Local Smoothing and Well-Posedness Results for KP-II Type Equations Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨ at der Rheinischen Friedrich-Wilhelms-Universit¨ at Bonn vorgelegt von Habiba Kalantarova aus Baku Bonn 2014
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Local Smoothing and Well-PosednessResults for KP-II Type Equations
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakultat
der
Rheinischen Friedrich-Wilhelms-Universitat Bonn
vorgelegt von
Habiba Kalantarova
aus Baku
Bonn 2014
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultat der
Rheinischen Friedrich-Wilhelms-Universitat Bonn am Institut fur Angewandte Mathe-
matik.
1. Gutachter: Prof. Dr. Herbert Koch
2. Gutachter: Prof. Dr. Sebastian Herr
Tag der Promotion: 26.01.2015
Erscheinungsjahr: 2015
Acknowledgements
I would like to express my deepest gratitude to my advisor Prof. Herbert Koch for his
continuous support, motivation, and guidance throughout my study and research and
for his patience during the correction of this thesis.
I would like to thank the rest of my thesis committee: Prof. Sebastian Herr (also for
many helpful conversations during his stay at MI) , Prof. Mete Soner, and Prof. Thomas
Martin.
I would also like to thank the members of the research group: Prof. Axel Grunrock,
Prof. Jeremy Marzuola, Prof. Junfeng Li, Dr. Stefan Steinerberger, Dr. Dominik John,
Dr. Tobias Schottdorf, Dr. Angkana Ruland and Shaoming Guo.
My cordial thanks go to my Bonn family: Dr. Orestis Vantzos (for always being there
for me), Catalin Ionescu (for all the sound advice), Irene Paniccia, Dr. Joao Carreira
and Dr. Branimir Cacic for making Bonn a very pleasant place.
I would also like to sincerely thank Ms. Karen Bingel for helping to survive in Bonn.
Last but not least I would like to thank my parents Prof. Varga Kalantarov and Gandaf
Kalantarova and my sister, my oldest friend Nargiz Kalantarova for all the support and
A KPII 83A.1 Derivation of the explicit formula for the soliton Q . . . . . . . . . . . . . 83
B (gKP-II)3 85
Bibliography 89
5
Chapter 1
Introduction
In this thesis, we study the qualitative properties of the solution of the Cauchy problem
for the Kadomtsev-Petviashvili II (KP-II) equation
∂tu+ ∂3xu+ 3∂−1
x ∂2yu+ 6u∂xu = 0,
and the well posedness of the Cauchy problem for the generalized Kadomtsev-Petviashvili
II equation with cubical nonlinearity ((gKP-II)3)
∂tu+ ∂3xu+ 3∂−1
x ∂2yu− 6u2∂xu = 0
that satisfy initial conditions with low regularity.
When the sign in front of 3∂−1x ∂2
yu term is minus in the above two equations they are
called the KP-I and the (gKP-I)3 equations respectively. Despite their formal similarity,
the KP-I and the KP-II equations differ significantly with respect to their underlying
mathematical structure. The KP-I, the KP-II and the (gKP-II)3 equations are inte-
grable Hamiltonian systems and consequently possess infinitely many conservation laws.
The KP-I and the (gKP-I)3 equations have conservation laws with positively defined
quadratic parts. This allows the corresponding Sobolev type norms to be controlled
by the KP-I flow and the use of energetic methods to analyze these equations. On the
other hand, the KP-II equation has conservation laws that do not have positively de-
fined quadratic parts. In order to study the KP-II and the (gKP-II)3 equation harmonic
analysis methods have been used starting with [2].
The KP equation came as a natural generalization of the Korteweg-de Vries (KdV)
equation from one to two spatial dimensions,
1
2 Chapter 1 Introduction
∂tu+ ∂3xu+ 6u∂xu = 0, (t, x) ∈ R× R. (1.1)
It was first introduced in 1970 by B. B. Kadomtsev and V. I. Petviashvili [14]. They
derived the equation as a model to study the evolution of long ion-acoustic waves of small
amplitude propagating in plasmas under the effect of long transverse perturbations.
These equations were later derived by other researchers in other physical settings as
well. The KP equations have been obtained as a reduced model in ferromagnetics [30],
Bose-Einstein condensates [31] and string theory [7].
The KdV equation has remarkable solutions, called solitons. Solitons are solutions that
are localised and maintain their form for long periods of time and depend upon variables
x and t only through x− ct where c is a fixed constant. Substituting u(t, x) = Q(x− ct)into (1.1) one obtains the ordinary differential equation
− cQ′ +Q(3) + 6QQ′ = 0
which is satisfied by the following family of solutions
Q =c
2sech2
(c1/2
2x).
Figure 1.1: Graph of a soliton solution of the KdV equation.
Moreover the other solitons and radiations can pass through them without destroying
their form, [35].
Chapter 1 Introduction 3
Figure 1.2: Interaction of two solitons.
The soliton solutions of the KdV equation considered as solutions of the KP equations
are called the line solitons.
Figure 1.3: Graph of a line soliton.
The line solitons for KP-I are stable if they have small speed [27] and unstable if they
have large speed [26], [36]. However, for the KP-II equation heuristic analysis [14] and
inverse scattering [32] suggest that the line soliton is stable.
4 Chapter 1 Introduction
In Chapter 3, we present the results of our attempt to solve this problem. We conjectured
a perturbed solution of the form
u(t, x, y) = Q(x− t, y) + εw(t, x− t, y),
but T. Mizumachi in [23] showed that our hope was naive. The line soliton is more
strongly perturbed than we hoped. In [23], T. Mizumachi proved the stability of line
solitons for exponentially localized perturbations.
The (gKP-II)3 equation is a model for the evolution of sound waves in antiferromagnets
[30]. The well posedness of this equation has been previously studied in [13], [15], [9]
and in references therein. In Chapter 4, we prove global well posedness of the Cauchy
problem for the (gKP-II)3 equation with initial condition in the space defined by the
following norm
‖u‖`∞12
`p0(L2) := supλλ1/2
(∑k
‖uλ,k‖pL2(R2)
)1/p.
This extends the result in [9]. The fundamental idea of the proof is due to J. Bourgain
[2]. We construct function spaces based on the linear part of the dispersive equation
we study. Instead of Bourgain spaces we use Up (due to H. Koch-D. Tataru, [18]) and
V p (due to N. Wiener, [34]) function spaces, which are more useful in the analysis of
nonlinear dispersive partial differential equations at critical regularity. This reduces our
problem to proving multilinear estimates on the constructed spaces.
Chapter 2
Basic notions and function spaces
In this chapter, we review certain definitions and properties of the function spaces that
are used throughout this work. The content of this chapter can be found in many
sources. The author has consulted [20] and [16] for Section 2.1, [4], [16], [28] and [29]
for Section 2.2, [16], [29] and [1] for Section 2.3, [28] for Section 2.4, and finally [10] and
[17] for Section 2.5.
2.1 The Fourier Transform
Definition 2.1. Let f ∈ L1(Rn). The Fourier transform of f , denoted by f , is defined
as
f(ξ) = (2π)−n2
∫e−i(x,ξ)f(x)dx, ξ ∈ Rn,
where
(x, ξ) :=n∑i=1
xiξi.
We will use the notation F(f) and f interchangeably.
F is a bounded linear map from L1(Rn) to L∞(Rn). The virtue of the Fourier transform
is that it converts constant coefficient linear partial differential operators into multipli-
cation with polynomials.
We summarize the fundamental properties of the Fourier transform in the following
proposition.
5
6 Chapter 2 Basic notions and function spaces
Proposition 2.2. If f, g ∈ L1(Rn), then
(i) F(f(· − x0))(ξ) = e−i(ξ,x0)f(ξ),
(ii) F(ei(·,ξ0)f(·))(ξ) = f(ξ − ξ0),
(iii) F(f)(ξ) = f(−ξ),
(iv) For (f ∗ g)(y) =∫
Rn f(y − x)g(x)dx, we have f ∗ g = (2π)n2 f g,
(v) F(∂xjf)(ξ) = iξj f(ξ),
(vi) F(xjf)(ξ) = i∂ξj f(ξ),
(vii)∫f(x)g(x)dx =
∫f(ξ)g(ξ)dξ.
Definition 2.3 (Schwartz function). A function φ ∈ C∞(Rn) is called rapidly decreasing
or Schwartz function if for all multiindices α, β (i.e. α, β ∈ Zn+) there exist constants
cα,β such that
ρα,β(φ) := supx∈Rn
|xα∂βφ(x)| ≤ cα,β.
We call the Frechet space of all Schwartz functions with the topology given by the family
of semi-norms ρα,β the Schwartz space and denote it by S(Rn). The natural topology on
S(Rn) is as follows: a sequence of functions φj converges to zero if for all multi-indices α,
β, xα∂βφj converges uniformly to zero. A complete metric inducing the same topology
on S(Rn) can be defined by
d(φ, ψ) =∑α,β
2−|α|−|β|ρα,β(φ− ψ)
1 + ρα,β(φ− ψ).
Note that C∞0 (Rn) is dense in S(Rn) in the above defined metric topology.
Remark 2.4. The map φ 7→ φ is an isomorphism on S(Rn) with the inverse
φ = (2π)−n2
∫ei(x,ξ)φ(ξ)dξ, x ∈ Rn.
Theorem 2.5 (Plancherel’s Theorem). If φ and ψ are in S(Rn), then
∫Rnφ(x)ψ(x)dx =
∫Rnφ(ξ)ψ(ξ)dξ.
Chapter 2 Basic notions and function spaces 7
Definition 2.6 (Tempered distributions). We define the space of tempered distributions
S ′(Rn) to be the dual space of the Schwartz space.
Note that for every tempered distribution u there exists N ∈ N and a constant C = Cα,β
such that
|u(φ)| ≤ C∑
|α|,|β|≤N
sup |xα∂βφ|, φ ∈ S(Rn).
Then the definition of the Fourier transform can be further naturally extended to the
tempered distributions by
u(φ) = u(φ), φ ∈ S(Rn).
Theorem 2.7. The Fourier transform F extends to a unitary map from L2(Rn) to itself
and thus the following identity of Parseval holds
‖u‖L2(Rn) = ‖u‖L2(Rn).
Furthermore since Lp ⊂ S′(Rn) the Fourier transform is also defined for all such spaces.
2.2 Sobolev Spaces
Definition 2.8. Let Ω be a nonempty open set in Rn, 1 ≤ p ≤ ∞ and s be a nonnegative
integer. The Sobolev space W s,p consists of all locally summable functions u : Ω → Rsuch that for each multiindex α with |α| ≤ s, ∂αu exists in the weak sense and belongs
to Lp(Ω). W s,p is a normed space equipped with the norm
‖u‖W s,p :=
(∑
|α|≤s∫
Ω |∂αu|pdx
) 1p if 1 ≤ p <∞ ,∑
|α|≤s esssupΩ|∂αu| if p =∞.
Remark 2.9. Among the spaces W s,p, particular importance is attached to W s,2 because
they are Hilbert spaces. We denote them by Hs.
Definition 2.10 (Fractional Hs−Sobolev spaces). Let s ∈ R. We say that u ∈ Hs(Rn)
if u ∈ S ′(Rn) has a locally integrable Fourier transform and
‖u‖2Hs :=∫
Rn(1 + |ξ|2)s|u(ξ)|2dξ <∞.
8 Chapter 2 Basic notions and function spaces
In the following X → Y denotes a continuous embedding of X into Y , and X ⊂⊂ Y
denotes a compact embedding.
Proposition 2.11. If
1 < p ≤ q ≤ ∞ and 0 ≤ t ≤ s <∞
are such that
n
p− s ≤ n
q− t,
and such that at least one of the two inequalities
q ≤ ∞, n
p− s ≤ n
q− t
is strict, then
W s,p(Rn) →W t,q(Rn).
Next, we recall the definitions of the homogeneous Sobolev spaces which are commonly
used, because of the symmetry properties they have.
Definition 2.12 (Homogeneous Sobolev Space). We call the space Hs equipped with
the following semi-norm
‖u‖2Hs :=
∫Rn|ξ|2s|u(ξ)|2dξ <∞ (2.1)
the homogeneous Sobolev space.
Definition 2.13 (Non-isotropic Homogeneous Sobolev space). Let s1, s2 ∈ R. Hs1,s2(R2)
is the space of tempered distributions with
‖u‖Hs1,s2 :=(∫
R2
|ξ|2s1 |η|2s2 |u(ξ, η)|2dξdη) 1
2
<∞. (2.2)
Chapter 2 Basic notions and function spaces 9
2.3 Besov Spaces
The Littlewood-Paley theory is a method of decomposing a function into a sum of in-
finitely many frequency localised components, that have almost disjoint frequency sup-
ports. In the following we present one of the standard ways of setting up the Littlewood-
Paley theory. We start with introducing a dyadic partition of unity. Let φ(ξ) be a real
radial bump function such that
φ(ξ) =
1 if |ξ| ≤ 1,
0 if |ξ| > 2,
and χ(ξ) = φ(ξ)−φ(2ξ). Then χ(ξ) is supported onξ ∈ Rn : 1
2 ≤ |ξ| ≤ 2
and satisfies
∑k∈Z
χ(2−kξ) = 1.
We define the Littlewood-Paley projection Pk by
Pkf(ξ) = χ(ξ/2k)f(ξ)
in frequency space, or equivalently in physical space by
Pkf = fk = mk ∗ f,
where mk(x) = 2nkm(2kx) and m(x) is the inverse Fourier transform of χ. Then ∀f ∈L2(Rn) we have
f =∑k∈Z
Pkf.
We sum up the crucial properties of the Littlewood-Paley projections in the following
theorem.
Theorem 2.14. The Littlewood-Paley projections have the following properties:
(i) [Almost Orthogonality] The operators Pk are selfadjoint. Furthermore, the family
Pkfk is almost orthogonal in L2(Rn) in the following sense
10 Chapter 2 Basic notions and function spaces
Pk1Pk2 = 0 whenever |k1 − k2| ≥ 2
and
‖f‖L2 ≈∑k
‖Pkf‖2L2 ,
which is an easy consequence of Parseval’s Identity.
(ii) [Lp−boundedness] Let J ⊂ Z and 1 ≤ p ≤ ∞. Then the following estimate holds
true
‖PJf‖Lp . ‖f‖Lp .
(iii) [Finite band property] Let k be an integer. For any 1 ≤ p ≤ ∞
‖∂Pkf‖Lp . 2k‖f‖Lp ,
2k‖Pkf‖Lp . ‖∂f‖Lp .
(iv) [Bernstein inequalities] For any 1 ≤ p ≤ q ≤ ∞ we have
‖Pkf‖Lq . 2kn(1/p−1/q)‖f‖Lp , ∀k ∈ Z,
‖P≤0f‖Lq . ‖f‖Lp .
Remark 2.15. The Bernstein inequality is a remedy for the failure of
Wnp,p(Rn) ⊂⊂ L∞(Rn).
The Littlewood-Paley theory has proven to be invaluable in studying partial differential
equations. It allows us to decompose the data into pieces, solve the problem on each
piece, and then ”sum” these solution components.
Remark 2.16. The definitions of Sobolev norms can alternatively be given and extended
to s ∈ R by using the Littlewood-Paley theory as follows
Chapter 2 Basic notions and function spaces 11
‖f‖W s,p ≈∥∥∥∑k∈Z
2ksPkf∥∥∥Lp,
‖f‖W s,p ≈∥∥∥∑k∈Z
(1 + 2k)sPkf∥∥∥Lp.
Definition 2.17 (Besov Spaces). Let s ∈ R and 1 ≤ p, q ≤ ∞. The Besov space is the
completion of C∞0 (Rn) with respect to the norm defined by
‖f‖Bsp,q :=
(‖P≤0f‖qLp +
∑∞k=1 2sqk‖Pkf‖qLp
)1/q if 1 ≤ q <∞,
sup‖P≤0f‖Lp , 2sk‖Pkf‖Lp if q =∞.
Definition 2.18 (Homogeneous Besov Spaces). Let s ∈ R and 1 ≤ p, q ≤ ∞. The
homogeneous Besov norm is defined by
‖f‖Bsp,q :=
(∑
k∈Z 2sqk‖Pkf‖qLp)1/q if 1 ≤ q <∞,
supk 2sk‖Pkf‖Lp if q =∞.
We collect the main Besov space embeddings in the following proposition.
If x < 0, then we choose the closed curve γ2 and repeat similar calculations.
In Region I if x > 0 then
∣∣∣∣∫γ1
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ = 2π∣∣∣∣ η2eixξ3
(ξ3 − ξ1)(ξ3 − ξ2)(ξ3 − ξ4)(ξ3 − ξ5)
∣∣∣∣=
2πη2∣∣∣−2τ +√τ2 + 12η2
2+ i
√4 + 3τ2 + 12η2
2
∣∣∣︸ ︷︷ ︸≥1
2√4 + 3τ2 + 12η2︸ ︷︷ ︸
≥2
∣∣∣−τ+√−4−3τ2−12η2
2
∣∣∣≤ π
3
and if x < 0
Chapter 3 The Kadomtsev-Petviashvili II equation 39
∣∣∣∣∫γ2
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ =2π|η2|
|ξ1 − ξ2||ξ1 − ξ3||ξ1 − ξ4||ξ1|
+2π|η2|
|ξ2 − ξ1||ξ2 − ξ3||ξ2 − ξ4||ξ2|+
2πη2
|ξ4 − ξ1||ξ4 − ξ2||ξ4 − ξ3||ξ4|+
2πη2
|ξ1||ξ2||ξ3||ξ4|
=2πη2
√τ2 + 12η2
∣∣∣2τ −√τ2 + 12η2
2− iε1 + i
√4 + 3τ2 + 12η2
2
∣∣∣︸ ︷︷ ︸≥ 1
2
2∣∣∣ τ−√τ2+12η2
2 − iε1
∣∣∣
+2πη2
√τ2 + 12η2
∣∣∣2τ +√τ2 + 12η2
2− iε2 + i
√4 + 3τ2 + 12η2
2
∣∣∣︸ ︷︷ ︸≥ 1
2
2∣∣∣ τ+√τ2+12η2
2 − iε2
∣∣∣
+2πη2∣∣∣−2τ +
√τ2 + 12η2
2+ iε1 − i
√4 + 3τ2 + 12η2
2
∣∣∣︸ ︷︷ ︸≥ 1
2
2√4 + 3τ2 + 12η2
∣∣∣√4+4τ2+12η2
2
∣∣∣
+2πη2∣∣∣∣ τ−√τ2+12η2
2 − iε1
∣∣∣∣ ∣∣∣∣ τ+√τ2+12η2
2 − iε2
∣∣∣∣ ∣∣∣∣−τ+i√
4+3τ2+12η2
2
∣∣∣∣ ∣∣∣∣ τ+i√
4+3τ2+12η2
2
∣∣∣∣≤ 8π.
In Regions II, III and VII if x > 0
∣∣∣∣∫γ1
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ = 2π∣∣∣∣ η2eixξ3
(ξ3 − ξ1)(ξ3 − ξ2)(ξ3 − ξ4)(ξ3 − ξ5)
∣∣∣∣=
πη2∣∣∣√−1+√
1+12η2
2 + i
√1+√
1+12η2
2
∣∣∣2(1+√
1+12η2
2
)≤ 2π,
and if x < 0 then
40 Chapter 3 The Kadomtsev-Petviashvili II equation
∣∣∣∣∫γ2
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ =2πη2
|ξ1 − ξ2||ξ1 − ξ3||ξ1 − ξ4||ξ1|
+2πη2
|ξ2 − ξ1||ξ2 − ξ3||ξ2 − ξ4||ξ2|+
2πη2
|ξ4 − ξ1||ξ4 − ξ2||ξ4 − ξ3||ξ4|+
2πη2
|ξ1||ξ2||ξ3||ξ4|
=2πη2
(−1 +√
1 + 12η2)∣∣∣∣−√−1+
√1+12η2
2 − iε1 + i
√1+√
1+12η2
2
∣∣∣∣2+
2πη2
(−1 +√
1 + 12η2)∣∣∣∣√−1+
√1+12η2
2 − iε2 + i
√1+√
1+12η2
2
∣∣∣∣2+
2πη2∣∣∣∣√−1+√
1+12η2
2 + iε1 − i√
1+√
1+12η2
2
∣∣∣∣2 (1 +√
1 + 12η2)
+2πη2∣∣∣∣√−1+
√1+12η2
2 + iε1
∣∣∣∣ ∣∣∣∣√−1+√
1+12η2
2 − iε2
∣∣∣∣ (1+√
1+12η2
2
)≤ 8π.
In Regions V and VIII the argument sequence that lead us to deduce the boundedness
of integrals I1 and I4 and the fact that in these regions |η| < 10 imply the boundedness
of I6.
In Regions IV, VI and IX calculations are similar. Here we illustrate the calculations
for the Region IV.
If x > 0 then we have
∣∣∣∣∫γ1
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ = 2π∣∣∣∣ η2eixξ3
(ξ3 − ξ1)(ξ3 − ξ2)(ξ3 − ξ4)(ξ3 − ξ5)
∣∣∣∣≤ 2πη2
|ξ3 − ξ1|︸ ︷︷ ︸> 1
2|τ |1/3
|ξ3 − ξ2|︸ ︷︷ ︸>|τ |1/3
|ξ3 − ξ4|︸ ︷︷ ︸>|τ |1/3
|ξ3 − ξ5|︸ ︷︷ ︸> 3η2
2|τ |
< 3π,
and if x < 0 then
Chapter 3 The Kadomtsev-Petviashvili II equation 41
∣∣∣∣∫γ2
η2eixξ
ξ(ξ4 + ξ2 − (τ − iε)ξ − 3η2)dξ
∣∣∣∣ ≤ 2πη2
|ξ1 − ξ2|︸ ︷︷ ︸>|τ |1/3
|ξ1 − ξ3|︸ ︷︷ ︸> 1
2|τ |1/3
|ξ1 − ξ4|︸ ︷︷ ︸> 1
2|τ |1/3
|ξ1|︸︷︷︸> 3η2
2|τ |
+2πη2
|ξ2 − ξ1|︸ ︷︷ ︸>|τ |1/3
|ξ2 − ξ3|︸ ︷︷ ︸>|τ |1/3
|ξ2 − ξ4|︸ ︷︷ ︸>|τ |1/3
|ξ2|︸︷︷︸>|τ |1/3
+2πη2
|ξ4 − ξ1|︸ ︷︷ ︸> 1
2|τ |1/3
|ξ4 − ξ2|︸ ︷︷ ︸>|τ |1/3
|ξ4 − ξ3|︸ ︷︷ ︸>|τ |1/3
|ξ4|︸︷︷︸>|τ |1/3
+2πη2
|ξ1|︸︷︷︸> 3η2
2|τ |
|ξ2|︸︷︷︸>|τ |1/3
|ξ3|︸︷︷︸>|τ |1/3
|ξ4|︸︷︷︸>|τ |1/3
< 9π.
3.1.2 T ∗T Principle
Theorem 3.6. Assume w is a solution of
∂tw + ∂3xw − ∂xw + 3∂−1
x ∂2yw = f + ∂xg + η∂−1
x h︸ ︷︷ ︸=F
, (3.20)
where f , g and h have compact supports in t > 0. Assume further that
Fx,y(w(t, x, y))→ 0 as t→ −∞.
Then
‖w(0, x, y)‖L2(R2) . ‖f‖L1xL
2ty
+ ‖g‖L1xL
2ty
+ ‖h‖L1xL
2ty.
The proof of this theorem is an application of Lemma 2.2 of [8]. For the sake of com-
pleteness we state the lemma here.
Lemma 3.7. Let H be a Hilbert space, X a Banach space, X∗ the dual of X, and Da vector space densely contained in X. Assume that T : D → H is a linear map and
T ∗ : H → D∗ is its adjoint, defined by
42 Chapter 3 The Kadomtsev-Petviashvili II equation
〈T ∗v, f〉D = 〈v, Tf〉 ,∀f ∈ D, ∀v ∈H ,
where D∗ is the algebraic dual of D, 〈φ, f〉D is the pairing between D∗ and D (with
f ∈ D and φ ∈ D∗), and 〈·, ·〉 is the scalar product in H (conjugate linear in the first
argument). Then the following conditions are equivalent:
(1) There exists a ∈ [0,∞), such that for all f ∈ D
‖Tf‖ ≤ a‖f‖X .
(2) R(T ∗) ⊂ X∗, and there exists a ∈ [0,∞), such that for all v ∈H ,
‖T ∗v‖X∗ ≤ a‖v‖.
(3) R(T ∗T ) ⊂ X∗, and there exists a ∈ [0,∞), such that for all f ∈ D
‖T ∗Tf‖X∗ ≤ a2‖f‖X ,
where ‖ · ‖ denotes the norm in H . The constant a is the same in all three parts. If one
of (all) those conditions is (are) satisfied, the operators T and T ∗T extend by continuity
to bounded operators from X to H and from X to X∗, respectively.
Proof of Theorem 3.6. The solution w of (3.20) can be written as
w = w1 + w2 + w3
where wi, for i = 1, 2, 3 are the solutions of the following inhomogeneous equations,
respectively,
∂tw1 + ∂3xw1 − ∂xw1 + 3∂−1
x ∂2yw1 = f, (3.21)
∂tw2 + ∂3xw2 − ∂xw2 + 3∂−1
x ∂2yw2 = ∂xg, (3.22)
∂tw3 + ∂3xw3 − ∂xw3 + 3∂−1
x ∂2yw3 = ∂−1
x ∂yh, (3.23)
where
Chapter 3 The Kadomtsev-Petviashvili II equation 43
Fxy(wi(t, x, y))→ 0 as t→ −∞.
We start by studying (3.21). We take the Fourier transform of it with respect to the
space variables x and y
∂tw1(t, ξ, η)− iξ3w1 − iξw1 + 3iη2
ξw1 = f(t, ξ, η). (3.24)
Then we solve the resulted ordinary differential equation (3.24) for each fixed ξ and η.
(3.24) ⇒ w1(t, ξ, η) =∫ t
−∞e
(iξ3+iξ−3i η2
ξ)(t−t′)
f(t′, ξ, η)dt′
⇒ w1(t, x, y)
=∫ ∞−∞
∫ ∞−∞
∫ t
−∞e
(iξ3+iξ−3i η2
ξ)(t−t′)+iξx+iηy
f(t′, ξ, η)dt′dξdη
=:∫ t
−∞e(t−t′)Sf(t′, x, y)dt′.
We define the operator
T1 : L1xL
2ty → L2
xy,
by
T1f =∫ 0
−∞e−t
′Sf(t′, x, y)dt′.
We have
〈T ∗1 v, f〉 = 〈v, T1f〉
=∫ ∞−∞
∫ ∞−∞
v
∫ 0
−∞e−t
′Sf(t′, x, y)dt′dxdy
(we use Plancherel’s theorem)
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞ve
(iξ3+iξ−3i η2
ξ)t′f(t′, ξ, η)dt′dξdη
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞et′Svf(t′, x, y)dt′dxdy.
44 Chapter 3 The Kadomtsev-Petviashvili II equation
This implies that
T ∗1 : L2xy → L∞x L
2ty
and it is defined by
T ∗1 v = etSv.
Hence
T ∗1 T1f =∫ 0
−∞e(t−t′)Sf(t′, x, y)dt′.
The local smoothing estimate (3.11) proved in Theorem 3.2 implies the boundedness of
the operator T ∗1 T1 and using the Lemma 3.7 we infer the boundedness of T1 and T ∗1 .
The boundedness of T1 gives us
‖w1(0, x, y)‖L2xy
. ‖f‖L1xL
2ty. (3.25)
Next, we treat the equation (3.22) in a similar way.
We define the operator
T2 : L1xL
2ty → L2
xy,
by
T2g =∫ 0
−∞e−tS∂xg(t)dt.
Then
Chapter 3 The Kadomtsev-Petviashvili II equation 45
〈T ∗2 v, g〉 = 〈v, T2g〉
=∫ ∞−∞
∫ ∞−∞
v
∫ 0
−∞e−tS∂xg(t, x, y)dtdxdy
=∫ ∞−∞
∫ ∞−∞
v
∫ 0
−∞e−tS∂xg(t, x, y)dtdxdy
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞−iξve(iξ3+iξ−3i η
2
ξ)tg(t, ξ, η)dtdξdη
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞−∂x(etSv)g(t, x, y)dtdxdy.
Hence
T ∗2 v = −∂x(etSv),
and
T ∗2 T2g = −∂x∫ 0
−∞e(t−t′)S∂xg(t′, x, y)dt′.
As in the study of the operator T ∗1 T1, the boundedness of the operator T ∗2 T2 follows
from the smoothing estimate (3.11), which implies the boundedness of the operator T2.
Thus we have
‖w2(0, x, y)‖ . ‖g‖L1xL
2ty. (3.26)
Finally, we study the equation (3.23).
We define
T3 : L1xL
2ty → L2
xy,
by
T3h =∫ 0
−∞e−tS∂−1
x ∂yh(t, x, y)dt.
We have
46 Chapter 3 The Kadomtsev-Petviashvili II equation
〈T ∗3 v, h〉 = 〈v, T3h〉
=∫ ∞−∞
∫ ∞−∞
v
∫ 0
−∞e−tS∂−1
x ∂yh(t, x, y)dtdxdy
=∫ ∞−∞
∫ ∞−∞
v
∫ 0
−∞
e−tS∂−1x ∂yh(t, x, y)dtdxdy
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞
η
ξve
(iξ3+iξ−3i η2
ξ)th(t, ξ, η)dtdξdη
=∫ ∞−∞
∫ ∞−∞
∫ 0
−∞∂−1x ∂y(etSv)h(t, x, y)dtdxdy.
Then
T ∗3 v = ∂−1x ∂ye
tSv,
and
T ∗3 T3h = ∂−1x ∂y
∫ 0
−∞e(t−t′)S∂−1
x ∂yh(t′)dt′.
Again (3.11) implies the boundedness of the operator T ∗3 T3, then by using the Lemma
3.7 we conclude the boundedness of the operator T3, which gives us
‖w3(0, x, y)‖L2xy
. ‖h‖L1xL
2ty. (3.27)
We combine (3.25), (3.26) and (3.27) and obtain the desired result
‖w(0, x, y)‖L2xy≤ ‖w1(0, x, y)‖L2
xy+ ‖w2(0, x, y)‖L2
xy+ ‖w3(0, x, y)‖L2
xy
. ‖f‖L1xL
2ty
+ ‖g‖L1xL
2ty
+ ‖h‖L1xL
2ty.
Definition 3.8. We say u ∈ U2S if and only if e−·Su ∈ U2 and
‖u‖U2S
= ‖e−·Su‖U2 .
Chapter 3 The Kadomtsev-Petviashvili II equation 47
Proposition 3.9. Assume ψ ∈ U2S. Then the following estimates hold true
‖ψ‖L∞x L2ty
. ‖ψ‖U2S, (3.28)
‖∂xψ‖L∞x L2ty
. ‖ψ‖U2S, (3.29)
‖∂−1x ∂yψ‖L∞x L2
ty. ‖ψ‖U2
S. (3.30)
Proof. Let φ be a U2−atom. Then there exist tkKk=0 ∈ Z and φkK−1k=0 ⊂ L
2 with
K−1∑k=0
‖φk‖2L2 = 1 and φ0 = 0
such that
φ =K∑k=1
1I[tk−1,tk)φk−1.
Let
ψ =K−1∑k=0
ψk(t)
where
ψk(t) := etSφk(tk) on [tk, tk+1).
The boundedness of T ∗1 defined in the proof of Theorem 3.6 gives us
‖ψk‖L∞x L2ty≤ c‖ψ(tk)‖L2
xy= c‖φk(tk)‖L2
xy
on [tk, tk+1).
Thus
‖ψ‖2L∞x L2ty≤
K−1∑k=0
‖ψk‖2L∞x L2ty≤ c2
K−1∑k=0
‖φk(tk)‖2L2xy≤ c2,
48 Chapter 3 The Kadomtsev-Petviashvili II equation
which implies that (3.28) is true for ψ =∑K−1
k=0 1I[tk,tk+1)etSφk(tk), where φ is an arbitrary
U2 atom. Since the constant c is independent of φ then (3.28) holds for any φ ∈ U2.
Similarly the boundedness of operators T ∗2 and T ∗3 in the proof of Theorem 3.6 gives us
‖∂xψj‖L∞x L2ty
. ‖φj(tj)‖L2xy
(3.31)
and
‖∂−1x ∂yψj‖L∞x L2
ty. ‖φj(tj)‖L2
xy(3.32)
respectively.
Summing over all j’s the estimates (3.31) and (3.32) we get the estimates (3.29) and
(3.30), respectively.
3.1.3 Miura Transformation
The Miura transformation is an explicit nonlinear transformation that relates solutions
of the KdV equation and the mKdV equation, [22]:
If v is a solution of the mKdV equation
∂tv + ∂3xv − 6v2∂xv = 0,
then u given by the Miura transformation
u = ±∂xv − v2
satisfies the KdV equation
∂tu+ ∂3xu+ 6u∂xu = 0.
In this section we use the idea of [24] and [21] of using the properties of the following
generalisation of the Miura transformation
M c±(v) = ±∂xv + ∂−1
x vy − v2 +c
2
Chapter 3 The Kadomtsev-Petviashvili II equation 49
that exploits the Galilean invariance of the KP-II equation and maps the solution of the
mKP-II equation
vt + vxxx + 3∂−1x vyy − 6v2vx + 6vx∂−1
x vy = 0 (3.33)
into the solution of the KP-II equation (3.2) by
u(t, x, y) = M c±(v)(t, x− 3ct, y).
As it is observed in [24], the kink Φc
Φc(x, y) =c1/2
2tanh
(c1/2
2x
)
is related to the line soliton of KP-II
Qc(x, y) =c
2sech2
(c1/2
2x
), c > 0
through the following relation
M c+(Φc) = Qc.
One can also easily check that
M c−(Φc) = 0.
In [3], the authors show that one can relate mKdV solutions near kink solutions to either
KdV solutions near 0 or to KdV solutions near a soliton. As expected the same relations
can be generalised to the case of KP-II and mKP-II equations’ solutions.
Proposition 3.10. Let v be a solution of the mKP-II equation linearized at Φc(x+ c2 t)
in a moving frame:
∂tv +c
2∂xv + ∂3
xv + 3∂−1x ∂2
yv + 6∂xΦc∂−1x ∂yv − 6∂x(Φ2
cv) = 0 (3.34)
and u be a solution of the KP-II equation linearized at zero
50 Chapter 3 The Kadomtsev-Petviashvili II equation
After rearranging the terms of the identity (2) above and adding the term (ξ2 −ξ1)3 − (η2−η1)2
ξ2−ξ1 to both sides of it we obtain
Chapter 4 The cubic generalized Kadomtsev-Petviashvili II equation 69
ξ31 −
η21
ξ1− ξ3
2 +η2
2
ξ2+ (ξ2 − ξ1)3 − (η2 − η1)2
ξ2 − ξ1
= (ξ − ξ2)3 − (η − η2)2
ξ − ξ2− (ξ − ξ1)3 +
(η − η1)2
ξ − ξ1+ (ξ2 − ξ1)3 − (η2 − η1)2
ξ2 − ξ1.
By the algebraic resonance identity, we have
ω := ξ1ξ2(ξ1 − ξ2)
3 +
∣∣∣η1ξ1 − η2ξ2
∣∣∣2|ξ2 − ξ1|2
= (ξ − ξ2)(ξ − ξ1)(ξ1 − ξ2)
3 +
∣∣∣η−η1ξ−ξ1 −η−η2ξ−ξ2
∣∣∣2|ξ2 − ξ1|2
(4.9)
which implies
sgn(ξ1ξ2) = sgn((ξ − ξ1)(ξ − ξ2)). (4.10)
It follows from the definitions (4.4) and (4.5) that
µ ≤ |ξ1| ≤ 2µ, λ ≤ |ξ2| ≤ 2λ, (4.11)
and
− 12
+ l ≤ η1
µξ1<
12
+ l, −12
+ k ≤ η2
λξ2<
12
+ k. (4.12)
Furthermore, since (l, k) ∈ Qj,λµm,n we have
−12µ+ 2jmλ ≤ η1
ξ1<
12µ+ 2j(m+ 1)λ,(
−12
+ 2jn)λ ≤ η2
ξ2<
(12
+ 2j(n+ 1))λ,
where |m− n| < 8. Using these data we want to estimate L given by (4.6). First
we estimate the denominator of the integrand
70 Chapter 4 The cubic generalized Kadomtsev-Petviashvili II equation
[|∇φ(τ − τ2, ξ − ξ2, η − η2)|2|∇φ(τ − τ1, ξ − ξ1, η − η1)|2
− 〈∇φ(τ − τ1, ξ − ξ1, η − η1),∇φ(τ − τ2, ξ − ξ2, η − η2)〉2]1/2
= [|∇ψ(ξ − ξ2, η − η2)−∇ψ(ξ − ξ1, η − η1)|2
+ |∇ψ(ξ − ξ2, η − η2)|2|∇ψ(ξ − ξ1, η − η1)|2
− 〈∇ψ(ξ − ξ2, η − η2),∇ψ(ξ − ξ1, η − η1)〉2]1/2
≥ [|∇ψ(ξ − ξ2, η − η2)−∇ψ(ξ − ξ1, η − η1)|2]1/2
which gives us
L2((τ1, ξ1, η1), (τ2, ξ2, η2))
≤∫
Σ
dHd−2
[|∇ψ(ξ − ξ2, η − η2)−∇ψ(ξ − ξ1, η − η1)|2]1/2. (4.13)
Next we provide more explicit determination of the interval of integration through
a detailed study.
Without loss of generality we may assume that ξ1 < ξ2.
Note that if ω > 0, then (4.9) implies that
0 < (ξ − ξ2)(ξ − ξ1)(ξ1 − ξ2) ≤ 13ω (4.14)
⇒ (ξ − ξ2)(ξ − ξ1) < 0,
⇒ ξ ∈ (ξ1, ξ2).
Combining the above result with (4.10), we have ξ1 < 0 < ξ2. Since µ < |ξ− ξ2| <2µ and λ < |ξ − ξ1| < 2λ, in this case the interval of integration is restricted to
(ξ2 − µ, ξ2).
On the other hand if ω < 0, then again from (4.9) it follows that
13ω ≤ (ξ − ξ2)(ξ − ξ1) (ξ1 − ξ2)︸ ︷︷ ︸
<0
< 0
⇒ (ξ − ξ2)(ξ − ξ1) > 0,
⇒ ξ ∈ ξ < ξ1 ∪ ξ > ξ2.
Chapter 4 The cubic generalized Kadomtsev-Petviashvili II equation 71
Thus in this case the interval of integration is (ξ2, ξ2 + µ). Substituting this infor-