classification. Primary 35B35, 37K45;
Secondary 35Q35.
Key words and phrases. line solitary waves, transverse linear stability.
1
2 TETSUMIZUMACHIANDYUSUKESHIMABUKURO
1. Introduction
Inthispaper,westudytransverselinearstability oflinesolitary waves fortheBenney-Luke
equation
(1.1) ∂2Φ ∆Φ+a∆2Φ b∆∂2Φ+(∂ Φ)(∆Φ)+∂ ( Φ 2)= 0 on R R2.
t − − t t t |∇ | ×
The Benney-Luke equation is an approximation model of small amplitude long water waves
with finite depth originally derived by Benney and Luke [4] as a model for 3D water waves.
Here Φ = Φ(t,x,y) corresponds to a velocity potential of water waves. We remark that (1.1)
is an isotropic model for propagation of water waves whereas KdV, BBM and KP equations
are unidirectional models. See e.g. [6, 7] for the other bidirectional models of 2D and 3D
water waves.
The parameters a, b are positive and satisfy a b = τˆ 1/3, where τˆ is the inverse Bond
− −
number. If we think of waves propagating in one direction, slowly evolving in time and
having weak transverse variation, then the Benney-Luke equation can be formally reduced to
the KP-II equation if 0 < a < b and to the KP-I equation if a > b > 0. More precisely, the
Benney-Luke equation (1.1) is reduced to
2f +(b a)f +3f f +f =0
x˜t˜ − x˜x˜x˜x˜ x˜ x˜x˜ y˜y˜
in the coordinate t˜= ǫ3t, x˜ = ǫ(x t) and y˜ = ǫ2y by taking terms only of order ǫ5, where
−
Φ(t,x,y) = ǫf(t˜,x˜,y˜). See e.g. [22] for the details. In this paper, we will assume 0 < a < b,
which corresponds to the case where the surface tension is weak or negligible.
The solution Φ(t) of the Benney-Luke equation (1.1) formally satisfies the energy conser-
vation law
(1.2) E(Φ(t),∂ Φ(t)) = E(Φ ,Ψ ) for t R,
t 0 0
∈
where
E(Φ,Ψ) := Φ 2+a(∆Φ)2+Ψ2+b Ψ 2 dxdy,
R2 |∇ | |∇ |
Z
and (1.1) is globally well-posed in(cid:8)the energy class (H˙2(R2) H˙1((cid:9)R2)) H1(R2) (see [40]).
∩ ×
The Benney Luke equation (1.1) has a 3-parameter family of line solitary wave solutions
(1.3) Φ(t,x,y) = ϕ (xcosθ+ysinθ ct+γ), c> 1, γ R, θ [0,2π),
c
− ± ∈ ∈
where
2(c2 1) α c2 1
c
ϕ (x) = − tanh( x), α = − ,
c cα 2 c bc2 a
c r
−
and
c2 1 α x
qc(x) := ϕ′c(x)= c− sech2 2c
is a solution of (cid:0) (cid:1)
3c
(1.4) (bc2 a)q (c2 1)q + q2 = 0.
− c′′− − c 2 c
Stability of solitary waves to the 1-dimensional Benney-Luke equation are studied by [38]
for the strong surface tension case a > b > 0 and by [30] for the weak surface tension case
b > a > 0. If a > b > 0, then (1.1) has a stable ground state for c satisfying 0 < c2 < 1
ASYMPTOTIC LINEAR STABILITY OF THE BENNEY-LUKE EQUATION IN 2D 3
([33, 39]). See also [23] for the algebraic decay property of the ground state. In view of
[42, 43], line solitary waves for the 2-dimensional Benney-Luke equation are expected to be
unstable in this parameter regime. On the other hand if 0 < a < b and c := √1+ǫ2 is close
to 1 (the sonic speed), then ϕ (x ct) is expected to be transversally stable because q (x)
c c
−
is similar to a KdV 1-soliton and line solitons of the KP-II equation is transversally stable
([21, 27, 28]).
The dispersion relation for the linearization of (1.1) around 0 is
1+a(ξ2+η2)
ω2 = (ξ2+η2)
1+b(ξ2+η2)
foraplanewavesolutionΦ(t,x,y) = ei(xξ+yη ωt). Ifb > a > 0,then ω 1andlinesolitary
−
|∇ | ≤
waves travel faster than the maximum group velocity of linear waves. Measuring the size of
perturbations with an exponentially weighted norm biased in the direction of motion of a line
solitary wave, we can observe that perturbations which are decoupled from the line solitary
wave decay as t . In the 1-dimensional case, small solitary waves are exponentially
→ ∞
linearly stable in the weighted space and λ = 0 is an isolated eigenvalue of the linearized
operator (see [30]). In our problem, however, the value λ = 0 is not an isolated eigenvalue.
This is because line solitary waves do not decay in the transverse direction. Indeed, for any
size of line solitary waves of (1.1), there appears a curve of continuous spectrum that goes
through λ = 0 and locates in the stable half plane (Theorem 2.1). The curve of continuous
eigenvalues hastodowithperturbationsthatpropagatetoward thetransversedirection along
the crest of the line solitary wave (Theorem 2.3). If line solitary waves are small, the rest of
the spectrum locates in a stable half plane λ C λ β < 0 (Theorem 2.4). For the
{ ∈ | ℜ ≤ − }
KP-II equation, the spectrum of the linearized operator around a 1-line soliton near λ = 0
can be obtained explicitly thanks to the integrability of the equation (see [2, 9, 28]). In this
paper, wewill usethe Lyapunov-Schmidt methodto findresonant eigemodes of the linearized
operator.
To prove non-existence of unstable modes for the linearized operator around small line
solitary waves, we make use of the KP-II approximation of the the linearized operator of
(1.1) on long length scales and make use of the transverse linear stability of line solitons for
theKP-IIequation. For 1-dimensionallong wave models, non-existence of unstablemodesfor
the linearized operator around solitary waves has been proved by utilizing spectral stability
of KdV solitons. See e.g. [12, 27, 24, 34, 36] and [30] for the 1-dimensional Benney-Luke
equation. We expect that the KP-II approximation of the linearized operator is useful to
other 2-dimensional long wave models such as KP-BBM and Boussinesq systems with no
surface tension (see e.g. [10]).
Nowletusintroduceseveralnotations. For anoperatorA,wedenotebyσ(A)thespectrum
andby D(A) andR(A)thedomain andthe rangeof theoperator A, respectively. For Banach
spaces V and W, let B(V,W) be the space of all linear continuous operators from V to W
and T = sup Tu for T B(V,W). We abbreviate B(V,V) as B(V). For
k kB(V,W) kxkV=1k kW ∈
4 TETSUMIZUMACHIANDYUSUKESHIMABUKURO
f (Rn) and m (Rn), let
′
∈S ∈ S
( f)(ξ) = fˆ(ξ)= (2π) n/2 f(x)e ixξdx,
− −
F Rn
Z
( 1f)(x)= fˇ(x) = fˆ( x),
−
F −
and (m(D)f)(x) = (2π) n/2(mˇ f)(x). We denote f,g by
−
∗ h i
m
f,g = f (x)g (x)dx
j j
h i R
j=1Z
X
for Cm-valued functions f = (f , ,f ) and g = (g , ,g ).
1 m 1 m
··· ···
Let L2(R2) = L2(R2;e2αxdxdy), L2(R) = L2(R;e2αxdx) and let Hk(R2) and Hk(R) be
α α α α
Hilbert spaces with the norms
1/2
u = ∂ku 2 + ∂ku 2 + u 2 ,
k kHαk(R2) k x kL2α(R2) k y kL2α(R2) k kL2α(R2)
(cid:16) 1/2 (cid:17)
u = ∂ku 2 + u 2 .
k kHαk(R) k x kL2α(R) k kL2α(R)
(cid:16) (cid:17)
We use a . b and a = O(b) to mean that there exists a positive constant such that a Cb.
≤
VariousconstantswillbesimplydenotedbyC andC (i N)inthecourseofthecalculations.
i
∈
We denote x = √1+x2 for x R.
h i ∈
2. Statement of the result
Since(1.1)is isotropicandtranslation invariant, wemay assumeθ = γ = 0in(1.3)without
loss of generality. Let Ψ = ∂ Φ, A = I a∆ and B = I b∆. Then in the moving coordinate
t
− −
z =x ct, the Benney-Luke equation (1.1) can be rewritten as
−
∂ Φ = c∂ Φ+Ψ,
t z
(2.1)
(∂tΨ = c∂zΨ+B−1A∆Φ B−1(Ψ∆Φ+2 Φ Ψ),
− ∇ ·∇
Let r (z) = cq (z). Linearizing (2.1) around (Φ,Ψ)= (ϕ (z),r (z)), we have
c c c c
−
Φ Φ
(2.2) ∂ = ,
t
Ψ L Ψ
(cid:18) (cid:19) (cid:18) (cid:19)
c∂ 1
z
= +V , = ,
L L0 L0 B 1A∆ c∂
(cid:18) − z(cid:19)
0 0
(2.3) V = B 1 , v = 2r (z)∂ +r (z)∆, v = 2q (z)∂ +q (z).
− − v v 1,c c′ z c 2,c c z c′
(cid:18) 1,c 2,c(cid:19)
We studylinear stability of (2.2) in a weighted space X := H1(R2) L2(R2). Let (η)u(z) =
α × α L
e iyη (eiyηu(z)) for η R. Note that V is independent of y. For each small η = 0, the
−
L ∈ 6
operator (η) has two stable eigenvalues.
L
ASYMPTOTIC LINEAR STABILITY OF THE BENNEY-LUKE EQUATION IN 2D 5
Theorem 2.1. Let 0 < a < b and k N. Fix c > 1 and α (0,α ). Then there exist a
c
∈ ∈
positive constant η , λ(η) C ([ η ,η ]),
0 ∞ 0 0
∈ −
ζ(,η) C ([ η ,η ];Hk(R) Hk 1(R)), ζ (,η) C ([ η ,η ];Hk (R) Hk 1(R))
· ∈ ∞ − 0 0 α × α− ∗ · ∈ ∞ − 0 0 −α × −−α
such that
(η)ζ(z, η) = λ( η)ζ(z, η), (η) ζ (z, η) = λ( η)ζ (z, η),
∗ ∗ ∗
L ± ± ± L ± ∓ ±
(2.4) λ(η) = iλ η λ η2+O(η3),
1 2
−
(2.5) ζ(,η) = ζ +iλ ηζ +O(η2) in Hk(R) Hk 1(R),
· 1 1 2 α × α−
(2.6) ζ (,η) = ζ iλ ηζ +O(η2) in Hk (R) Hk 1(R),
∗ · 2∗− 1 1∗ −α × −−α
(2.7)
λ(η) = λ( η), ζ(z,η) = ζ(z, η), ζ (z,η) = ζ (z, η) for η [ η ,η ] and z R,
∗ ∗ 0 0
− − − ∈ − ∈
where λ and λ are positive constants, A = 1 a∂2, B = 1 b∂2 and
1 2 0 − z 0 − z
ζ = qc , ζ = z∞∂cqc ,
1 2
r ∂ r
(cid:18) c′(cid:19) (cid:18)R− c c (cid:19)
z
B ∂ r 2q ∂ q q ∂ q A q
ζ = c − 0 c c− c c c− c′ c c , ζ = 0 c′ .
1∗ B z ∂ q −∞ 2∗ B r
0 c c R ! (cid:18)− 0 c(cid:19)
−∞
Remark 2.1. We remark that (0) isRa linearized operator of the 1-dimensional Benney-Luke
L
equation around ϕ (z) and ζ and ζ belong to the generalized kernel of (0). More precisely,
c 1 2
L
(0)ζ = 0, (0)ζ = ζ , (0) ζ =ζ , (0) ζ = 0,
L 1 L 2 1 L ∗ 1∗ 2∗ L ∗ 2∗
ker ( (0)) = span ζ ,ζ , ker ( (0)) = span ζ ,ζ .
g L { 1 2} g L { 1∗ 2∗}
The eigenvalue λ = 0 for (0) splits into two stable eigenvalues λ( η) for (η) with η = 0.
L ± L 6
In the exponentially weighted space L2(R), the value λ = 0 is an isolated eigenvalue of
α
(0) and there exists a β > 0 such that
L
σ( (0)) 0 λ C λ β
L \{ } ⊂ { ∈ | ℜ ≤ − }
provided c > 1 and c is sufficiently close to 1. See Lemma 2.1, Theorem 2.3 and Appendix B
in [30].
Remark 2.2. We expect that ζ (,η) and ζ+(,η) (k = 1, 2) do not belong to L2(R) as is the
k · k ·
same with continuous resonant modes for the KP-II equation. This is a reason why we study
spectral stability of in the exponentially weighted space X.
L
We will prove Theorem 2.1 by using the Lyapunov Schmidt method in Section 6.
Let (η ) be the spectral projection onto the subspace corresponding to the continuous
0
P
eigenvalues λ(η) and (η ) = I (η ). Let Z = (η )(H1(R2) L2(R2)). If
is spectrally{stabl}e−,ηt0h≤eηn≤eη0t isQexp0onenti−allPy st0able. Q 0 α × α L
L Z
|
Corollary 2.2. Let 0 < a < b, c > 1 and α (0,α ). Consider the operator in the space
c
∈ L
X = H1(R2) L2(R2). Assume that there exist positive constants β and η such that
α × α 0
(H) σ( ) λ λ β ,
Z
L| ⊂ { |ℜ ≤ − }
6 TETSUMIZUMACHIANDYUSUKESHIMABUKURO
where is the restriction of the operator on Z. Then for any β < β, there exists a
Z ′
L| L
positive constant C such that
(2.8) et (η ) Ce β′t for any t 0.
L 0 B(X) −
k Q k ≤ ≥
The semigroup estimate (2.8) follows from the assumption (H) and the Geahart-Pru¨ss
theorem [15, 37] which tells us that the boundedness of C0-semigroup in a Hilbert space is
equivalent to the uniform boundedness of the resolvent operator on the right half plane. See
also [17, 18].
Time evolution of the continuous eigenmodes etλ(η)g(z,η) can be considered as
{ }−η0≤η≤η0
a linear approximation of non-uniform phase shifts of modulating line solitary waves. For the
KP-II equation, modulations of the local amplitude and the angle of the local phase shift of a
line soliton are described by a system of Burgers’ equations (see [28, Theorems 1.4 and 1.5]).
In this paper, we find the first order asymptotics of solutions for the linearized equation (2.2)
is described by a wave equation with a diffraction term and it tends to a constant multiple
of the x-derivative of the line solitary wave as t .
→ ∞
Theorem2.3. Let0 < a < b, c > 1, αbeasinTheorem2.2and(Φ ,Ψ ) H2(R2) H1(R2).
0 0 ∈ α × α
Assume (H). Then a solution of (2.2) with (Φ(0),∂ Φ(0)) =(Φ ,Ψ ) satisfies
t 0 0
∂ Φ(t,z,y) q (z)
z (H W f)(y) c′ = O(t 1/4) as t ,
t t −
∂ Φ(t,z,y) − ∗ ∗ r (z) → ∞
(cid:13)(cid:13)(cid:18) t (cid:19) (cid:18) c′ (cid:19)(cid:13)(cid:13)L2α(Rz)L∞(Ry)
(cid:13) (cid:13)
where(cid:13)f(y) = cB Ψ A ∂ Φ ,q , H (y) = (4(cid:13)πλ t) 1/2e y2/4λ2t, κ = λ1 dE(q ,r ) and
h 0 0− 0 z 0 ci t 2 − − 1 2 dc c c
W (y) = (2κ ) 1 for y [ λ t,λ t] and W (y) = 0 otherwise.
t 1 − 1 1 t
∈ −
We remark that if f(y) is well localized and f(y)dy = 0, then H W f(y)
R t t
6 ∗ ∗ ≃
(2κ ) 1 f(y)dy on any compact intervals in y as t . The first order asymptotics
1 − R R → ∞
of solutions to (2.2) suggests that the local phaseshift of line solitary waves propagate mostly
R
at constant speed toward y = .
±∞
If c is close to 1, then the assumption (H) is valid and the spectrum of is similar to that
L
of the linearized KP-II operator around a line soliton. To utilize the spectral property of
the linearized operators of the KP-II equation around 1-line solitons, we introduce the scaled
parameters and variables
(2.9) λ = ǫ3Λ, c2 = 1+ǫ2, zˆ= ǫz, yˆ= ǫ2y, ξ = ǫξˆ, η = ǫ2ηˆ,
and translate the solitary wave profile q (x) as
c
1 αˆ zˆ 1
(2.10) q (z) = ǫ2θ (zˆ), θ (zˆ) = sech2 ǫ , αˆ = .
c ǫ ǫ ǫ
c 2 √bc2 a
(cid:18) (cid:19) −
Let
αˆ
αˆ = (b a) 1/2, θ (zˆ) = sech2( 0zˆ),
0 − 0
− 2
1
= (b a)∂3 ∂ +∂ 1∂2+3∂ (θ ) .
LKP −2{ − zˆ − zˆ zˆ− yˆ zˆ 0· }
ASYMPTOTIC LINEAR STABILITY OF THE BENNEY-LUKE EQUATION IN 2D 7
We remark that the operator is the linearization of the KP-II equation
KP
L
3
(2.11) 2∂ u+(b a)∂3u+∂ 1∂2u+ ∂ (u2) = 0
t − x x− y 2 x
around its line soliton solution θ (x t). The linearized operator has continuous eigen-
0 KP
− L
values λ (η) = iη √1+iγ η which has to dowith dynamics of modulatingline solitons (see
KP √3 1
[9, 28] and Section 3.1).
In the low frequency regime, we can deduce the eigenvalue problem
u u
(2.12) = λ
L v v
(cid:18) (cid:19) (cid:18) (cid:19)
to ∂ u= Λ∂ u provided ǫ is sufficiently small. More precisely, we have the following.
KP zˆ zˆ
L
Theorem 2.4. Let c = √1+ǫ2, α = αˆǫ and αˆ (0,αˆ /2). Then there exist positive
0
∈
constants ǫ , η , βˆ and a smooth function λ (η) such that if ǫ (0,ǫ ), then
0 0 ǫ 0
∈
(2.13) σ( ) λ (η) η [ ǫ2η ,ǫ2η ] λ C λ βˆǫ3 ,
ǫ 0 0
L \{ | ∈ − } ⊂ { ∈ | ℜ ≤ − }
(2.14) lim ǫ 3λ (ǫ2η) λ (η) = O(η3) for η [ η ,η ],
− ǫ KP 0 0
ǫ 0 | − | ∈ −
↓
(2.15) et (ǫ2η ) Ke βˆǫ3t for any t 0,
L 0 B(X) −
k Q k ≤ ≥
where K is a constant that does not depend on t.
3. Resonant modes of the linearized operator
In this section, we will prove the existence of resonant continuous eigenvalues of near
L
λ = 0 and show that the resonant eigenvalues and resonant eigenmodes for are similar to
L
those for the linearized KP-II operator provided line solitary waves are small.
KP
L
3.1. Spectral stability in the KP-II scaling regime. First, werecall somespectralprop-
erties of the linearized KP-II equation around 1-line solitons. Let us consider the eigenvalue
problem of the linearized operator of (2.11) around θ . Let
0
1 3
= (b a)∂3 ∂ +∂ 1∂2 , = ∂ (θ ),
LKP,0 −2{ − z − z z− y} LKP LKP,0− 2 z 0·
1 η2
(η) = ∂ (b a)∂2 1+3θ + ∂ 1.
LKP −2 z{ − z − 0} 2 z−
Formally,wehave (u(z)eiyη)= eiyη( (η)u)(z). Theoperator isspectrallystable
KP KP KP,0
L L L
in exponentially weighted spaces.
Lemma 3.1. Let αˆ (0,αˆ ) and βˆ = αˆ 1 (b a)αˆ2 . Then
∈ 0 0 2{ − − }
(3.1) (Λ ) 1 ( Λ+βˆ ) 1 for Λ satisfying Λ> βˆ .
k −LKP,0 − kB(L2αˆ(R2)) ≤ ℜ 0 − ℜ − 0
8 TETSUMIZUMACHIANDYUSUKESHIMABUKURO
Moreover, there exists a positive constant C such that if Λ> βˆ ,
0
ℜ −
1+j
βˆ − 2
(3.2) ∂j(Λ ) 1 C Λ+ 0 for j = 1,2,
k z −LKP,0 − kB(L2αˆ(R2)) ≤ ℜ 2 !
2/3
βˆ −
(3.3) (Λ ) 1 C Λ+ 0 .
k −LKP,0 − kB(L2αˆ(R2)) ≤ (cid:12) 2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. By the Plancherel theorem, (cid:12) (cid:12)
(cid:12) (cid:12)
(3.4) g 2 = e2αx g(x,y)2dxdy = gˆ(ξ +iα,η)2dξdη
k kL2α(R2) R2 | | R2| |
Z Z
for any g C (R2) and an operator m(D):= 1 mˇ f is bounded on L2(R2) if and only if
∈ 0 2π ∗ α
(3.5) m(D) B(L2(R2)) = sup m(ξ +iα,η) < .
k k α (ξ,η) R2| | ∞
∈
Suppose f L2(R2) and that u is a solution of
∈ α
(Λ )u = f.
KP,0
−L
Then
fˆ(ξ,η)
uˆ(ξ,η) = ,
Λ (ξ,η)
KP,0
−L
where (ξ,η) = i (b a)ξ3+ξ ξ 1η2 . Since
LKP,0 2{ − − − }
1 αˆη2
(ξ +iαˆ,η) = 3(b a)αˆξ2+2βˆ + βˆ ,
ℜLKP,0 − 2 − 0 ξ2+αˆ2 ≥ − 0
(cid:26) (cid:27)
it follows from (3.5) that for j = 0,1,2 and Λ with Λ> βˆ ,
0
ℜ −
ξ +iαˆ j
k∂zj(Λ−LKP,0)−1kB(L2αˆ(R2) =(ξ,sηu)pR2 Λ K| P,0(ξ+| iαˆ,η) .
∈ | −L |
Thus we have (3.1) and (3.2). Moreover, we have (3.3) because (ξ + iαˆ,η) .
KP,0
|ℑL |
(ξ +iαˆ,η) 3/2. (cid:3)
KP,0
{−ℜL }
Let γ = 4 (b a)/3, xˆ = αˆ0x and
1 − 2
p iη
λ (η) = 1+iγ η,
KP 1
√3
2(b a) p
g0(x,η) = γ √1−+iγ η∂x2 e−√1+iγ1ηxˆsechxˆ ,
1 1
(cid:16) (cid:17)
i
g (x,η) = ∂ e√1 iγ1ηxˆsechxˆ .
0∗ η x −
(cid:16) (cid:17)
Using Lemma 2.1 in [28] and the change of variable
αˆ 1
0
x x, y y, η γ η,
1
7→ 2 7→ γ 7→
1
ASYMPTOTIC LINEAR STABILITY OF THE BENNEY-LUKE EQUATION IN 2D 9
we have for η R 0 ,
∈ \{ }
(η)g (x, η) = λ ( η)g (x, η),
KP 0 KP 0
L ± ± ±
(η) g (x, η) = λ ( η)g (x, η),
LKP ∗ 0∗ ± KP ∓ 0∗ ±
g (x,η)g (x,η)dx = 1, g (x,η)g (x, η)dx = 0.
R 0 0∗ R 0 0∗ −
Z Z
To resolve the singularity of g (x,η) and the degeneracy of g (x,η) at η = 0, we decompose
0 0∗
them into their real parts and imaginary parts. Let
1
g (x,η) = g (x,η)+g (x, η), g (x,η) = g (x,η) g (x, η) ,
0,1 0 0 0,2 0 0
− iη{ − − }
1 η
g (x,η) = g (x,η)+g (x, η) , g (x,η) = g (x,η) g (x, η) .
0∗,1 2{ 0∗ 0∗ − } 0∗,2 2i{ 0∗ − 0∗ − }
Then
g (x,η)g (x,η)dx = δ for j, k = 1, 2.
0,j 0∗,k jk
R
Z
Moreover, we see that g (x,η) and g (x,η) are even in η and that for k = 1, 2 and
0,k 0∗,k
αˆ (0,αˆ ),
0
∈
(3.6) g (,η) g (,0) + g (,η) g (,0) = O(η2),
k 0,k · − 0,k · kL2αˆ k 0∗,k · − 0∗,k · kL2−αˆ
√3 x
(3.7) g (x,0) = θ (x), g (x,0) = θ (x)+ +αˆ 1 θ (x),
0,1 − 2 0′ 0,2 0 2 −0 0′
αˆ x (cid:16) (cid:17) αˆ
0 0
(3.8) g (x,0) = (x θ (x )+2θ (x ))dx , g (x,0) = θ (x).
0∗,1 2√3 1 0′ 1 0 1 1 0∗,2 2 0
Z−∞
Let (η ) be the spectral projection to resonant modes g (x, η)eiyη defined
PKP 0 { 0 ± }−η0≤η≤η0
by
1 η0
(η )f(x,y) = a (η)g (x,η)eiyηdη,
KP 0 0,k 0,k
P √2π
kX=1,2Z−η0
a (η) = ( f)(x,η) g (x,η)dx,
0,k R Fy · 0∗,k
Z
and let (η ) = I (η ). By Lemma 3.1 in [28], the operator (η ) and (η )
KP 0 KP 0 KP 0 KP 0
Q −P P Q
are bounded on L2(R2) for αˆ (0,αˆ ). Moreover, we have the following.
αˆ ∈ 0
Proposition 3.2. Let αˆ (0,αˆ ) and η be a positive number satisfying αˆ( √1+iγη 1)=
∈ 0 ∗ 2 ℜ ∗−
αˆ. For any η (0,η ), there exists a positive number b such that
0
∈ ∗
sup (Λ ) 1 (η ) < .
Λ bk −LKP − QKP 0 kB(L2αˆ(R2)) ∞
ℜ ≥−
Proof. By Proposition 3.2 in [28], there exist positive constants b and C such that
1
ketLKPQKP(η0)kB(L2αˆ(R2)) ≤ Ce−b1t.
If Λ b > b , then
1
ℜ ≥ − −
1
k(Λ−LKP)−1QKP(η0)kB(L2αˆ) ≤Z0∞ke−ΛtetLKPQKP(η0)kB(L2αˆ(R2))dt . b1−b .
10 TETSUMIZUMACHIANDYUSUKESHIMABUKURO
(cid:3)
3.2. Resonant modes. Inthissubsection,wewillprovetheexistenceofcontinuousresonant
modes of near λ = 0 by using the Lyapunov Schmidt method. Let
L
A(η) = 1+aη2 a∂2, B(η) = 1+bη2 b∂2,
− z − z
c∂ 1
z
(η) = ,
L0 B(η) 1A(η)(∂2 η2) c∂
(cid:18) − z − z(cid:19)
0 0
(η) = (η)+V(η), V(η) = B(η) 1 ,
0 −
L L − v (η) v (η)
(cid:18) 1,c 2,c (cid:19)
v (η) = 2r ∂ +r (∂2 η2), v (η) = 2q ∂ +q .
1,c c′ z c z − 2,c c z c′
If eiyη(u (z),u (z)) is a solution of (2.12), then
1 2
u u
1 1
(3.9) (η) =λ
L u u
(cid:18) 2(cid:19) (cid:18) 2(cid:19)
or equivalently,
(3.10) A(η)(∂2 η2) (λ c∂ )2B(η) u v (η)u v (η)(λ c∂ )u = 0,
{ z − − − z } 1 − 1,c 1 − 2,c − z 1
(3.11) u = (λ c∂ )u .
2 z 1
−
We will find solutions of (3.9) in H1(R) L2(R) for small η. Using the change of variables
α × α
(2.9) and (2.10) and dropping the hats in the resulting equation, we have
(3.12) F(U,Λ,ǫ,η) := 2L (η)U ΛT (ǫ,η)U +ǫ2Λ2B (η)∂ 1U = 0,
ǫ − 1 ǫ z−
where U(z) = ∂ u (z/ǫ) and
z 1
1 η2
L (η) = ∂ (bc2 a)∂2 1+3cθ + T (ǫ,η),
ǫ −2 z{ − z − ǫ} 2 2
T (ǫ,η) = 2cB (η) ǫ2(2θ +θ ∂ 1), T (ǫ,η) = A (η)+ǫ2(bc2 a)∂2+cǫ2θ ∂ 1,
1 ǫ − ǫ ǫ′ z− 2 { ǫ − z ǫ} z−
A (η) = 1+aǫ2(ǫ2η2 ∂2), B (η) = 1+bǫ2(ǫ2η2 ∂2).
ǫ − z ǫ − z
Let L (η) be an operator on L2(R) with D(L )= H3(R) for an αˆ (0,αˆ ) and
ǫ αˆ ǫ αˆ ∈ ǫ
(∂ 1f)(z) = ∞f(z )dz for f L2(R).
z− − 1 1 ∈ αˆ
Zz
We remark that F(U,Λ,0,η) = 2 (η)U 2ΛU and the translated eigenvalue problem
KP
L −
(3.12) is similar to the eigenvalue problem of the KP-II equation provided ǫ is sufficiently
small. For small η = 0, (3.9) has two eigenvalues in the vicinity of 0.
6
First, we will find an approximate solution of (3.12). Let U(η) = U +ηU +η2U +O(η3),
0 1 2
Λ(η) = iΛ0 η Λ0 η2+O(η3) and formally equate the powers of η in (3.12). Then
1,ǫ − 2,ǫ
(3.13) L (0)U = 0,
ǫ 0
i
(3.14) L (0)U = Λ0 T (ǫ,0)U ,
ǫ 1 2 1,ǫ 1 0
(3.15) 2L (0)U = T (ǫ,0)+Λ0 T (ǫ,0) ǫ2(Λ0 )2B (0)∂ 1 U +iΛ0 T (ǫ,0)U .
ǫ 2 − 2 2,ǫ 1 − 1,ǫ ǫ z− 0 1,ǫ 1 1
(cid:8) (cid:9)