The defocusing nonlinear Schrödinger equation with step-like oscillatory data

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Doctoral Thesis in Mathematics

The defocusing nonlinear

Schrödinger equation with step-like oscillatory data

SAMUEL FROMM

Stockholm, Sweden 2021

kth royal institute of technology

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The defocusing nonlinear

Schrödinger equation with step-like oscillatory data

SAMUEL FROMM

Doctoral Thesis in Mathematics KTH Royal Institute of Technology Stockholm, Sweden 2021

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Monday the 14th of June 2021, at 2:00 p.m. in U1, Brinellvägen 28A, Kungliga Tekniska Högskolan, Stockholm.

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Printed by: Universitetsservice US-AB, Sweden 2021

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iii

Abstract

The thesis at hand consists of three papers as well as an introduc- tory chapter and a summary of results. The topic of the thesis is the study of the defocusing nonlinear Schrödinger equation with step-like oscillatory data.

Paper A studies the Cauchy problem for the defocusing nonlinear Schrödinger equation on the line with step-like oscillatory boundary conditions. More precisely, the solution is required to approach a sin- gle exponential as x → −∞ and to decay to zero as x → +∞. We prove existence of a global solution and show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. We also compute the long-time asymptotics of the solution and apply the results to a related initial-boundary value problem on the half-line.

Paper B studies an initial-boundary value problem for the defocusing nonlinear Schrödinger equation on the half-line with asymptotically oscillatory boundary conditions. More precisely, the solution is required to approach a single exponential on the boundary as t → +∞

and to decay to zero as x → +∞. We construct a solution of the problem in a sector close to the boundary and compute its long-time behaviour.

Paper C studies a similar problem as Paper B but instead of the nonlinear Schrödinger equation we study the Gerdjikov–Ivanov equation. We give necessary conditions for the existence of a solution of the associated initial-boundary value problem under asymptotically oscillatory boundary conditions.

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iv

Sammanfattning

Denna avhandling består av tre artiklar, ett introduktionskapitel och ett kapitel som sammanfattar avhandlingens huvudresultat.

I Artikel A studeras Cauchy problemet för den icke-linjära Schrö- dingerekvationen med trappliknande oscillerande randvärden. Vi visar att det finns en global lösning och att lösningen kan uttryckas genom ett Riemann-Hilbert problem. Vi beräknar vidare lösningens beteende för stora tider och tillämpar våra resultat på ett relaterat begynnelse- randvärdesproblem.

Artikel B behandlar ett begynnelse-randvärdesproblem för den icke- linjära Schrödingerekvationen med asymptotiskt periodiska randvär- den. Vi konstruerar en lösning i närheten av randen och beräknar lös- ningens beteende för stora tider.

Artikel C behandlar ett liknande problem som Artikel B, men istäl- let för den icke-linjära Schrödingerekvationen studerar vi Gerdjikov–

Ivanov-ekvationen. Vi anger nödvändiga villkor för att en lösning till begynnelse-randvärdesproblemet med asymptotiskt periodiska rand- värden ska existera.

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Acknowledgements

First and foremost, I would like to express my gratidude towards my advisor Jonatan Lenells. I am thankful for the guidance, support, and advise he has given me over the course of my PhD and for introducing me to the research area.

Second, I would like to thank my coworker Ronald Quirchmayr. I am thankful for many interesting discussions which not only made working on our project more productive but also more enjoyable.

Third, I want to thank my colleagues and friends at KTH who turned my time there into a unique experience. In particular, and in no particular order, I would like to thank Nasrin, Julian, Lena, Martina, Parikshit, Thomas, Federico, Tomas, Scott, Gustav, Eric, Eric, Isabel, David, Gerard, and Oliver.

I would also like to thank my friends from Karlsruhe, where this mathematical journey arguably started.

Finally, I wish to thank my family for their support and encouragement throughout the years.

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1 Introduction

The purpose of this chapter is to give an overview of the main results of the thesis and to illustrate some of the techniques used to obtain these results.

We point out that, in this chapter, we sometimes omit mathematical details in favour of a more accessible and concise presentation.

We begin by explaining the title of the thesis.

1.1 The nonlinear Schrödinger equation

The nonlinear Schrödinger (NLS) equation is a nonlinear partial differential equation given by

iut+ u_xx± 2|u|²u = 0. (1.1.1) Here u(x, t) is a complex function of two real variables x and t, often referred to as the space and time coordinates, and the subscripts indicate partial derivatives with respect to the respective variable. In the case that the plus sign is chosen in (1.1.1), the equation is referred to as the focusing NLS equation, whereas the version with the minus sign is referred to as the defocusing NLS equation. The names represent the different characters of certain solutions of the two versions of (1.1.1) which appear in the context of nonlinear optics [8]. Besides optics, the NLS equation plays an important role in the study of water waves [9]. This thesis studies the defocusing NLS equation.

From a mathematical standpoint, one of the most interesting properties of the NLS equation is that it is integrable, as was shown by Zakharov and Shabat [10] in 1972. While it is difficult to give a rigorous definition of integrability within the theory of partial differential equations that covers

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2 CHAPTER 1. INTRODUCTION

all examples of interest, one possible definition is that the solution (or, more precisely, the solution of a well-posed initial or boundary value problem) can be obtained by means of only linear operations. What this essentially means is that we can write down a recipe for computing the solution of (1.1.1) (subject to some inital or boundary conditions). In the case of the NLS equation, the integrability is provided by means of the so-called inverse scattering transform (IST).

1.2 The initial value problem

Before we continue talking about solving the NLS equation, let us first introduce the precise problem considered in the thesis. While there are many solutions of equation (1.1.1), we are interested in solutions of the initial value problem

iu_t+ u_xx− 2|u|²u = 0, x ∈R, t > 0, (1.2.1a)

u(x, 0) = u₀(x), x ∈R, (1.2.1b)

under the additional assumption that the solution vanishes as x → +∞ and approaches an oscillatory plane wave u^b(x, t) as x → −∞, i.e. ,

u(x, t) ∼

(u^b(x, t), x → −∞,

0, x → +∞. (1.2.2)

Here a plane wave solution of (1.2.1a) is a solution of the form u^b(x, t) = αe^2iβx+iωt

where α > 0 and β ∈ R are two parameters. The constant ω is given by ω := −4β²− 2α² and is determined by the requirement that u^b(x, t) should solve the equation (1.2.1a). Letting t = 0 in (1.2.2), it follows that the initial data u₀(x) must satisfy

u₀(x) ∼

(αe^2iβx, x → −∞,

0, x → +∞. (1.2.3)

The boundary conditions (1.2.2) are of a “step-like oscillatory” type.

The “oscillatory” part comes from the exponential term e^2iβx+iωt, which

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1.3. THE INVERSE SCATTERING TRANSFORM 3

0

x

Figure 1. A picture of a step-function.

has absolute value 1 but oscillates. The boundary conditions are called

“step-like” because the absolute value of a function u(x, t) satisfying (1.2.2) (for fixed t) resembles a step-function, see Figure 1. This explains the title of the thesis.

For a motivation in the context of earlier work of why we study the particular problem (1.2.1) we refer to the introduction of Paper A. We note however that asymptotically oscillatory boundary conditions are of interest from a point of view of applications [2].

1.3 The inverse scattering transform

One of the main goals of the thesis is to show that the initial value problem (1.2.1) described in the previous section has a solution for any values of α > 0 and β ∈R, assuming that the convergence in (1.2.3) is “fast enough”.

The proof is based on the celebrated inverse scattering transform (IST). As mentioned in Section 1.1 it gives a recipe for solving the problem (1.2.1).

The IST is sometimes also referred to as a “nonlinear Fourier transform”.

Thus, before introducing the IST, let us first recall how one can use the Fourier transform to solve the initial value problem

iu_t+ u_xx = 0, x ∈R, t > 0, (1.3.1a)

u(x, 0) = u₀(x), x ∈R, (1.3.1b)

for the linear Schrödinger equation. For simplicity, we assume vanishing boundary conditions, that is, we seek a smooth complex function u(x, t) for x ∈R and t > 0 such that u solves (1.3.1) and such that u(x, t) decays as x → ±∞. Note that this corresponds to the problem introduced in Section 1.2, but instead of step-like boundary data we have vanishing boundary data and instead of the defocusing NLS we have the linear Schrödinger equation.

We also point out that for small u₀, the problem (1.3.1) can be viewed as an

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“approximation” of (1.2.1), since the nonlinear term |u|²u present in (1.2.1a) can be considered “small”.

Using the Fourier transform u(k, t) =ˆ

Z ∞

−∞

u(x, t)e^2ikxdx,

the initial value problem (1.3.1) can be transformed (assuming sufficient regularity) into

iˆu_t− 4k²u = 0,ˆ x ∈R, t > 0, u(k, 0) = ˆˆ u0(k), k ∈R.

The latter problem is an ordinary differential equation whose solution is given by

u(k, t) = eˆ ^−4ik²^tu(k, 0) = eˆ ^−4ik²^tuˆ₀(k).

A solution u of the original problem can now be obtained by applying the inverse Fourier transform to ˆu, i.e.,

u(x, t) = 1 π

Z ∞

−∞

u(k, t)eˆ ^−2ikxdk = 1 π

Z ∞

−∞

uˆ0(k)e^{−2ikx−4ik}²^tdk where

uˆ₀(k) = Z ∞

−∞

u₀(x)e^2ikxdx.

Thus the Fourier transform provides a recipe to compute the solution of the problem (1.3.1): Transform the problem, solve the (easier) transformed problem, and finally apply the inverse transform to obtain a solution of the original problem. A summary of the procedure is shown in Figure 2.

The IST presents a similar picture. However, there is no “general” transform. Instead, the IST is different for different equations, although the general ideas are the same.

The term “inverse-scattering transform” was first introduced by Ablowitz, Kaup, Newell, and Segur [1] in 1974. However, influental work in the same direction is due to Gardner, Greene, Kruskal, and Miura [5] (1967), Lax [6]

(1968), and Zakharov and Shabat [10] (1971).

In the following we will give a brief overview of the most important ideas of the IST. The same ideas are used to prove existence for our problem (1.2.1) in the thesis.

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1.3. THE INVERSE SCATTERING TRANSFORM 5

u(x, t) = _π¹ ^R_−∞^∞ u(k, t)eˆ ^−2ikxdk

u(x, 0) = u₀(x) uˆ₀(k) =^R_−∞^∞ u₀(x)e^2ikxdx u(k, t) = eˆ ^−4ik²^tu0(k)

iu_t+ u_xx= 0 e^−4ik²^t

F⁻¹

F

Figure 2. An overview of how the Fourier transform can be used to solve the linear problem. Here F and F⁻¹ stand for the Fourier transform and its inverse, respectively.

We observe first that (1.2.1a) is the compatibility condition of two auxil- iary linear matrix-valued equations, called the Lax pair of (1.2.1a) and given by

(φx+ ikσ₃φ = Uφ,

φt+ 2ik²σ3φ = Vφ, (1.3.3) where k ∈C is a complex variable, called spectral parameter, and the matrices U and V are defined by

U := 0 u u¯ 0

!

, V := −i|u|² 2ku + iu_x 2k ¯u − i¯u_x i|u|²

! .

Here “(1.2.1a) is the compatibility condition of (1.3.3)” means that u is a solution of (1.2.1a) if and only if φ_xt= φ_tx, that is, if and only if there exists a function φ solving both equations in (1.3.3) simultaneously. The Lax pair (1.3.3) allows us to essentially replace the study of a nonlinear equation with the study of two linear (and thus generally much easier to study) equations.

Using the Lax pair (1.3.3) one can now introduce the so-called “reflection” coefficient, which corresponds to ˆu₀(k) in the linear case (the name

“reflection coefficient” stems from physics where it describes how much of a wave is reflected when the medium through which the wave travels changes).

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In order to define the reflection coefficient one first defines two solu- tions φ₁(x, k) and φ₂(x, k) of the x-part of (1.3.3) (i.e., the first equation of (1.3.3)), where the appearances of u on the right-hand side are replaced by u₀. Furthermore, one chooses to normalize φ₁(x, k) at −∞ and φ₂(x, k) at +∞ in a suitable way (depending on the respective limiting values of u0). These solutions are usually defined as solutions of Volterra integral equations. Since φ₁ and φ₂ are two solutions of the same linear equation, their column vectors are linearly dependent, which implies that there exists a unique matrix-valued function s(k) such that

φ1(x, k) = φ₂(x, k)s(k).

The reflection coefficient r(k) is then defined as the quotient of two of the entries of the matrix s(k). We will not go into details here, but the main takeaway is that the reflection coefficient only depends on the initial data u0(x) and is given (somewhat) explicitly (again the reader should have the linear case in mind). The map u₀(x) 7→ r(k) is sometimes referred to as the direct scattering transform.

The first main result of the thesis, see Theorem 2.1 in paper A, derives properties of the direct scattering transform and of the reflection coefficient r(k).

1.4 Riemann–Hilbert problems

Before we consider the solution of the inverse problem, i.e., the construction of u(x, t) in terms of the reflection coefficient r(k), we take a quick excursion into a subfield of complex analysis, namely that of Riemann–Hilbert (RH) problems (named after Bernhard Riemann and David Hilbert).

In its simplest form, a RH problem can be stated as follows: Given a contour Σ in the complex plane which divides the plane into two regions Σ₊ and Σ₋ (we may think of Σ as the real line or as the unit circle) and a function J (k) defined on the contour Σ, find a complex-valued function M (k) which is analytic in Σ+ and Σ−, has continuous boundary values on Σ, and satisfies the jump relation

M₊(k) = M₋(k) + J (k), k ∈ Σ, (1.4.1) where M_± denote the boundary values of M onto Σ from within Σ_±.

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1.5. THE INVERSE SCATTERING TRANSFORM - CONTINUATION7

Alternatively, one may study a multiplicative version, in which (1.4.1) is replaced by

M₊(k) = M₋(k)J (k). (1.4.2) We note that (1.4.1) and (1.4.2) are related through the logarithm function (since applying the exponential function to (1.4.1) yields a jump of the form (1.4.2)). Looking at (1.4.2), we see that if M is a solution to the problem then so is any constant multiple of M . Thus, to ensure uniqueness, one adds a normalization condition, which in the case of (1.4.2) might take the form

M (k) → 1 as k → ∞ where k ∈C.

The solution of the first problem (1.4.1) (with the normalization condi- tion M (k) = O(k⁻¹) as k → ∞) is given by

M (k) = 1 2πi

Z

Σ

J (z)

z − kdz, (1.4.3)

assuming that the contour Σ and the jump function J are sufficiently well behaved. This is a consequence of the so-called Sokhotski-Plemelj formulas.

The RH problems appearing in this thesis are 2 × 2-matrix valued RH problems. The formulation of a matrix valued RH problem is similar to the formulation given above, but M and J are now matrices and the relevant jump condition is given by (1.4.2) (as an identity of matrices).

We will not give more details on matrix valued RH problems but only mention that there is a well-developed theory for them. Under relatively mild assumptions on the contour and on the jump function, the solution (if it exists) can be expressed in terms of the solution of a singular integral equation involving a Cauchy type operator (similar to the operator appearing in (1.4.3)). Furthermore, a solution is known to exist, for example, if the norm of this Cauchy operator is “small”.

1.5 The inverse scattering transform - Continuation

Let us return to the IST. Recall that the direct problem is given by the map u₀ 7→ r, where the reflection coefficient r can be constructed by solving a

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linear Volterra integral equation. However, even more is true. One can show that there exists a solution of the following RH problem:

• m₀(x, k) is analytic for k ∈C \ R.

• m₀(x, ·) has continuous boundary values on R from the upper and lower half-planes denoted by m₀₊and m₀₋, respectively. Furthermore, these boundary values satisfy the jump relation

m₀₊(x, k) = m₀₋(x, k)v₀(x, k) for k ∈R, where the jump matrix v₀ is given by

v₀(x, k) = 1 − |r(k)|² r(k)e^−2ikx

−r(k)e^2ikx 1

!

. (1.5.1)

• m₀(x, k) = I + O(k⁻¹) as k → ∞.

Here m₀(x, k) is a matrix valued function onC \ R (and the contour is given simply by Σ =R). The solution m0 can be explicitly written down in terms of φ₁ and φ₂ (which were defined in Section 1.3). Furthermore, it holds that

u0(x) = 2i lim

k→∞k[m0(x, k)]₁₂, (1.5.2) where [m₀(x, k)]₁₂ denotes the 12-entry of the matrix function m₀.

Equation (1.5.2) provides a solution to the inverse problem, that is, it provides a way to construct u₀(x) from the reflection coefficient r(k). In analogy with the Fourier transform, one now “time evolves” the reflection coefficient, and hence also the jump matrix (1.5.1) in the above RH problem.

Applying (1.5.2) to the time evolved RH problem yields a way to obtain the solution u(x, t) for any time t > 0. This leads to the second main result of the thesis, see Theorem 2.2 in Paper A, which reads as follows.

Theorem 1.1. The Cauchy problem (1.2.1) with “good enough” initial data u0 has a global solution which can be constructed as follows:

Let r be the reflection coefficient corresponding to u₀, and define a jump matrix v(x, t, ·) by

v(x, t, k) := 1 − |r(k)|² r(k)e^−2i(kx+2k²^t)

−r(k)e^2i(kx+2k²^t) 1

!

. (1.5.3)

For each x ∈R, t > 0, the RH problem

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1.6. THE NONLINEAR STEEPEST DESCENT METHOD 9

• m(x, t, k) is analytic for k ∈C \ R.

• m(x, t, ·) has continuous boundary values on R from the upper and lower half-planes denoted by m₊ and m₋, respectively. Furthermore, these boundary values satisfy the jump relation

m₊(x, t, k) = m₋(x, t, k)v(x, t, k) for k ∈R.

• m(x, t, k) = I + O(k⁻¹) as k → ∞.

has a unique solution m(x, t, ·) and u(x, t) := 2i lim

k→∞k[m(x, t, k)]12, (1.5.4) is a solution of the Cauchy problem (1.2.1) with initial data u₀.

The first part of the proof consists of showing that a solution of the above RH problem actually exists. The main ingredient here is a so-called

“vanishing lemma”, which shows that the homogeneous version of the RH problem (i.e., when we replace the condition m(x, t, k) = I + O(k⁻¹) as k → ∞ with m(x, t, k) = O(k⁻¹) as k → ∞) only has the trivial (zero) solution. Once existence has been established, the proof proceeds as follows.

In order to show that the limit defined in (1.5.4) actually solves our problem, one shows that m satisfies a Lax pair similar to (1.3.3), but with u on the right-hand side replaced by the limit (1.5.4). As mentioned in Section 1.3, this implies that the function defined in (1.5.4) is a solution of (1.2.1) (due to the compatibility condition). To show that u satisfies the initial value (1.2.1b) one simply observes that m(x, 0, k) and m₀(x, k) solve the same RH problem (compare (1.5.3) for t = 0 with (1.5.1)). By uniqueness, they must be equal and comparing (1.5.2) and (1.5.4) yields that u(x, 0) = u₀(x).

To summarize, we arrive at a similar picture as we have seen in Figure 2 for the linear case, see Figure 3.

1.6 The nonlinear steepest descent method

The third main result of the thesis consists of computing the behaviour of the solution u(x, t) (which exists as a consequence of Theorem 1.1 above) for large t. The derivation of asymptotics is based on a nonlinear steepest descent analysis of the associated RH problem.

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u(x, t)

u(x, 0) = u₀(x) r(k)

e^−2ik²^tr(k)

iu_t+ u_xx −2|u|²u = 0 e^−2ik²^t (solution via RH problem)

inverse problem

direct problem

Figure 3. An overview of the how the IST can be used to solve the nonlinear problem.

The nonlinear steepest descent method is a generalization to RH problems of the classical steepest descent method for exponential integrals. It was introduced by Percy Deift and Xin Zhou [4] in 1993. Here, we will dis- cuss the classical steepest descent method. The nonlinear steepest descent method is based on similar ideas.

Let us go back to the linear equation considered in Section 1.3. We showed that the solution is given explicitly by

u(x, t) = 1 π

Z ∞

−∞

uˆ₀(k)e^{−2ikx−4ik}²^tdk where

uˆ₀(k) = Z ∞

−∞

u₀(x)e^2ikxdx.

While this provides an integral representation for the solution, it is in general not possible to compute these integrals in closed form. However, we can still extract detailed information about the solution (in particular about its behaviour for large t) from this integral representation.

Let x = ξt, where ξ ∈ R. We would like to compute the behaviour of u(ξt, t) for large t. To this end we write θ(ξ, k) = −2kξ − 4k², so that

u(ξt, t) = 1 π

Z ∞

−∞

uˆ0(k)e^itθ(ξ,k)dk. (1.6.1)

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1.6. THE NONLINEAR STEEPEST DESCENT METHOD 11

Im θ > 0

Im θ < 0

Im θ > 0 k0

Figure 4. The contour C in the complex plane is represented by a thick line. The region where Im θ > 0 is shaded and the level set where Im θ = 0 is represented by a dashed line.

The main idea of the steepest descent method is to view the integral appearing on the right-hand side of (1.6.1) as a complex contour integral. By deforming the contour (in this caseR) in a suitable way, we can then com- pute the asymptotics of the integral for large values of t. In order to do so we first compute

d

dkθ(ξ, k) = −2ξ − 8k.

The zero k₀ = −ξ/4 of _dk^dθ is called a critical point.

Now let us assume that the function ˆu0(k) is analytic (this is for instance the case if u₀ has compact support). Using Cauchy’s integral theorem, we deform the path of integration to a contour C, where C = k₀+Re^−iπ/4, see Figure 4. The integral in (1.6.1) thus becomes

u(ξt, t) = 1 π

Z

C

uˆ0(k)e^itθ(ξ,k)dk.

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left sector

middle sector

right sector

0

x

x/t = 4β − 2α x/t = 4β + 4α

Figure 5. The left, middle, and right asymptotic sectors in the xt-plane.

The main observation now is that

|e^itθ(ξ,k)| = e−t Im θ(ξ,k)

(note that −t Im θ(ξ, k) < 0 for t > 0 and Im θ(ξ, k) > 0). Thus, for large t, the exponential in the integrand of (1.6.1) gets very small, assuming that k 6= k₀. Hence we expect that the main contribution to the large t limit of (1.6.1) comes from the part of the integral close to the critical point k₀. This can indeed be made rigorous and in this way one can obtain an expansion of the integral (1.6.1) for large t.

Deift and Zhou were able to adapt this method to RH problems. We will not give details but the main idea is similar to the one described above.

Furthermore, they introduced “analytic approximations”, so that one can transform the integral (up to a small error) even if the integrand (in the above case ˆu0) is not analytic.

The third main result of the thesis is proved using these methods. We compute the behaviour of the solution u(x, t) of (1.2.1) for large t. It turns out that there are three asymptotics sectors, referred to as the left, middle, and right sector (see Figure 5). In the left sector, the leading term in the asymptotics is given by the plane wave u^b(x, t) = αe^2iβx+iωt multiplied by a factor which approaches 1 as x/t → −∞. In the right sector, the leading term is of order t^−1/2 and includes a factor which vanishes as x/t → +∞.

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1.7. A RELATED INITIAL-BOUNDARY VALUE PROBLEM 13

left sector

middle sector

right sector 0

x t x/t = 4β − 2α

x/t = 4β + 4α

Figure 6. The left, middle, and right asymptotic sectors in the xt-plane in the case when 4β − 2α > 0. The t-axis is represented by a dashed line.

Note that this behaviour is consistent with (1.2.2). The asymptotics in the middle sector are also computed.

1.7 A related initial-boundary value problem

The initial value problem (1.3.1) is related to an initial-boundary value problem on the half-line. Here half-line refers to the positive real axis. The formulation of the problem is similar to the one given in (1.2.1). How- ever, instead of x ∈R we only look at x ≥ 0. Furthermore, the boundary condition (1.2.2) as x → −∞ is not present anymore. Instead one intro- duces boundary conditions along the t-axis. If one introduces conditions for u(0, t), similarly to those introduced in (1.2.1b) along the x-axis, we speak of Dirichlet boundary conditions. If u(0, t) is replaced by u_x(0, t), we speak of Neumann boundary conditions.

In this thesis, we are interested in initial-boundary value problems on the half-line with asymptotically oscillatory boundary conditions. One major difficulty in the study of such problems is that for a well-posed problem not all boundary values are known. For instance, if one assumes that the Dirichlet data are given, then the Neumann value has to be computed. Here well-posed means that the problem has a unique solution and that this solution depends continuously on the data.

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It turns out, however, that we can avoid the above difficulty (in our specific case) by simply viewing the half-line problem as a restriction of the problem on the whole line R. To illustrate this idea further, let us revisit the asymptotics derived for the problem on the line. A picture of the asymptotic sectors in the case when 4β − 2α > 0 is shown in Figure 6. We find that if α and β are such that 4β − 2α > 0, then the t-axis lies in the left sector. As mentioned in the previous section, the asymptotics in this sector are (up to a factor of absolute value one) given by the plane wave u^b(x, t) = αe^2iβx+iωt. Hence if we restrict the solution of the problem on the line to the half-line, the boundary values will be given (for large t) by u^b(0, t) = αe^iωt multiplied by a constant of absolute value one. It thus follows that we have found a solution of the initial-boundary value problem with asymptotically oscillatory boundary conditions. This is the content of the last main result of Paper A, see Theorem 2.6 in Paper A.

In Paper B, we continue the study of this initial-boundary value problem in its own right, without assuming that the solution is obtained by restriction from a solution on the line. Paper B utilizes similar methods as Paper A. In fact, the initial-boundary value problem considered in Paper B also has step- like oscillatory data in the sense that the initial data u₀(x) vanishes as x →

∞, the boundary data u(0, t) approaches an oscillatory single exponential as t → ∞, and the initial and boundary values are compatible at the origin.

Paper C presents first results towards the implementation of the techniques of Paper A and Paper B for another integrable equation, the so called Gerdjikov–Ivanov equation. Thus it can be regarded as a starting point for future work.

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2 Summary of Results

2.1 Paper A

Paper A studies the Cauchy problem for the defocusing nonlinear Schrödinger equation under the assumption that the solution vanishes as x → +∞ and approaches an oscillatory plane wave as x → −∞. The exact problem was introduced in Section 1.2, see (1.2.1).

The first result of the paper derives properties of the reflection coefficient and relates the initial data to the solution of a RH problem.

The second result shows that the Cauchy problem with step-like initial data (1.2.1) has a “global” solution. It also provides an expression for the solution in terms of the solution of a RH problem.

The third result computes the long-time behaviour of the solution u(x, t).

Three asymptotic sectors are identified and we compute the leading and subleading terms in the asymptotics as t → ∞.

Finally, the long-time asymptotics are used to provide a solution of a related initial-boundary value problem on the half-line.

2.2 Paper B

Paper B studies an initial-boundary value problem for the defocusing nonlinear Schrödinger on the half-line with asymptotically oscillatory boundary conditions. In [3] it was shown that there exists a solution u of the focusing NLS equation satisfying

u(0, t) ∼ αe^iωt and u_x(0, t) ∼ ce^iωt as t → ∞ (2.2.1)

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16 CHAPTER 2. SUMMARY OF RESULTS

if and only if the parameter triple (α, ω, c) belongs to one of two explicitly given families. For the defocusing NLS equation necessary conditions for the existence of a solution of the half-line problem satisfying (2.2.1) were derived in [7]. One of the parameter configurations derived in [7] (see [7, Eq. (2.3c)]) is given by

n

α, ω, c = iα^p−2α²− ω: α > 0, ω < −3α²^o. (2.2.2) The main result of Paper B states that for any parameter triple contained in (2.2.2), we may construct a solution of the defocusing NLS equation (1.1.1) with boundary values satisfying (2.2.1), at least in a sector close to the boundary. Furthermore, we compute the leading and subleading term in the long-time asymptotics of the constructed solution.

In order to show existence we utilise Riemann–Hilbert methods. We provide a class of independent spectral data (corresponding to the reflection coefficient r in the introduction), and show that these spectral data give rise to a solution of (1.1.1) in a sector close to the t-axis. The long-time asymptotics follow from an application of the Deift–Zhou nonlinear steepest descent method.

2.3 Paper C

Paper C studies the Gerdjikov–Ivanov equation iu_t+ u_xx+ iu²u¯_x+1

2|u|⁴u = 0. (2.3.1) on the half-line with boundary values of the form (2.2.1). Similarly to the results from [3, 7] mentioned in Paper B, we give necessary conditions on the parameters (α, ω, c) for the existence of a solution of equation (2.3.1) with boundary values of the form (2.2.1). Thus Paper C can be seen as a basis for future work: it takes a first step towards the implementation of the techniques of Paper A and Paper B to the Gerdjikov–Ivanov equation.

Contributions of the author

The results in Paper A were obtained in collaboration with J. Lenells and R. Quirchmayr. The contributions of the authors were approximately equal and all authors took an active part in the writing.

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References

[1] Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Se- gur. The inverse scattering transform - Fourier analysis for nonlinear problems. Studies in Applied Mathematics, 53(4):249–315, 1974.

[2] J. L. Bona, W. G. Pritchard, and L. R. Scott. An evaluation of a model equation for water waves. Philos. Trans. Roy. Soc. London, A 302:457–510, 1981.

[3] A. Boutet de Monvel, Kotlyarov V., and D. Shepelsky. Decaying long- time asymptotics for the focusing NLS equation with periodic boundary condition. Int. Math. Res. Not. IMRN, 3:547–577, 2009.

[4] P. Deift and X. Zhou. A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the mKdV equation. Ann.

of Math., 137(2):295–368, 1993.

[5] Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura. Method for solving the Korteweg-Devries equation.

Phys. Rev. Lett., 19:1095–1097, Nov 1967.

[6] Peter D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Communications on Pure and Applied Mathematics, 21(5):467–

490, 1968.

[7] J. Lenells. Admissible boundary values for the defocusing nonlinear Schrödinger equation with asymptotically t-periodic data. J. Diff. Eq., 136:3–63, 2015.

[8] G. B. Whitham. Linear and Nonlinear Waves. John Wiley and Sons, 1974.

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18 REFERENCES

[9] V.E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys, 9:190–194, 1968.

[10] V.E. Zakharov and A.B. Shabat. Exact theory of two-dimensional self- focusing and one-dimensional self-modulation of waves in nonlinear me- dia. Soviet Journal of Experimental and Theoretical Physics, 34:62, 1972.

The defocusing nonlinear Schrödinger equation with step-like oscillatory data

The defocusing nonlinear

Schrödinger equation with step-like oscillatory data

The defocusing nonlinear

Schrödinger equation with step-like oscillatory data

Contents

Acknowledgements

1 Introduction

1.1 The nonlinear Schrödinger equation

1.2 The initial value problem

1.3 The inverse scattering transform

1.4 Riemann–Hilbert problems

1.5 The inverse scattering transform - Continuation

1.6 The nonlinear steepest descent method

1.7 A related initial-boundary value problem

2 Summary of Results

2.1 Paper A

2.2 Paper B

2.3 Paper C

Contributions of the author

References