Applications of Complex Variables to Spectral Theory and ......The KP, KdV Equations and Re ectionless Schr odinger Operators The Kaup{Broer System and 2+1 Dimensional Generalization

Applications of Complex Variables to Spectral Theoryand to Completely Integrable Nonlinear PDEs:

Application to Solitons and Nonlinear Periodic Waves

Patrik Vaclav Nabelek

The University of Arizona: Program in Applied Mathematics

April 10, 2018

Patrik Vaclav Nabelek (UA:PAM) Applications of Complex Variables April 10, 2018 1 / 47

Acknowledgements

I would like to thank my advisors Vladimir Zakharov and KenMcLaughlin for their advice and support.

I would like the thank my committee members Nick Ercolani andDoug Pickrell.

I would like to thank my family for their support.

I would like to thank my friends for their support, and for makingmy time in graduate school more fun.


The First Sighting of a Soliton

“I was observing the motion of a boat which was rapidly drawn along anarrow channel by a pair of horses, when the boat suddenly stopped –not so the mass of water in the channel which it had put in motion; itaccumulated round the prow of the vessel in a state of violentagitation, then suddenly leaving it behind, rolled forward with greatvelocity, assuming the form of a large solitary elevation, a rounded,smooth and well-defined heap of water, which continued its coursealong the channel apparently without change of form or diminution ofspeed. I followed it on horseback, and overtook it still rolling on ...,preserving its original figure some thirty feet long and a foot to a footand a half in height. Its height gradually diminished, and after a chaseof one or two miles I lost it ...”

– John Scott Russel, 1844


Overview of Talk

In this talk we will:

1. Give an overview of the traveling wave solutions to the KdVequation.

2. Produce the multi soliton solutions to the KP equation and theKdV equation using the dressing method.

3. Produce the multi soliton solutions to the Kaup–Broer system andan integrable 2+1 dimensional generalization for the KB system.(Work with Dr. Zakharov)

4. Discuss a Riemann–Hilbert problem approach to inverse spectraltheory, prove uniqueness, and discuss application of theRiemann–Hilbert to the KdV equation. (Work with Dr.McLaughlin)


Table of Contents

1 Introduction:Overview of New Results and Traveling Waves

2 The Dressing MethodThe KP, KdV Equations and Reflectionless Schrodinger OperatorsThe Kaup–Broer System and 2+1 Dimensional Generalization

3 Spectrum of Periodic Schrodinger Operators and KdV Equation


Result on Kaup–Broer System

One of our main results on the Kaup–Broer system is that if χn solve

N∑n=1

(δmn +

cme−φ(zm)

zn − z−1m

)χn = cme

−φ(zm), (1)

φ =

(λ− 1

λ

)x+

1

2

(λ2 − 1

λ2

)t. (2)

Then if

B(x, t) = − ∂

∂x

N∑n=1

χn, s(x, t) = − log

(1−

N∑n=1

χnzn

)(3)

the functions ϕ = s and η = B − 12sxx are N–soliton solutions to the

Kaup–Broer system

ηt + ϕxx + (ηϕx)x +1

4ϕxxxx = 0, (4)

ϕt +1

2ϕ2x + η = 0. (5)


Riemann–Hilbert Problem for Infinite Gap Potentials

1. For a periodic initial condition q0(x) ∈ L∞(R), the solution to theKdV equation

qt − 6qqx + qxxx = 0 (6)

with q(x, 0) = q0(x) is determined by solving a Riemann–Hilbertproblem. This has been done for finite gap solutions, we do it forinfinite gap solutions. The main challenge was the establishmentof a uniqueness theorem.

2. This can be thought of as a nonlinear Fourier transform in thesense that the space and time dependence will enter in somesimple phase factors e2ikx+8ik3t solving qt + qxxx = 0.

3. At each time t0, the Schrodinger operator L = − ∂2

∂x2+ q(x, t0) are

determined by their spectrum which is constant in time is a halfline with possibly an infinite number of intervals (gaps) removes,and a point on the circle over each band gap which evolves in time.


Traveling Wave Solutions

1. Since the KdV equation qt − 6qqx + qxxx = 0 is translationinvariant, it must conserve momentum. This is related to theexistence traveling wave solutions.

2. The traveling wave solutions are the building blocks for morecomplicated solutions, as will be the subject of the latter chapters.

3. This process is analogous to building solutions to linear equationsvia superposition, and is known as “nonlinear superposition.”


Explicit Traveling Wave Solutions to the KdV

1. The soliton solutions are given by

q(x, t) = − c2

sech2

(√c

4(x− ct− x0)

). (7)

2. The periodic cnoidal wave solutions are given by

q(x, t) = 2℘(x− ct− x0 + iβ)− c

6(8)

where ℘ is the Wierstrass elliptic function with half periods α, iβ.


The Cnoidal Wave in Nature

http://www.amath.washington.edu/%7Ebernard/kp/waterwaves/panama.jpg


Table of Contents





The Dressing Method for KP and KdV Equations

In this subsection we will:

1. Introduce the dressing method by application to the KP equation.

2. Use a special case of the dressing method to produce N–solitonsolutions to the KP equation.

3. Discuss the reduction of the method to the KdV equation, anddiscuss N–soliton solutions.


The Lax Pair for the KdV Equation

1. Consider the system

Lϕ = Eϕ, ϕt = Aϕ (9)

where

L = − ∂2

∂x2+ q(x, t), (10)

A = −4∂3

∂x3+ 6q(x, t)

∂

∂x+ 3qx(x, t). (11)

2. The compatibility condition is still the Lax equation

Lt = [A,L]. (12)

3. This is isospectral and equivalent to

qt − 6qqx + qxxx = 0. (13)


The Lax Pair for the KP Equation

1. We now consider the more general linear system

Lψ = 0, ψt = Aψ, (14)

L = − ∂2

∂x2− α ∂

∂y+ u(x, y, t), (15)

A = −4∂3

∂x3+ 6u(x, y, t)

∂

∂x+ 3w(x, y, t). (16)

2. The compatibility condition is the Lax equation

Lt = [A,L]. (17)

3. This is equivalent to a system involving u and w that reduces to

(ut − 6uux + uxxx)x + 3α2uyy = 0. (18)


Deformed Lax Operators

1. Introduce the differential operators

Dx =∂

∂x+ iλ, Dy =

∂

∂y+ αλ2, Dt =

∂

∂t+ 4iλ3, (19)

L1 = −D2x − αDy + u(x, y, t), (20)

L2 = −4D3x + 6u(x, y, t)Dx + 3w −Dt. (21)

2. If χ satisfies Ljχ = 0 and ψ = eφ(λ)χ in terms of

φ(λ) = iλx+ αλ2y + 4iλ3t (22)

then Lψ = 0, ψt = Aψ.

3. The compatibility condition for the Lax equation is the KPequation, so if Ljχ = 0 then u solves the KP equation.


The ∂ Operator

1. If λ = x+ iy, then

∂

∂λ=

1

2

(∂

∂x− i ∂

∂y

),

∂

∂λ=

1

2

(∂

∂x+ i

∂

∂y

). (23)

2. The operator ∂∂λ is the usual complex differential.

3. The equation ∂f∂λ

= 0 is the Cauchy–Riemann equations.


The Nonlocal ∂ Problem

1. Determine χ as the solution to the nonlocal ∂ problem

∂χ

∂λ(λ) =

∫CR0(λ, η)eφ(η)−φ(λ)χ(η)dA(η) (24)

normalized so that χ→ 1 as λ→∞, or the equivalent integralequation

χ(λ) = 1 +1

π

∫∫C2

R0(ξ, η)eφ(η)−φ(ξ)χ(η)

λ− ξdA(η)dA(ξ). (25)

2. Equation (24) involves the spectral parameter λ, it is ageneralization of the analytical characterization of the Jostsolutions appearing in general spectral problems.

3. Equation (24) and its normalization produces a function χ, but uand w in Lj have yet to be defined. The goal is to define u and win terms of χ so that the compatibility condition Ljχ(λ) = 0 issatisfied for all λ.


The Nonlocal ∂ Problem for Ljχ

1. The key-point is that the choice of φ and Dx,y,t used to define Ljhave been defined in such a way that Ljχ(λ) also solve thenonlocal ∂ problem

∂(Ljχ)

∂λ(λ) =

∫CR0(λ, η)eφ(η)−φ(λ)(Ljχ)(η)dA(η). (26)

2. With the choices

u = 2iχ0x, w = 4iχ0xx − 4χ1x − 2iuχ0 (27)

we have Ljχ→ 0 as λ→∞.

3. Then Ljχ solves the homogenous integral equation

Ljχ(λ) =1

π

∫∫C2


λ− ξdA(η)dA(ξ) (28)

and therefore Ljχ ≡ 0.


KP N–Solitons

1. The specific choice of

R0(λ, η) = π

N∑n=1

cnδ(λ− kn)δ(η − wn). (29)

allows us to produce N–soliton solutions.

2. The ∂ problem is then solved by

χ(λ) = 1 +

N∑n=1

χn(x, y, t)

λ− kn, (30)

N∑n=1

(δmn +

cmeφ(wm)−φ(km)

kn − wm

)χn = cme

φ(wm)−φ(km). (31)

3. The N–soliton solution is given by

u(x, y, t) = 2i∂

∂x

N∑n=1

χn. (32)


KdV and Schrodinger Equation

1. The choice R0(λ, η) = R0(λ)δ(λ+ η) produces χ that isindependent of y. In terms of N–solitons this amounts to takingwm = −km so that eφ(wm)−φ(km) = e−2ikmx−8ik3mt.

2. At each time t, the potential q(x, t) leads to L = − ∂2

∂x2+ q(x, t)

with a reflectionless continuous spectrum [0,∞), and discretespectrum k2

1, k22, · · · , k2

N.3. Examples of N–solitons are depicted in movies and the following

slides.


KdV Two Soliton Collision


KdV Three Soliton Collision






Kaup–Broer Solitons

1. The Kaup–Broer is closely related to the KdV equation, in factthe Kaup–Broer equation reduces to the KdV in the case whenwaves move in one direction.

2. Unlike the KdV equation, the Kaup–Broer system can havecounter propagating multisolitons.

3. In this subsection we will adapt the method in the previoussubsection to the Kaup–Broer system, this involves firstconsidering a 2+1 dimensional generalization of the Kaup–Broersystem.

4. We will derive N–soliton solutions to the Kaup–Broer system, andcompare and contrast with the N–soliton solutions to the KdVequation.


Manakov Triple for 2+1 Dimensional System

1. We now consider the system

st + 14

(s2u − suu

)+ 1

4

(s2v − svv

)+ ρ = 0 (33)

Bt + 14(Buu + 2(suB)u + 2suu) (34)

+ 14(Bvv + 2(svB)v − 2asvv) = 0, (35)

ρuv = 12Buu + 1

2Bvv. (36)

2. This system is equivalent to

Lψ = 0, ψt = Mψ =⇒ Lt = [M,L] +QL, (37)

L =∂2

∂u∂v+A

∂

∂v+B + 1, (38)

M = 14

∂2

∂u2+ 1

4

∂2

∂v2+ F

∂

∂v+G, (39)

Av = −2Fu = suv, Gv = 12Bu, Q = Fv − 1

2Au. (40)


The Nonlocal ∂ Problem

1. We again consider

∂χ

∂λ(λ) =

∫CR0(λ, η)eφ(η)−φ(λ)χ(η)dA(η). (41)

2. This time we consider

φ = λu+a

λv +

(αλ2 +

β

λ2

)t. (42)

3. Choose the operators

Du =∂

∂u+ λ, Dv =

∂

∂v+a

λ, Dt =

∂

∂t+ αλ2 +

β

λ2. (43)


A More Complicated ∂ Problem for Ljχ

1. We choose the operators

L1χ =DuDvχ+ADvχ+ (B − a)χ (44)

=χuv + λχv +a

λχu +Aχv +A

a

λχ+B, (45)

L2χ =DuDvχ+ADvχ+ (B − a)χ (46)

=χuv + λχv +a

λχu +Aχv +A

a

λχ+B. (47)

2. This time Ljχ satisfy the nonlocal ∂ problem

∂Ljχ

∂λ(λ) = γjδ(λ) +

∫CR0(λ, η)eφ(η)−φ(λ)Ljχ(η)dA(η), (48)

γ1 = πa(χu(0) +Aχ(0)), γ2 = −πa(2βχv(0) + Fχ(0)) (49)


Showing Ljχ ≡ 0

1. This time, to guarantee Ljχ ≡ 0 we need to check both γj = 0 inaddition to Ljχ→ 0 as λ→∞.

2. From the ∂ inversion formula

Ljχ(λ) =γjλ

+1

π

∫∫C2


λ− ξdA(η)dA(ξ) (50)

we see that γj = 0 is equivalent to the condition that Ljχ isregular at 0.

3. Explicitly, these conditions are satisfied if

A = su, F = 2βsv, B = −χ0v, G = −2αχ0u (51)

where s = − log(χ(0)) and χ(λ) = 1 + χ0

λ +O(λ−2).


N–Solitons to the 2+1 Dimensional K–B system

1. To produce N–solitons choose

R0(λ, η) = π

N∑n=1

cnδ(λ− zn)δ(η − wn). (52)

2. The ∂ problem is then solved by

χ(λ) = 1 +

N∑n=1

χn(u, v, t)

λ− zn, (53)

N∑n=1

(δmn +

cmeφ(wm)−φ(zm)

zn − wm

)χn = cme

φ(wm)−φ(zm). (54)

3. The N–soliton solution is given by

B(u, v, t) = − ∂

∂v

N∑n=1

χn, s(u, v, t) = − log

(1−

N∑n=1

χnzn

)(55)


Reduction to 1+1 Dimensions

1. If R0(λ, η) = δ(λ− η−1)R0(λ) then χ is y independent.

2. In terms of the 2+1 dimensional N–solitons, this reductionamounts to taking wn = z−1

n .

3. Then ϕ = s and η = B − 12sxx satisfy the non dimensional

Kaup–Broer system

ϕt +1

2ϕ2x + η = 0 (56)

ηt + ϕxx + (ηϕx)x +1

4ϕxxxx = 0. (57)


2+1 Dimensional Kaup–Broer one Soliton

B(x,y,-0.5)

0

0.025

0.050

0.075

0.100

0.125

B(x,y,0)

0

0.025

0.050

0.075

0.100

0.125

B(x,y,0.5)

0

0.025

0.050

0.075

0.100

0.125

s(x,y,-0.5)

0

0.25

0.50

0.75

1.00

1.25

s(x,y,0)

0

0.25

0.50

0.75

1.00

1.25

s(x,y,0.5)

0

0.25

0.50

0.75

1.00

1.25


1+1 D Kaup Broer System Two Soliton Collisions

η(x,t)

0

0.1

0.2

0.3

0.4

η(x,t)

0

0.2

0.4

0.6


Table of Contents





Motivation for the Problem

1. Our goal here is a method for expressing the solution to theCauchy problem for the KdV equation with a general periodicinitial condition as a Riemann–Hilbert problem.

2. One motivation for this is to make connection to theRiemann–Hilbert problems of Trogdon and Olver for finite gappotentials, and the Riemann–Surface theory of Novikov, Dubrovin,Matveev and Its for finite gap potentials.

3. Infinite gap potentials have been studied by Marchenko, McKean,Trubowitz, Kappeler among others. Benefits of our approach are:

I It is related only to Bloch–Floquet functions, or equivalentlyBaker–Akheizer functions on Riemann–Surfaces.

I Proof uses only elementary complex variables theorems.I It is analogous to Riemann–Hilbert formulation of IST.I It is a nonlinear Fourier transform, in the sense that the space and

time dependence enters through e2i√λx+8i

√λ3t.


Inverse Spectral Theory

1. The potential q ∈ L∞(R) with q(x+ T ) = q(x) is determined by:

I The nondegenrate band ends Ek such that the spectrum of L breaksup as a disjoint union

σ(L) =

G−1⋃j=0

[E2j , E2j+1] ∪ [E2G ,∞) (58)

where G can be either finite or countably infinite.I The Dirichlet eigenvalues µnk

∈ [E2k−1, E2k].I The signature σnk

= −sgn(log(|y2′(T, µnk)|)).

2. Because the KdV evolution is isospectral, it is equivalent to anevolution of µnk

, σnk. This evolution is described by the Dubrovin

equation.


Floquet Solutions to Hill’s Equations

1. Introduce solutions y1, y2 to Hill’s equation

−yxx(x) + q(x)y(x) = λy(x), (59)

y1(0, λ) = y2x(λ) = 1, y1x(λ) = y2(λ) = 0.

2. Define the following ∆(λ) = y1(T, λ) + y2x(T, λ).

3. The functions

ψ±(x, λ) = y1(x, λ) +ρ(λ)±1 − y1(T, λ)

y2(T, λ)y2(x, λ) (60)

factor as ψ±(x, λ) = ρ(λ)±xT−1p±(x, λ) where p± are periodic and

ρ(λ) =∆(λ) +

√∆(λ)2 − 4

2, ρ(λ)−1 =

∆(λ)−√

∆(λ)2 − 4

2. (61)


Asymptotic Expansion of Bloch–Floquet Functions

Lemma

ψ±(x, λ) = e±i√λx

(1± 1

2i√λ

∫ x

0q(t)dt+O(λ−1)

), (62)

andψ±′(x, λ) = e±i

√λx(±i√λ+O(1)

), (63)

as λ→∞ valid for λ ∈ Ωs where

Ωs := λ ∈ C : | arg λ| > s. (64)


Baker–Akheizer Function

Let Σ ⊂ C× C be the the curve defined by

w2 = P (λ) = λ

G∏k=1

T 4

n4π4(λ− E2k−1)(λ− E2k). (65)

The projection π((λ,w))→ λ has two inverses under composition,

π−1+ (λ) = (λ,

√P (λ)), π−1

− (λ) = (λ,−√P (λ)). (66)

Corollary

Corollary to Uniqueness Proof For every x there is a uniquemeromorphic function ψ(x, p) on Σ such that ψ±(x, λ) = ψ(x, π−1

± (λ))with G simple poles at (µnk

, σnkP (µnk

)),

ψ(x, π−1± (λ)) = e±i

√λx(1 +O(

√λ−1

)) (67)

as λ→∞ with λ ∈ Ωs, and |ψ(x, π−1± (λ))| ≤Mec|λ|

2for λ ∈ C \ D.


Definition for Riemann–Hilbert Problem I

Definition

The matrix Floquet solution Ψ is given by

Ψ(x, λ) =

(ψ−(x, λ) ψ+(x, λ)

ψ−′(x, λ) ψ+′(x, λ)

). (68)

Definition

Let A be the 2× 2 matrix valued function

A(λ) :=

√f0(λ)

4√

∆(λ)2 − 4

(f−(λ) 0

0 f+(λ)

)(69)

f±(λ) :=

∞∏n=1

σn=±1

T 2

n2π2(µn − λ), f0(λ) :=

∞∏n=1σn=0

T 2

n2π2(µn − λ). (70)


Definition for Riemann–Hilbert Problem II

Definition

Φ(x, λ) := Ψ(x, λ)A(λ)eiσ3√λx. (71)

Definition

Let V : R+ \ Ej2Gj=0 → SL(2,C) be given by

V (λ) :=

(−1)k+m(λ)−1

(0 i f

+(λ)f−(λ)

if−(λ)f+(λ)

0

)λ ∈ (E2k−2, E2k−1)

(−1)k+m(λ)−1e2iσ3√λx λ ∈ (E2k−1, E2k)

. (72)

m(λ) := |k ∈ N : µnk≤ λ, σnk

= 0| (73)


Riemann–Hilbert Problem

For each x ∈ R find a 2× 2 matrix valued function Φ(x, λ) such that:

1. Φ(x, λ) is a holomorphic function of λ for λ ∈ C \ R+.

2. Φ±(x, λ) are continuous functions of λ ∈ R+ \ Ej2Gj=0 that have

at worst quartic root singularities on Ek2Gk=0.

3. Φ±(x, λ) satisfy the jump relation Φ+(x, λ) = Φ−(x, λ)V (x, λ).

4. Φ(x, λ) has an asymptotic description of the form

Φ(x, λ) =

(1 1

−i√λ i√λ

)(I +O

(√λ−1))

A(λ) (74)

as λ→∞ with λ ∈ Ωs for some 0 < s < π8 .

5. There exist positive constants c, and M such that|ϕij(x, λ)| ≤Mec|λ|

2for all λ ∈ D.


Uniqueness Proof

1. For standard Riemann–Hilbert problems, one uses the fact thatΦ(λ)Φ−1(λ) is bounded and entire in conjunction with uniformasymptotics to use Liouvilles theorem to prove Φ(λ)Φ−1(λ) = I.

2. In the infinite gap case, the asymptotic behavior is not uniformwhich requires use of:

Theorem (Phragmen–Lindelof)

Let α, β ∈ R be such that β − α < 2π let p > 0 satisfy p < π2|β−α| .

Choose a branch of arg so that (α, β) ⊂ Im(arg). Let f(λ) be aholomorphic function for

λ ∈ Υα,β := λ ∈ C : α < arg(λ) < β (75)

that extends to a continuous function on Υα,β. If f(λ) is bounded forλ ∈ ∂Υα,β and |f(λ)| ≤Mec|z|

pfor λ ∈ Υα,β then f(λ) is bounded for

λ ∈ Υα,β.


Recovery of the Potential

Theorem

Let x ∈ R be fixed, then Φ(x, λ) solves our Riemann–Hilbert problem ifand only if

Φ(λ) =

(1 0

α(x) 1

)Φ(x, λ) (76)

for some constant α(x). Moreover, the potential q(x) evaluated at xcan be recovered from any solution Φ(x, λ) of our Riemann–Hilbertproblem by

q(x) = 2i∂

∂xlimλ→∞

√λ(

1− ei√λx[Φ(x, λ)A(λ)−1]11

). (77)


The final step is to prove that if we use the new definition

Definition

Let V : R+ \ Ej2Gj=0 → SL(2,C) be given by

V (λ) :=

(−1)k+m(λ)−1

(0 i f

+(λ)f−(λ)

if−(λ)f+(λ)

0

)λ ∈ (E2k−2, E2k−1)

(−1)k+m(λ)−1e2iσ3√λx+8iσ3

√λ3t λ ∈ (E2k−1, E2k)

. (78)

m(λ) := |k ∈ N : µnk≤ λ, σnk

= 0| (79)

for the jump matrix, then we can produce KdV solutions from asolution the Riemann–Hilbert problem by

q(x, t) = 2i∂

∂xlimλ→∞

√λ(

1− ei√λx[Φ(x, t, λ)A(λ)−1]11

). (80)


1. If we consider the Bloch–Floquet functions ψ±(x, t, λ) nowcorresponding time time dependent potential q(x, t), then theysolve

ψ±t (x, t, λ) + α±(t, λ)ψ±(x, t, λ) = Aψ±(x, t, λ), (81)

where

α±(t, λ) = (4λ+ 2q(0, t))ρ(λ)±1 − y1(T, t, λ)

y2(T, t, λ)− qx(0, t). (82)

2. If we define

e±(t, λ) = exp

(∫ t

0α±(τ, λ)dτ

)(83)

then e±(t, λ)ψ±(x, t, λ) satisfies the Lax pair system, and limits to

ei√λx+4i

√λ3t as λ→∞ for λ ∈ Ωs, and has fixed singular behavior.

3. Removing the time dependence gives the phase in the jump.


Future Work

1. In the first part of this defense, we produced N–solitons by solvinga nonlocal ∂ problem equivalent to an N dimensional linearequation. We are now working on studying the case where thelinear equation are replaced with integral equations.

2. Solutions to the KP and KdV equations produced in this mannerdescribe periodic solutions, dispersive shockwaves, and otherbounded solutions. This also leads to a theory of band structureunrelated to periodicity, and could potentially be used to modelnonperiodic semiconductors and insulators.

3. In the second part of this defense we formulate the inverse spectraltheory of periodic potentials, and the Cauchy problem for theKdV equation.

4. This could be used to study ‘Gibbs phenomena’ for a high energycutoff producing a finite gap potential from an infinite gappotential, and also the ‘nonlinear Talbot effect’ for the KdVequation for square wave initial condition.


Thank you for your Time!


Documents

Applications of Complex Variables to Spectral Theory and ......The KP, KdV Equations and Re ectionless Schr odinger Operators The Kaup{Broer System and 2+1 Dimensional Generalization