Local Smoothing and Well-Posedness Results for KP-II Type Equations

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

Rheinischen Friedrich-Wilhelms-Universit¨at Bonn am Institut f¨ur Angewandte Mathe- matik.

1. Gutachter: Prof. Dr. Herbert Koch 2. Gutachter: Prof. Dr. Sebastian Herr Tag der Promotion: 26.01.2015

Erscheinungsjahr: 2015

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 Gr¨unrock, Prof. Jeremy Marzuola, Prof. Junfeng Li, Dr. Stefan Steinerberger, Dr. Dominik John, Dr. Tobias Schottdorf, Dr. Angkana R¨uland 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. Jo˜ao Carreira and Dr. Branimir ´Ca´ci´c 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 understanding.

3

Acknowledgements 3

1 Introduction 1

2 Basic notions and function spaces 5

2.1 The Fourier Transform . . . 5

2.2 Sobolev Spaces . . . 7

2.3 Besov Spaces . . . 9

2.4 Bourgain Spaces . . . 12

2.5 U^p and V^p spaces. . . 13

2.5.1 U^p spaces . . . 13

2.5.2 V^p spaces . . . 14

3 The Kadomtsev-Petviashvili II equation 17 3.1 Linear Theory. . . 18

3.1.1 A Local Smoothing Estimate Part I . . . 19

3.1.2 T^∗T Principle. . . 41

3.1.3 Miura Transformation . . . 48

3.1.4 A Local Smoothing Estimate Part II . . . 56

3.2 Notes and References. . . 58

4 The cubic generalized Kadomtsev-Petviashvili II equation 61 4.1 The Symmetries of the (gKP-II)₃ Equation . . . 63

4.2 Function Spaces. . . 64

4.3 Multilinear Estimates . . . 65

4.4 Global well-posedness for small data . . . 79

A KPII 83 A.1 Derivation of the explicit formula for the soliton Q . . . 83

B (gKP-II)3 85

Bibliography 89

5

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+∂_x³u+ 3∂_x⁻¹∂_y²u+ 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+∂³_xu+ 3∂_x⁻¹∂_y²u−6u²∂xu= 0 that satisfy initial conditions with low regularity.

When the sign in front of 3∂_x⁻¹∂_y²u term is minus in the above two equations they are called the KP-I and the (gKP-I)3equations 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)3equation 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

∂tu+∂_x³u+ 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 xand tonly throughx−ctwherecis a fixed constant. Substitutingu(t, x) =Q(x−ct) into (1.1) one obtains the ordinary differential equation

−cQ⁰+Q⁽³⁾+ 6QQ⁰ = 0 which is satisfied by the following family of solutions

Q= c

2sech²c^1/2 2 x

.

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].

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.

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

kuk_`∞ 1 2

`^p₀(L²):= sup

λ

λ^1/2X

k

ku_λ,kk^p_L2(

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 U^p (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.

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 Section2.3, [28] for Section2.4, and finally [10] and [17] for Section2.5.

2.1 The Fourier Transform

Definition 2.1. Letf ∈L¹(Rⁿ). The Fourier transform of f, denoted by ˆf, is defined as

fˆ(ξ) = (2π)⁻ⁿ² Z

e^−i(x,ξ)f(x)dx, ξ∈Rⁿ, where

(x, ξ) :=

n

X

i=1

xiξi.

We will use the notationF(f) and ˆf interchangeably.

F is a bounded linear map fromL¹(Rⁿ) toL^∞(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

Proposition 2.2. If f, g∈L¹(Rⁿ), then

(i) F(f(· −x₀))(ξ) =e^−i(ξ,x0)fˆ(ξ), (ii) F(e^i(·,ξ0)f(·))(ξ) = ˆf(ξ−ξ₀), (iii) F(f)(ξ) = ˆf(−ξ),

(iv) For(f∗g)(y) =R

Rⁿf(y−x)g(x)dx, we have f[∗g= (2π)ⁿ²fˆg,ˆ (v) F(∂_xjf)(ξ) =iξjfˆ(ξ),

(vi) F(x_jf)(ξ) =i∂ξjf(ξ),ˆ (vii) R

f(x)ˆg(x)dx=Rfˆ(ξ)g(ξ)dξ.

Definition 2.3(Schwartz function). A functionφ∈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) := sup

x∈Rⁿ

|x^α∂^βφ(x)| ≤cα,β.

We call the Fr´echet space of all Schwartz functions with the topology given by the family of semi-normsρ_α,β the Schwartz space and denote it byS(Rⁿ). The natural topology on S(Rⁿ) is as follows: a sequence of functionsφ_j converges to zero if for all multi-indicesα, β,x^α∂^βφj converges uniformly to zero. A complete metric inducing the same topology on S(Rⁿ) can be defined by

d(φ, ψ) =X

α,β

2^{−|α|−|β|} ρα,β(φ−ψ) 1 +ρ_α,β(φ−ψ).

Note thatC₀^∞(Rⁿ) is dense inS(Rⁿ) in the above defined metric topology.

Remark 2.4. The mapφ7→φˆis an isomorphism on S(Rⁿ) with the inverse

φˇ= (2π)⁻ⁿ² Z

e^i(x,ξ)φ(ξ)dξ, x∈Rⁿ. Theorem 2.5 (Plancherel’s Theorem). If φ and ψ are in S(Rⁿ), then

Z

Rⁿ

φ(x)ψ(x)dx= Z

Rⁿ

φ(ξ) ˆˆ ψ(ξ)dξ.

Definition 2.6(Tempered distributions). We define the space of tempered distributions S⁰(Rⁿ) to be the dual space of the Schwartz space.

Note that for every tempered distributionuthere existsN ∈Nand a constantC =C_α,β such that

|u(φ)| ≤C X

|α|,|β|≤N

sup|x^α∂^βφ|, φ∈S(Rⁿ).

Then the definition of the Fourier transform can be further naturally extended to the tempered distributions by

ˆ

u(φ) =u( ˆφ), φ∈ S(Rⁿ).

Theorem 2.7. The Fourier transformF extends to a unitary map fromL²(Rⁿ)to itself and thus the following identity of Parseval holds

kˆuk_L2(Rⁿ)=kuk_L2(Rⁿ).

Furthermore sinceL^p ⊂S⁰(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 inRⁿ, 1≤p≤ ∞andsbe a nonnegative integer. The Sobolev space W^s,p consists of all locally summable functions u : Ω→ R such that for each multiindexα with|α| ≤s,∂^αu exists in the weak sense and belongs toL^p(Ω). W^s,p is a normed space equipped with the norm

kuk_W^s,p :=





 P

|α|≤s

R

Ω|∂^αu|^pdx _p¹

if 1≤p <∞ , P

|α|≤sesssup_Ω|∂^αu| ifp=∞.

Remark 2.9. Among the spacesW^s,p, particular importance is attached toW^s,2 because they are Hilbert spaces. We denote them by H^s.

Definition 2.10 (FractionalH^s−Sobolev spaces). Lets∈R. We say thatu∈H^s(Rⁿ) ifu∈ S⁰(Rⁿ) has a locally integrable Fourier transform and

kuk²_Hs :=

Z

Rⁿ

(1 +|ξ|²)^s|ˆu(ξ)|²dξ <∞.

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 ˙H^s equipped with the following semi-norm

kuk²_˙

H^s :=

Z

Rⁿ

|ξ|^2s|ˆu(ξ)|²dξ <∞ (2.1) the homogeneous Sobolev space.

Definition 2.13(Non-isotropic Homogeneous Sobolev space). Lets1, s2 ∈R. ˙H^s1,s2(R²) is the space of tempered distributions with

kuk_H˙s1,s2 :=

Z

R²

|ξ|^2s1|η|^2s2|ˆu(ξ, η)|²dξdη ¹₂

<∞. (2.2)

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. Letφ(ξ) be a real radial bump function such that

φ(ξ) =

( 1 if|ξ| ≤1, 0 if|ξ|>2, andχ(ξ) =φ(ξ)−φ(2ξ). Thenχ(ξ) is supported on

ξ∈Rⁿ: ¹₂ ≤ |ξ| ≤2 and satisfies

X

k∈Z

χ(2^−kξ) = 1.

We define the Littlewood-Paley projection P_k by

Pd_kf(ξ) =χ(ξ/2^k) ˆf(ξ) in frequency space, or equivalently in physical space by

P_kf =f_k=m_k∗f,

where m_k(x) = 2^nkm(2^kx) and m(x) is the inverse Fourier transform of χ. Then ∀f ∈ L²(Rⁿ) we have

f =X

k∈Z

P_kf.

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 {P_kf}_k is almost orthogonal in L²(Rⁿ) in the following sense

P_k1P_k2 = 0 whenever |k₁−k₂| ≥2 and

kfk_L2 ≈X

k

kP_kfk²_L2,

which is an easy consequence of Parseval’s Identity.

(ii) [L^p−boundedness] Let J ⊂Z and 1 ≤p ≤ ∞. Then the following estimate holds true

kP_Jfk_Lp.kfk_Lp.

(iii) [Finite band property] Let k be an integer. For any 1≤p≤ ∞

k∂P_kfk_L^p .2^kkfk_L^p, 2^kkP_kfk_L^p .k∂fk_L^p. (iv) [Bernstein inequalities] For any 1≤p≤q ≤ ∞we have

kP_kfk_Lq .2kn(1/p−1/q)kfk_Lp, ∀k∈Z, kP_≤0fk_L^q .kfk_L^p.

Remark 2.15. The Bernstein inequality is a remedy for the failure of

W^np,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 tos∈Rby using the Littlewood-Paley theory as follows

kfk_W˙s,p ≈

X

k∈Z

2^ksPkf L^p, kfk_W^s,p ≈

X

k∈Z

(1 + 2^k)^sP_kf L^p.

Definition 2.17 (Besov Spaces). Lets∈Rand 1≤p, q ≤ ∞. The Besov space is the completion ofC₀^∞(Rⁿ) with respect to the norm defined by

kfk_B^s_p,q :=











kP_≤0fk^q_Lp+P∞

k=12^sqkkP_kfk^q_Lp

1/q

if 1≤q <∞, sup{kP_≤0fk_L^p,2^skkP_kfk_L^p} ifq =∞.

Definition 2.18 (Homogeneous Besov Spaces). Let s ∈ R and 1 ≤ p, q ≤ ∞. The homogeneous Besov norm is defined by

kfk_B˙s

p,q :=









 P

k∈Z2^sqkkP_kfk^q_Lp

1/q

if 1≤q <∞,

sup_k2^skkP_kfk_L^p ifq=∞.

We collect the main Besov space embeddings in the following proposition.

Proposition 2.19. Assume thats−ⁿ_p =s₁−_pⁿ

1. Then

(i) B_p,q^s ,→B_p^s1_1,q1, if 1≤p≤p1 ≤ ∞, 1≤q≤q1≤ ∞, s, s1 ∈R, (ii) B_p,p^s ,→W^s,p,→B^s_p,2, if s∈R, 1< p≤2,

(iii) B_p,2^s ,→W^s,p,→B^s_p,p, if s∈R, 2≤p <∞.

The anisotropic Besov spaces are called Besov-Nikol’skii spaces in literature.

Definition 2.20 (Besov-Nikol’skii Spaces). Suppose S = (s1, s2, . . . , sn) ∈ Rⁿ, N = (N₁, N₂, . . . , N_n)∈Zⁿand 1≤p, q≤ ∞. The linear spaceB_p,q^S of tempered distributions equipped with the norm

kfk_BS p,q =



kP_{(N1≤0,N2≤0,...,Nn≤0)}fk^q_Lp+ X

N∈Zⁿ+

2^q(S·N)kP_Nfk^q_Lp





1/q

,

is called a Besov-Nikol’skii space.

2.4 Bourgain Spaces

In this section, we present Bourgain spaces (also known as Fourier restriction spaces, or X^s,b spaces). The Bourgain spaces are constructed based on the linear part of the dispersive equation.

Leth be a real valued polynomial and L=ih ¹_i∇

. We consider

∂tu−Lu= 0. (2.3)

Taking the space-time Fourier transform of (2.3) we get

[τ −h(ξ)]ˆˆ u(τ, ξ) = 0.

Then ˆu(τ, ξ) is supported in{(τ, ξ) :τ =h(ξ)} which is called the characteristic hyper- surface of the space-time frequency space R×Rⁿ.

Hence

ˆ

u(τ, ξ) =δ(τ −ˆh(ξ))ˆu0(ξ), whereδ is the Dirac delta function defined by

δ(φ) =φ(0).

Now we consider a nonlinear perturbation of (2.3)

∂_tu−Lu−N(u) = 0. (2.4)

Note that if one multiplies a solution of (2.4) by suitably short time cutoff function, then for many types of nonlinearities and initial data the localised Fourier transform concentrates near the characteristic hypersurface. Because Bourgain spaces are built on the linear parts of dispersive equations, they reflect this dispersive smoothing effect.

Definition 2.21 (X^s,b spaces). Let h : Rⁿ → R be a continuous function, and let s, b∈R. The spaceX_τ=h(ξ)^s,b (R×Rⁿ), abbreviatedX^s,b(R×Rⁿ) or simplyX^s,b, is then defined to be the closure of the Schwartz functionsS_t,x(R×Rⁿ) under the norm

kukX_τ^s,b_=h(ξ)(R×Rⁿ):=k(1 +|ξ|²)^s/2(1 +|τ−h(ξ)|²)^b/2u(τ, ξ)kˆ _L2

τL²_ξ(R×Rⁿ).

Observe that if we take b = 0, then the X^s,b space is L²_tH_x^s, and if we take h = 0 the X^s,b space is simplyH_t^bH_x^s.

Lemma 2.22 (The Basic Properties of X^s,b spaces).

(i) X^s,b spaces are Banach spaces,

(ii) X_τ=h(ξ)^s0,b0 ,→X_τ=h(ξ)^s,b whenever s⁰ ≥sand b⁰ ≥b, (iii)

X_τ=h(ξ)^s,b ∗

=X_{τ=−h(−ξ)}^−s,−b ,

(iv) TheX^s,b spaces are invariant under translations in space and time, (v) kuk

X_{τ=−h(−ξ)}^s,b =kuk

X_τ=h(ξ)^s,b .

2.5 U

and V

spaces

In this section, we give a brief summary of the theory of U^p and V^p function spaces covered in detail in [10] and [17]. These spaces are useful in the analysis of nonlinear dispersive partial differential equations and have better properties than X^s,b spaces especially at critical regularity. TheU^p spaces have been introduced by H. Koch and D.

Tataru in [18], [19] and theV^p spaces have been introduced by N. Wiener in [34].

2.5.1 U^p spaces

Let

Z ={(t₀, t₁, . . . , t_K)| −∞=t₀ < t₁ < . . . < t_K =∞}

and

Z₀ ={(t₀, t₁, . . . , t_K)| −∞< t₀ < t₁ < . . . < t_K <∞}

be the sets of finite partitions.

Definition 2.23. Let 1≤p <∞. Assume{t_k}^K_k=0 ∈ Z and {φ_k}^K−1_k=0 ⊂L² with

K−1

X

k=0

kφ_kk^p_L2 = 1 and φ₀ = 0.

The function a:R→L² given by

a=

K

X

k=1

χ_[tk−1,tk)φk−1

is called aU^p-atom.

The atomic spaceU^p is defined as

U^p:=





 u=

∞

X

j=1

λ_ja_j |a_j U^p−atom,λ_j ∈C such that

∞

X

j=1

|λ_j|<∞





 ,

with norm

kuk_U^p := inf







∞

X

j=1

|λ_j| |u=

∞

X

j=1

λjaj, λj ∈C, aj U^p-atom





 . Proposition 2.24 (Properties of U^p spaces). Let 1≤p < q <∞.

(i) U^p is a Banach space, (ii) U^p ,→U^q ,→L^∞(R;L²),

(iii) Every u∈U^p is right continuous, (iv) limt→−∞u(t) = 0, limt→∞u(t) exists,

(v) The closed subspace of all continuous U^p functions, denoted by Uc^p, is a Banach space.

2.5.2 V^p spaces

Definition 2.25. The V^p space is the normed space of all functions v : R→ L² such that limt→±∞v(t) exist and for which the norm

kvk_V^p := sup

{t_k}^K_k=0∈Z K

X

k=1

kv(t_k)−v(t_k−1)k^p_L2

!¹_p

is finite with v(−∞) = lim_t→−∞v(t) andv(∞) = 0.

V₋^p denotes the closed subspace of allv∈V^p with limt→−∞v(t) = 0.

Proposition 2.26 (Properties of V^p space). Let 1≤p < q <∞.

(i) Define

kvk_V^p

0 := sup

{t_k}^K_k=0∈Z0

K

X

k=1

kv(t_k)−v(tk−1)k^p

L²

! .

If v : R → L² and kvk_V^p

0 < ∞, then v has left and right limits at every point.

Moreover

kvk_V^p =kvk_V^p

0.

(ii) The closed subspaces of all right-continuous V^p and V₋^p functions are denoted by Vrc^p andV_−,rc^p , respectively.

(iii) U^p ,→V_−,rc^p .

(iv) V^p ,→V^q and V₋^p ,→V₋^q. (v) V_−,rc^p ,→U^q.

Proposition 2.27 (Duality). Let u∈U^p, v∈V^p0 andt={t_k}^K_k=0∈ Z. Define

Bt(u, v) :=

K

X

k=1

hu(t_k)−u(tk−1), v(t_k)i,

where h·,·i denotes L² inner product. There exists a unique number B(u, v), such that for all ε >0 there exists t∈ Z such that for every t⁰⊃t

|B_t⁰(u, v)−B(u, v)|< ε.

is satisfied. Furthermore the associated bilinear form B : (u, v)7→B(u, v) satisfies

|B(u, v)| ≤ kuk_U^pkvk_Vp0.

Theorem 2.28. Let 1< p <∞. Then

(U^p)^∗ =V^p0, in the sense that the operator

T :V^p0 →(U^p)^∗, defined by

T(v) :=B(·, v) is an isometric isomorphism.

Proposition 2.29. Let 1 < p < ∞, u ∈ U^p be continuous and v, v^∗ ∈ V^p0. Suppose thatv(s) =v^∗(s) except for countably many points. Then

B(u, v) =B(u, v^∗).

Proposition 2.30. Suppose that 1< p < ∞, v ∈V^p0 and u∈V₋¹ is absolutely contin- uous on compact intervals. Then

B(u, v) =− Z ∞

−∞

hu⁰(t), v(t)idt.

The Kadomtsev-Petviashvili II equation

In this chapter, we present the results of our attempt to solve the problem of the stability of line solitons

Q_c(x, y) = c

2sech² c^1/2x 2

!

, c >0 (3.1)

for the Kadomtsev-Petviashvili II (KP-II) equation

∂tu+∂_x³u+ 6u∂xu+ 3∂_x⁻¹∂_y²u= 0, (3.2) whereu=u(t, x, y) is a real valued function and

(∂_x⁻¹u)(x) :=− Z ∞

x

u(s)ds. (3.3)

The validity of the conserved quantities of the KP-II equation requires the following two constraints on the initial data

Z ∞

−∞

u(x, y)dx= 0, (3.4)

Z ∞

−∞

Z x

−∞

u(x⁰, y)dx⁰dx= 0. (3.5)

17

The solution that evolves from the initial data satisfying (3.4) and (3.5) preserves these constraints for all time, [33].

3.1 Linear Theory

In this section, we study the linear equation

∂tw+∂_x³w−∂xw+ 6∂x(Qw) + 3∂⁻¹_x ∂_y²w=F, (3.6) whereQ is the line soliton defined by (3.1) withc= 1.

The linear equation (3.6) results from linearization of (3.2) aroundQin a moving coor- dinate system

x→x−t.

First, we derive a local smoothing estimate for the solution of the linearized problem (3.6) without the potential term

∂_tw+∂_x³w−∂_xw+

XX6∂_x(Qw) + 3∂XXX ⁻¹_x ∂_y²w=F. (3.7) Next, we estimate the initial data in terms of the inhomogenous data usingT^∗Tprinciple, [8]. Then, we prove estimates relating the solutions of the homogeneous linearized equation with and without potential term inL²,L^∞andL¹spaces inx−direction using the mapping properties of Miura type transforms. Finally, we use properties of Miura maps and the local smoothing estimate obtained for (3.7) to prove the main result of this chapter, stated in the following theorem.

Theorem 3.1. [A Local Smoothing Estimate]

Let η be the Fourier variable corresponding to y andw be a solution of

∂tw+∂_x³w−∂xw+ 6∂x(Qw) + 3∂⁻¹_x ∂_y²w=f +∂xg+∂⁻¹_x ∂yh

| {z }

=F

, (3.8)

where f, g and h have compact supports in t≥0.

Then we have the following local smoothing estimate

kF_y(w)k_L∞

x L²_t +k∂_xF_y(w)k_L∞

x L²_t +kη∂_x⁻¹F_y(w)k_L∞ xL²_t

.kF_y(f)k_L1

xL²_t +kF_y(g)k_L1

xL²_t +kF_y(h)k_L1

xL²_t (3.9) provided that η6= 0.

3.1.1 A Local Smoothing Estimate Part I

We study here the linear problem without the potential term∂_x(Qw), namely the equa- tion (3.7). First, we prove the local smoothing estimate (3.9) for (3.7) with no restriction on η.

Theorem 3.2. Let w be a solution of

∂tw+∂_x³w−∂xw+ 3∂_x⁻¹∂_y²w=f+∂xg+∂_x⁻¹∂yh

| {z }

=F

, (3.10)

then

kF_y(w)k_L∞

x L²_tL²_η+k∂_xF_y(w)k_L∞

xL²_tL²_η+kη∂_x⁻¹F_y(w)k_L∞ xL²_tL²_η

.kF_y(f)k_L1

xL²_tL²_η+kF_y(g)k_L1

xL²_tL²_η+kF_y(h)k_L1

xL²_tL²_η, (3.11) where η is the Fourier variable corresponding to y variable and f, g, h have compact supports in t≥0.

Proof. We take the Fourier transform of (3.10) with respect tot,x and y

iτwˆ−iξ³wˆ−iξwˆ+ 3iη²

ξ wˆ= ˆf +iξˆg+η

ξˆh. (3.12)

Then we solve the above algebraic equation for ˆw and take its inverse Fourier transform with respect tox which formally can be written as

F_ty(w) = (2π)⁻¹²

Z fˆ+iξˆg+^η_ξˆh

τ−ξ³−ξ+ 3^η_ξ²e^ixξdξ. (3.13)

Then we calculate theL^∞_x L²_tL²_ηnorms of the above expression,∂xF_ty(w) andη∂_x⁻¹F_ty(w) which are simply the terms on the left hand side of the local smoothing estimate (3.11) that we want to prove. Before proceeding with the calculations we make the following two remarks which will help to make the integral on the right hand side of (3.13) well- defined.

Remark 3.3. Consider the Fourier transform ofF(=f+∂_xg+∂_x⁻¹∂_yh) with respect to t

Fˆ(τ) = Z ∞

0

F(t)e^−itτdt.

Note that forz=Re^iθ

F(z) =ˆ Z ∞

0

F(t)e^{−iRt(cosθ+isinθ)}dt

since t≥ 0 andR ≥ 0 the above integral is bounded only if θ∈ [π,2π]. ThenF(τ) is nothing but the restriction of a holomorphic function defined on the lower half plane to the real axis.

Remark 3.4. Let A be the antiderivative operator defined by (3.3). Assume that φ ∈ S(R).

Ifξ = 0, then

F[Aφ](ξ) =−(2π)^1/2 Z ∞

−∞

Z ∞ x

φ(s)dsdx.

Ifξ 6= 0, then

F[Aφ](ξ) = −(2π)^1/2 Z ∞

−∞

e^−ixξ Z ∞

x

φ(s)dsdx

= (2π)^1/2 iξ e^−ixξ

Z ∞ x

φ(s)ds

x=∞

x=−∞+ (2π)^1/2 Z ∞

−∞

1

iξe^−ixξφ(x)dx

= 1

iξ φ(ξ).ˆ

Note that the right hand side is also defined for every complexξwith positive imaginary part.

Thus we can write (3.12) as

i(τ −i0) ˆw−iξ³wˆ−iξwˆ+ 3i η²

ξ+i0wˆ = ˆf+iξˆg+ η ξ+i0ˆh.

Hence, in order to obtain (3.11) it is enough to show that the following 6 simpler integrals are uniformly bounded

I1 = lim

ε→0⁺

Z ∞

−∞

ξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ, I2 = lim

ε→0⁺

Z ∞

−∞

ξ²e^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ, I₃ = lim

ε→0⁺

Z ∞

−∞

ηe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ, I₄ = lim

ε→0⁺

Z ∞

−∞

ξ³e^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ, I₅ = lim

ε→0⁺

Z ∞

−∞

ηξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ, I6 = lim

ε→0⁺

Z ∞

−∞

η²e^ixξ

(ξ+iε)(ξ⁴+ξ²−(τ−iε)ξ−3η²)dξ.

Note thatI₄ only exists as improper Lebesgue integral, that is as the limit

ε→0lim⁺ lim

R→∞

Z R

−R

ξ³e^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ.

Let us denote the denominator ofIi fori= 1. . . . ,5 by

p(ξ) =p(ξ)−i0ξ.

Observe that p⁰(ξ) = 4ξ³+ 2ξ−τ and p⁰⁰(ξ) = 12ξ²+ 2 > 0 so p is a strictly convex function andp(ξ) has one nonnegative and one nonpositive real root and 2 complex roots that are conjugates. Simple algebraic calculations show that adding i0ξ top(ξ) pushes both real roots to the lower half plane. To be more precise the roots of p(ξ) which we denote byξ1,ξ2,ξ3 andξ4 have the following properties

Im(ξ1) =−ε₁ <0, (3.14)

Im(ξ₂) =−ε₂ <0, (3.15)

Im(ξ₃)>0, (3.16)

Im(ξ₄)<0, (3.17)

whereε₁ andε₂ are small positive numbers.

We continue by further analysing the polynomialp(ξ). We study the polynomialp(ξ) in the following 9 regions:

Region I:={(τ, η) :|τ| ≤ ¹₂ and |η| ≤ ¹₄}, Region II:={(τ, η) :|τ| ≤ ¹₂ and |η|> ¹₄},

Region III:={(τ, η) :|τ| ≥10,|η| ≥10 and 3^|η|_|τ|² ≥ |τ|^1/3}, Region IV:={(τ, η) :|τ| ≥10,|η| ≥10 and 3^|η|_|τ|² <|τ|^1/3}, Region V:={(τ, η) :|τ| ≥10,|η| ≤10 and|τ|<10|η|²}, Region VI:={(τ, η) :|τ| ≥10,|η| ≤10 and |τ| ≥10|η|²}, Region VII:={(τ, η) : ¹₂ <|τ|<10,¹₄ <|η|and |τ|<|η|}, Region VIII:={(τ, η) : ¹₂ <|τ|<10,¹₄ <|η|and |τ| ≥ |η|}, Region IX:={(τ, η) : ¹₂ <|τ|<10 and |η| ≤ ¹₄}.

In the region I, we approximate the real roots of p(ξ) by the roots of the quadratic polynomial q(ξ) =ξ²−τ ξ−3η². Because

(i) p⁰(0) =q⁰(0) andp⁰⁰(ξ)≥q⁰⁰(ξ) which suggest a picture as follows

Ξ y

q p

(ii) Asτ →0 andη→0, the real roots of p(ξ) approach to the roots of q(ξ).

Since

p(ξ) =q(ξ)(ξ²+τ ξ+ 1 +τ²+ 3η²) + (τ³+ 6η²τ)ξ+ 3η²τ²+ 9η⁴ then the roots of p(ξ) are as follows:

ξ1≈ τ−p

τ²+ 12η² 2 −iε1, ξ₂≈ τ+p

τ²+ 12η² 2 −iε₂, ξ₃≈ −τ +p

−4−3τ²−12η²

2 ,

ξ₄≈ −τ −p

−4−3τ³−12η²

2 .

The≈sign above denotes a constant bound on the error which can be shown to be strictly less than ⁴₅|η|using the fact (ii) and the numerical data obtained by Mathematica 8.

Moreover

Im(ξ1) =

( 0 ifη= 0 and τ ≥0, strictly negative otherwise,

Im(ξ2) =

( 0 ifη= 0 and τ <0, strictly negative otherwise,

and

Im(ξ3,4) ≥1.

In regions II, III and VII one can approximate the roots ofp(ξ) by the roots of the simpler quartic polynomialr(ξ) =ξ⁴+ξ²−3η². The numerical data obtained by Mathematica 8 suggests that

d({roots ofr(ξ)},{roots ofp(ξ)})<0.2 in Region II, d({roots ofr(ξ)},{roots ofp(ξ)})<3^1/4η^1/2 in Region III, d({roots ofr(ξ)},{roots ofp(ξ)})<0.15 in Region VII,

whereddenotes the distance function defined by

d({roots ofr(ξ)},{roots ofp(ξ)}) := inf

ζi:root ofr(ξ), ξi:root ofp(ξ)

d(ζi, ξi).

Then we have

ξ1 ≈ − s

−1 +p

1 + 12η²

2 −iε1, ξ2 ≈ s

−1 +p

1 + 12η² 2 −iε2, ξ₃ ≈i

s 1 +p

1 + 12η²

2 , ξ₄ ≈ −i

s 1 +p

1 + 12η²

2 .

In analysing the regions IV, V, VIII and IX the theorem stated below will prove to be useful.

Theorem 3.5. Let ai,x∈C for i= 1, . . . , n and

p(x) =a0+a1x+· · ·+anxⁿ.

(i) If there is a positive real number m such that

|a₀| ≥ |a₁|m+|a₂|m²+· · ·+|a_n|mⁿ (3.18)

then m is a lower bound for the size of all the roots of the polynomial p(x). For example

m= |a₀|

max{|a₀|,|a₁|+|a₂|+. . .+|a_n|}

is a solution of the inequality (3.18).

(ii) If

|a_n|Mⁿ≥ |a₀|+|a₁|M+. . .+|an−1|Mⁿ⁻¹ (3.19) thenM is an upper bound for the size of all the roots of p(x) and

M = max{1, 1

|a_n|(|a₀|+|a₁|+· · ·+|a_n−1|)}

is a solution of (3.19).

Proof. Let r be an arbitrary root of the polynomialp(x).

(i) If |r|< m, then

|a₀|=

n

X

j=1

a_jr^j

≤

n

X

j=1

|a_j||r|^j <

n

X

j=1

|a_j|m^j,

which is the contrapositive form of the statement (i).

(ii) If|r|> M, then

0 =

n

X

j=0

ajr^j

=|r|ⁿ

n

X

j=0

ajr^j−n

≥ |r|ⁿ



|a_n| −

n−1

X

j=0

|a_j||r|^j−n





> |r|ⁿ



|a_n| −

n−1

X

j=0

|a_j|M^j−n





= |r|ⁿ Mⁿ



|a_n|Mⁿ−

n−1

X

j=0

|a_j|M^j





which completes the proof, since it is the contrapositive of the statement we wanted to prove.

In region IV,p(ξ) has one root that has size smaller than 3_|τ|^η2 and the remaining roots have sizes larger than |τ|^1/3. Moreover

min

|ξ_i| ≥ 3 2

η²

|τ| and max

|ξ_i| <2|τ|^1/3 due to the Theorem3.5.

In region V again thanks to the Theorem 3.5,

min

|ξ_i| ≥0.29 and max

|ξ_i| <11.

In region VI, the roots ofp(ξ) can be approximated as follows

ξ1 ≈ −3η² τ −iε1, ξ2 ≈ sgn(τ)|τ|^1/3−iε2, ξ₃ ≈ sgn(τ) −|τ|^1/3

2 +

√3 2 |τ|^1/3i

! ,

ξ₄ ≈ sgn(τ) −|τ|^1/3

2 −

√3 2 |τ|^1/3i

! .

Moreover

ξ1−sgn(−τ)3η²

|τ|

<0.01 and |ξ_i−τ^1/3|<0.2 for i= 2,3,4.

In region VIII we have

min

|ξ_i| > 1

54 and max

|ξ_i| <5.

In region IX, p(ξ) has one root that has the same size with

−3|η|²

|τ| +α where

α≈ 81^|η|_|τ|⁸4 + 9^|η|_|τ|⁴2

|τ|+ 108^|η|_|τ|⁶3 + 6^|η|_|τ|².

Then we can decomposep(ξ) into dominant parts and a small remainder as follows

ξ⁴+ξ²−τ ξ−3η² =

ξ+3η² τ −α

Q(ξ)−ατ+ 3η²

τ −α 2

+ 3η²

τ −α 4

, where

Q(ξ) =ξ³−3η² τ −α

ξ²+

1 +

3η² τ −α

2 ξ−

τ+ 3η² τ −α+

3η² τ −α

3 .

It follows from the Theorem3.5thatm= 0.25 is a lower bound for the size of each root of Q(ξ).

Also note that in each region all the roots are distinct and therefore all the poles of Ii

fori= 1, . . . ,5 are simple.

We summarise the analysis of the roots of p(ξ) or in other words the analysis of the poles of the integrand ofI_i,i= 1, . . . ,5, in the following table

REGIONS Poles in upper half plane Poles in lower half plane

Region I

ξ₁≈ ^τ−

√

τ²+12η²

2 −iε₁

ξ3 ≈ ^−τ+

√

−4−3τ²−12η²

2 ξ2≈ ^τ+

√

τ²+12η²

2 −iε2

ξ₄≈ ^−τ−

√

−4−3τ³−12η² 2

ξ₁≈ −

q−1+√

1+12η²

2 −iε₁

Regions II, III ξ3 ≈i q

1+

√

1+12η²

2 ξ2≈

q−1+√

1+12η²

2 −iε2

and VII

ξ₄≈ −i q1+√

1+12η² 2

Region IV

3η²

2|τ| ≤ |ξ₁|<3^|η|_|τ|²

|τ|¹³ <|ξ₃|<2|τ|¹³

|τ|¹³ <|ξ₂|,|ξ₄|<2|τ|¹³

Region V 0.29≤ |ξ₃|<11 0.29≤ |ξ₁|,|ξ₂|,|ξ₄|<11

Region VI

ξ1≈ −^3η_τ² −iε1

ξ₃ ≈sgn(τ)

−^|τ|₂^1/3 +

√ 3

2 |τ|^1/3i

ξ₂≈sgn(τ)|τ|^1/3−iε₂ ξ₄≈sgn(τ)

−^|τ|₂^1/3 −

√ 3

2 |τ|^1/3i

Region VIII 1/54≤ |ξ₃|<5 1/54≤ |ξ₁|,|ξ₂|,|ξ₄|<5

Region IX

|ξ₁| ≈3|η|²/|τ|+α

|ξ₃| ≈ |τ|¹³

|ξ₂|,|ξ₄| ≈ |τ|¹³

Now that we have the necessary information about the roots of the polynomial p(ξ) we proceed to the calculations of bounds of integrals I_i, i = 1, . . . ,6. Note that the boundedness of the integralI2 follows from the application of Cauchy-Schwarz inequality and the boundedness of I₁ and I₄. Similarly, the boundedness of I₁ and I₆ imply boundedness of I₃ and the boundedness of I₄ and I₆ imply that I₅ is bounded due to Cauchy-Schwarz inequality. So it suffices to show that the integrals I1, I4 and I6 are bounded.

Claim 1: |I₁|is uniformly bounded.

Proof of Claim 1: We define closed curves γ₁ and γ₂ as illustrated below.

x y

Ç

-R Å

Γ₁

R r -r

,

x y

Ç

Å -R

Γ₂ R r -r

Note that ifx >0, then

Z

γ1

ξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ = Z −r

−R

ξe^ixξ

ξ⁴+ξ²−(τ−iε)ξ−3η²dξ +

Z 0 π

re^iθe^ixrcosθe^−xrsinθire^iθdθ (re^iθ)⁴+ (re^iθ)²−(τ −iε)(re^iθ)−3η²

| {z }

:=Ir

+ Z R

r

ξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ +

Z π 0

Re^iθe^ixRcosθe^−xRsinθiRe^iθdθ (Re^iθ)⁴+ (Re^iθ)²−(τ −iε)(Re^iθ)−3η²

| {z }

:=IR

.

Since x >0 we have|e^−xRsinθ| ≤1 and hence

|I_R| −→0 as R−→ ∞, and

ifη= 0 and τ = 0, |I_r| −→π asr −→0, otherwise |I_r| −→0 as r−→0.

Ifx <0 we couldn’t have argued this way. One alternative would be to choose the closed pathγ₂, then we get

Z

γ2

ξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ = Z r

R

ξe^ixξ

ξ⁴+ξ²−(τ −iε)ξ−3η²dξ +

Z π 0

re^iθe^ixrcosθe^−xrsinθire^iθdθ (re^iθ)⁴+ (re^iθ)²−(τ −iε)(re^iθ)−3η²

| {z }

=−Ir

+

Z −R

−r

ξe^ixξ

ξ⁴+ξ²−(τ−iε)ξ−3η²dξ +

Z 0

−π

Re^iθe^ixRcosθe^−xRsinθiRe^iθdθ (Re^iθ)⁴+ (Re^iθ)²−(τ −iε)(Re^iθ)−3η²

| {z }

:=IR

.

where|I_R| →0 asR → ∞, sinceθ∈(−π,0) and x <0.

Finding a uniform bound on |I₁|is thus equivalent to finding a uniform bound on Z

γi

ξe^ixξ

ξ⁴+ξ²−(τ−iε)ξ−3η²dξ.

Tedious but simple estimates on the location of the roots of the polynomial show that the following estimates hold.

Letgdenote the integrand of I₁ andn(γ₁;ξ_k) denote the index of γ₁ with respect toξ_k, then in the Region Iifx >0 we have