Initial and Boundary Value Problems in Two and Three Dimensions

Transcription

1 Initia and Boundary Vaue Probems in Two and Three Dimensions Konstantinos Kaimeris Trinity Coege, Cambridge. ADissertationsubmittedfor the degree of Doctor of Phiosophy at the University of Cambridge 30 August 009

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3 Acknowedgments iii First and foremost, I woud ike to express my gratitude to my supervisor Thanasis Fokas. His intuition, insight and precision have aways been a great inspiration in my attempt to become a researcher. My research and studies in the University of Cambridge were supported by a student schoarship from the John S. Latsis Ieians Schoarships Foundation. Aso, my tuition fees were paid by EPSRC and I was receiving a part-cost bursary from George & Marie Vergottis Cambridge Bursaries. I hope they are a satisfied by the outcome of their investments. Aso I woud ike to sincerey thank George Dassios for the knowedge and advice he shared with me, both mathematica and non-mathematica, but mosty for his whoehearted support in both my undergraduate and graduate studentship. At this point I wish to extend my thanks to the members of the Fokas group : First of a, Euan Spence and Michai Dimakos for the interesting and sometimes enthusiastic conversations we had regarding many mathematica ideas, specuations and facts. Furthermore, I woud ike to thank Dionyssios Mantzavinos, Anthony Ashton, and Jonatan Lenes. I have enjoyed a our discussions. Aso, thanks shoud go to Stefanos and Stergios who seem to aways have an eegant answer to my mathematica questions. Specia thanks must be attributed to the Kastritsi peope. Especiay to Psios and Bogias for the periods of reaxation and Totis, Andrew and Nikoas for the times of great fun. Kokos and Iias were the companions in my ife who were wiing to cope with any issue that seemed to be a dead-end and concude with something more than a hope. More specia thanks are attributed to Constantinos for the nice time we had in a the paces we have been and the greatest thanks go to Anatoi for supporting me in any aspect of my ife, ignoring the potentia cost. Finay, I woud ike to thank my parents and my sister for their contribution in my becoming an independent and responsibe person.

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5 Decaration This dissertation is based on research done atthedepartmentofappiedmathematics and Theoretica Physics from October 005 to June 009. This dissertation is the resut of my own work and incudes nothing which is the outcome of work done in coaboration except where specificay indicated in the text. Konstantinos Kaimeris Cambridge, 30 August 009 v

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7 Abstract This thesis: a presents the soution of severa boundary vaue probems BVPs for the Lapace and the modified Hemhotz equations in the interior of an equiatera triange; b presents the soution of the heat equation in the interior of an equiatera triange; c computes the eigenvaues and eigenfunctions of the Lapace operator in the interior of an equiatera triange for a variety of boundary conditions; d discusses the soution of severa BVPs for the non-inear Schrödinger equation on the haf ine. In 1967 the Inverse Scattering Transform method was introduced; this method can be used for the soution of the initia vaue probem of certain integrabe equations incuding the ceebrated Korteweg-de Vries and noninear Schrödinger equations. The extension of this method from initia vaue probems to BVPs was achieved by Fokas in 1997, when aunifiedmethodforsovingbvpsforbothintegrabenoninearpdes,asweasinear PDEs was introduced. This thesis appies the Fokas method to obtain the resuts mentioned earier. For inear PDEs, the new method yieds a nove integra representation of the soution in the spectra transform space; this representation is not yet effective because it contains certain unknown boundary vaues. However, the new method aso yieds a reation, known as the goba reation, which coupes the unknown boundary vaues and the given boundary conditions. By manipuating the goba reation and the integra representation, it is possibe to eiminate the unknown boundary vaues and hence to obtain an effective soution invoving ony the given boundary conditions. This approach is used to sove severa BVPs for eiptic equations in two dimensions, as we as the heat equation in the interior of an equiatera triange. The impementation of this approach: a provides an aternative way for obtaining cassica soutions; b for probems that can be soved by cassica methods, it yieds vii

8 nove aternative integra representations which have both anaytica and computationa advantages over the cassica soutions; c yieds soutions of BVPs that apparenty cannot be soved by cassica methods. In addition, a nove anaysis of the goba reation for the Hemhotz equation provides amethodforcomputingtheeigenvauesandtheeigenfunctionsofthelapaceoperator in the interior of an equiatera triange for a variety of boundary conditions. Finay, for the noninear Schrödinger on the haf ine, athough the goba reation is in genera rather compicated, it is sti possibe toobtainexpicitresutsforcertainboundary conditions, known as inearizabe boundary conditions. Severa such expicit resuts are obtained and their significance regarding the asymptotic behavior of the soution is discussed. viii

9 Contents 1 Introduction The probems Cassica theory and techniques Green s integra representation Separation of variabes The method of images/refections Conforma mapping The Fokas method Achievements of the thesis Structure of the thesis Linear eiptic equations in an equiatera triange The probems The Lapace Equation ix

10 x Contents..1 Symmetric Dirichet Probem The Genera Dirichet Probem The Modified Hemhotz Equation The Symmetric Dirichet Probem The Poincaré Probem The Generaized Hemhotz Equation The Symmetric Dirichet probem in the Equiatera Triange Eigenvaues for the Lapace operator in the interior of an equiatera triange Formuation of the probems The Dirichet Probem The Neumann Probem The Robin Probem The Obique Robin Probem The Poincaré Probem The obique Robin, Robin, Neumman and Dirichet eigenvaues as particuar imits of the Poincaré eigenvaues Eigenfunctions The heat equation in the interior of an equiatera triange. 99

11 CONTENTS xi 4.1 The Symmetric Dirichet Probem An exampe The Genera Dirichet Probem Expicit soiton asymptotics for the noninear Schrödinger equation on the haf-ine Formuation of the probems Spectra Theory Lax pair Bounded and Anaytic Eigenfunctions Spectra functions The goba reation The Riemann-Hibert probem Asymptotic behavior of the soutions Linearizabe Conditions Soitons Hump-shaped initia profies Exponentia initia profies Future work. 147 Bibiography 149

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13 Chapter 1 Introduction. 1.1 The probems. In this thesis the foowing PDEs are discussed: 1 The second order inear eiptic PDEs in two spatia dimensions q xx x, y+q yy x, y+4λqx, y =0 x, y D, where λ is a compex constant and D is some dimensiona domain with piecewise smooth boundary. For λ =0thisistheLapaceequation,λ>0theHemhotz equation, λ<0themodifiedhemhotzequationandotherwisethe generaized Hemhotz equation. The heat equation, which is a second order inear evoution PDE, in three dimensions q t q x1 x 1 q x x = f, x 1,x D, 0 <t<t, 1.1. where fx 1,x,t is a given function and D is some dimensiona domain with piecewise smooth boundary. 3 The noninear Schrödinger NLS equation on the haf ine iq t + q xx λ q q =0,λ= ±1, 0 <x<, t>

14 1. Introduction. For the first cass of equations severa casses of Boundary Vaue probems BVP are soved expicity, when D is an equiatera triange. Moreover, the Hemhotz equationλ > 0 provides the eigenvaues and eigenfunction of the associated Lapace operatorλ =0. The boundary vaue probems anayzed have the foowing boundary conditions Dirichet: qx, y =known, x, y D Neumman: q x, y =known, N x, y D Robin: q x, y χqx, y =known, χ constant, N x, y D obique Robin: sin δ q N x, y D q x, y+cosδ x, y χqx, y =known, T δ, χ constants, Poincaré: sin δ j q N x, y +cosδ j q T x, y χ jqx, y =known, δ j, χ j constants, x, y D, where q N = q N, N is the unit outward-pointing norma vector to D, q T = q T, T is the unit tangent vector to D; theterminoogy obiquerobin canbejustifiedby rewriting the reevant condition as q q sin δ, cos δ x, y, x, y χqx, y =0, T N x, y D thus it invoves the derivative of q in the direction making an ange δ with the tangent vector on the boundary, i.e. with every side of the equiatera triange; the Poincaré condition describes the case when there exist different obique Robin conditions in each piece of the piecewise smooth boundary, i.e. in each side of the equiatera triange, see Figure 1.1. Simiar considerations are vaid for the Initia Boundary Vaue probems IBVP for the heat equation in the equiatera triange; in this case we mainy anayze the Dirichet probem, i.e. qx 1,x,t=known, x 1,x,t T, where T = {x 1,x D, 0 <t<t}.

15 1. Cassica theory and techniques Figure 1.1: For the NLS equation, the foowing types of boundary conditions, the so-caed inearizabe, are discussed: q0,t=0; q x 0,t=0; q x 0,t χq0,t=0, χ R Furthermore, we wi anayze three casses ofinitiaboundaryvaueprobemsibvp; these probems invove one of the boundary conditions 1.1.4, as we as initia conditions characterized by the foowing functions: a a soiton evauated at t = 0; b a function describing a hump; and c an exponentia function. A these probems, i.e. BV and IBV for both inear and integrabe noninear PDEs are anayzed by the unified method, caed the Fokas method introduced in [1], in 1997; it was further deveoped by severa authors, see for exampe [], [3], [4], [5], [6], [7], [8] and the monograph [9]. 1. Cassica theory and techniques. In this section we review briefy the cassica theory for soving for λ R. We discuss ony the techniques which can be appied to the boundary vaue probems considered in the thesis. [10] provides an exceent survey of both these techniques and many other exact and approximate methods for soving boundary vaue probems for inear PDEs.

16 4 1. Introduction Green s integra representation. Green s theorem gives an integra representation of the soution of 1.1.1, invoving the fundamenta soution sometimes known as the free space Green s function and both the known and unknown boundary vaues. We note that a drawback for both Hemhotz and modified Hemhotz in -d is that the fundamenta soution is given as a specia function. In order to formuate, for instance, an integra representation of the soution of for the Dirichet probem one shoud first determine the Green s function for the corresponding domain, i.e., y +4λGy, x =δy x, y Ω Gy, x =0, y Ω Aternativey, if the eigenvaues and eigenfunctions of the Lapacian are known in Ω then the probem is soved since the Greens function can be constructed as an infinite sum of the eigenfunctions. 1.. Separation of variabes. Start with a given boundary vaue probem in a separabe domain one where Ω = {a 1 x 1 b 1 } {a x b } where x j are the co-ordinates under which the differentia operator is separabe. This method invoves the separation of the PDE into two ODEs and the derivation of the associated competeness reation i.e. transform pair depending on the boundary conditions for one of the ODEs. Then the soution of the boundary vaue probem is given as a superposition of eigenfunctions of this ODE. Some of the main imitations of this method for soving boundary vaue probems are the foowing:

17 1. Cassica theory and techniques. 5 It fais for BVPs with non-separabe boundary conditions for exampe, those which incude a derivative at an ange to the boundary. The appropriate transform depends on the boundary conditions and so the process must be repeated for different boundary conditions. The soution is not uniformy convergent on the whoe boundary of the domain since it is given as a superposition of eigenfunctions of one of the ODEs. In the author s opinion the best references on separation of variabes are: [11] voume 1 chapter 4 spectra anaysis of differentia operators, [1] paragraph 5.1 separabe coordinates, [13] chapter 4 spectra anaysis, chapter5transformsandswitchingbetween the aternative representations, [14] chapter 7 spectra anaysis chapter 8 paragraph transform methods, [15] paragraphs 4.4, 5.7, 5.8 transform methods, [16] and [17] The method of images/refections. This technique can be used to find either the Green s function or the eigenfunctions and eigenvaues. The domains on which this technique works are the haf pane, the infinite strip, the semi-infinite strip, the wedge of ange π/n, n Z +,therectangeandthree types of trianges the equiatera, the right isoscees and the right triange. This appies to Dirichet and Neumann boundary conditions, as we as some mixed boundary conditions where Dirichet conditions are posed on part of the boundary and Neumann conditions on the rest the mixed boundary conditions which are aowed for each domain are detaied in [18]. For a the domains except for the haf pane and wedge, an infinite number of images is required, and so the Green s function is given as an infinite sum. The extension of the method to Robin and obique Robin boundary conditions in the upper haf pane is given in [19] and [0]. The Green s function is given as the source, pus one image, pus an semi-infinite ine of images. Robin and obique Robin boundary

18 6 1. Introduction. conditions in a wedge of ange π/n, n Z + are considered in [1]. For the Robin probem the Greens function is given as a source point, pus infinite ines of images, pus infinite regions of images. The obique Robin probem can ony be soved if n is odd and under some restrictions on the ange of derivative in the boundary conditions this is to ensure no images ie inside the domain. For the four bounded domains mentioned above, the method of images can be used to find their eigenfunctions and eigenvaues under Dirichet, or Neumann, or some mixed Dirichet-Neumann boundary conditions thesameonesforwhichthegreen sfunction can be found by refecting to one of the whoe space [], [3] a paraeogram [4], a rectange [5], where one can use separation of variabes in cartesian co-ordinates, then refecting back. This refection technique does not work for Robin or more compex boundary conditions. Some references that have interesting resuts concerning the method of images in poar co-ordinates are [6] and [7] Conforma mapping. The Lapace equation has the unique property that the Dirichet and Neumann probems can be soved using conforma mapping, in particuarschwarz-christoffemapping. When the mapping function is given expicity, this gives an integra representation of the soution. However, this is not the case for the equiatera triange in section., where inversion of specia functions is invoved. The other cassica techniques, and the Fokas method, become competitive when more

19 1.3 The Fokas method. 7 genera boundary conditions, such as Robin, are prescribed, which cannot be soved by conforma mapping. Simiar advantages of the Fokas method appear in the modified Hemhotz and Hemhotz equations. 1.3 The Fokas method. The Fokas method has the foowing basic ingredients: 1 the goba reation, which is an agebraic equation that invoves certain transforms of a initia and boundary vaues; the existence of these transforms justifies the terminoogy goba reation. the integra representation of the soution, given in terms of the goba form of a the initia and boundary vaues. Firsty, we wi iustrate how the Fokas method works for inear PDEs: Given a PDE, construct a scaar differentia form which is cosed iff the PDE is satisfied. From this differentia form define two compatibe inear eigenvaue equations with scaar eigenfunctions, which are caed a Lax pair. On the one hand, by empoying Green s theorem, this differentia form yieds the goba reation, which is an agebraic equation couping the reevant spectra functions. On the other hand, the simutaneous spectra anaysis of both parts of the Lax pair yieds a scaar Riemann-Hibert probem, which consequenty yieds the reevant integra representation of the soution in terms of the spectra functions.

20 8 1. Introduction. Finay, the expicit soution of the associated probem is derived through the eimination of the unknown boundary vaues in the integra representation, by using appropriatey the goba reation. The situation in the noninear PDEs is conceptuay simiar, but more compicated. Now, we construct a matrix differentia form, which yieds a Lax pair containing matrix eigenfunctions. This impies that the spectra functions are not given expicity by the reevant initia and boundary vauesthey are given as the soutions of inear integra equations of the Voterra type. Furthermore, the integra representation of the soution is given through a matrix Riemann-Hibert probem which cannot be soved in cosed formits soution is characterized by a inear integra equation of Fredhom type. However, there exist certain cass of boundary conditions, caed inearizabe, for which the unknown spectra functions can be obtained through the agebraic manipuation of the goba reation. 1.4 Achievements of the thesis. Boundary vaue probems for q z =0andtheModifiedHemhotzequationweresoved in [8], [9], [30] and [31]. Soutions in terms of infinite series have been derived for severa probems of the Lapace, Hemhotz and modified Hemhotz equations in the interior of an equiatera triange in [3] and for the Lapace equation in the interior of a right isoscees triange in [33], empoying the Fokas method; this is to be contrasted to other techniques based on the eigenvaues of the reevant operators that yied the soution as a bi-infinite series. The eigenvaues of the Lapace operator for the Dirichet, Neumann and Robin probems in the interior of an equiatera triange were first obtained by Lamé in 1833 [34]. Competeness for the associated expansions for the Dirichet and Neumann probems was obtained in [3], [4], [35], [5] using group theoretic techniques. Competeness for the associated expansion for the Robin probem was achieved in [36] using a homotopy argu-

21 1.4 Achievements of the thesis. 9 ment. These resuts have been rederived by severa authors, see for exampe [37]-[38]. The cassica probem of the heat equation is soved in severa ways in separabe domains, but for non-separabe has been mainy reated with the resuts obtained for the modified Hemhotz equation, through the Lapace transform. Moreover, the Fokas method was extended to evoution PDEs in two spatia dimensions in [39] and [40]. The integra representations of the initia-boundary vaue probems on the haf ine, appied on the NLS, the sine-gordonsg and the Korteweg-de VriesKdV, were derived in [3] and []. Furthermore, the inearizabe boundaryconditionswereobtainedforeach one of the equations. These resuts were reviewed in [9]. Considering these probems, the main achievements of this thesis are: The soutions of the same probems with those considered in [3], for the Lapace and modified Hemhotz equations in the interior of an equiatera triangenonseparabe domain, are now given as an integraas opposed to an infinite sum in [3], [33], and a bi-infinite sum cassicay. Furthermore, a nove approach has been introduced which empoys the goba reation at the same time that the contours of the integra representation are being deformed. As a resut, the integrands of the reevant integras are exponentiay decaying functions; this has anaytica and numerica advantages. A specific choice for the contours of integration in the integra representation and Cauchy s theorem, yieds the soution in terms of an infinite series of the reevant residues, which provides a reationship between the discrete and the continuous spectrum of these probems. The integra representation of the generaized Hemhotz equation in the interior of aconvexpoygonisgivenforthefirsttime;thisisasothecaseforthesoution of the Dirichet probem in the interior of an equiatera triange. These resuts are interesting, in particuar taking into consideration the reation of this equation with

22 10 1. Introduction. certain evoution PDEs in higher dimensions. Regarding the eigenvaues of the Lapace operator a simpe, unified approach for rederiving the previous resuts is presented. Furthermore the eigenvaues for the obique Robin and certain Poincaré probemsarederivedforthefirsttime. The method introduced here is based on the anaysis of the goba reation, see [7]. In addition, combining these resuts with the integra representation of the soution of the Hemhotz equation, yieds the corresponding eigenfunctions. The soution of the heat equation in an equiatera triange is expressed as an integra in the compex Fourier space, i.e. the compex k 1 and k panes, invoving appropriate integra transforms of the known boundary conditions. Moreover, the soution is expressed in terms of an integra whose integrand decays exponentiay as k. Hence, it is possibe to evauate this integra numericay in an efficient and straightforward manner. The distribution of zeros of the spectra functions of the inearizabe boundary vaue probems for the NLS yieds the expicit asymptotic behavior of the soution. In particuar, it yieds the number of soitons generated from the given initia and boundary conditions. 1.5 Structure of the thesis. Chapter : Linear Eiptic Equations in an Equiatera Triange. We sove: Lapace equation in an equiatera triange for symmetric Dirichet the same function is prescribed in a three sides, as we as arbitrary Dirichet boundary conditions. modified Hemhotz equation in an equiatera triange for symmetric Dirichet and Poincaré boundaryconditions.

23 1.5 Structure of the thesis. 11 generaized Hemhotz equation in an equiatera triange for symmetric Dirichet boundary conditions. Particuar cases of the Poincaré probemyiedthesoutionofotherprobems,e.g. obique Robin, Robin and Neumann. Common characteristics appear in the soution of a the above probems. The soution is given in terms of integras that have exponentiay decaying integrands on the contours of integration. Chapter 3: Eigenvaues for the Lapace operator in the interior of an equiatera triange. We find expicity the eigenvaues of the Lapace operator for the Dirichet and the Neumann probems in the equiatera triange. We derive expicit formuae for the computation of the eigenvaues of the Lapace operator for the Robin, the obique Robin and certain Poincaré probems in the equiatera triange. The formuae for Poincaré probem, yied the reevant eigenvaues of a other probems, via particuar imits. We find the eigenfunctions of the Lapace operator for the Dirichet probem and aso indicate how the eigenfunctions for a other probems can be computed. Chapter 4: The heat equation in the interior of an equiatera triange. We sove the heat equation in the interior of an equiatera triange for symmetric Dirichet and arbitrary Dirichet boundary conditions. this is achieved by empoying simiar techniques with those used in Chapter.

24 1 1. Introduction. The soution is given in terms of integras that have exponentiay decaying integrands on the contours of integration. Chapter 5: Expicit soiton asymptotics for the noninear Schrödinger equation on the haf-ine. A review of the Fokas method is given, in connection with initia and boundary vaue probems for noninear integrabe PDEs on the haf ine; emphasis is paced in the NLS. The inearizabe boundary conditions, forwhichtheunknownspectrafunctionsare computed via agebraic manipuation of the goba reation, are derived; furthermore, for this cass of boundary conditions threeinitia-boundaryvaueprobems are anayzed. These probems are characterized by the foowing initia conditions: asoitonevauatedatt =0; afunctiondescribingahump; an exponentia function. The anaysis of the spectra functions yieds effectiveasymptoticresutsusingthe Deift-Zhou techniques for the asymptotic anaysis of the reevant Riemann-Hibert probem, see [41].

25 Chapter Linear eiptic equations in an equiatera triange. Beow, we describe the soutions of some boundary vaue probems for the basic eiptic equations using the Fokas method, introduced in [1]. For inear PDEs, this method invoves the foowing stepssee [9]: 1 Given a PDE, construct a differentia form which is cosed iff the PDE is satisfied. From this differentia form define two compatibe inear eigenvaue equations which, in anaogy with the theory of noninear integrabe PDEs, are caed a Lax pair. 3 Empoying Green s theorem in this differentia form yieds a reation between certain functions ˆq j k, caed the spectra functions; these functionsarecertainintegrasof the vaues of q and of its derivatives on the boundary of the domain. From now on we wi refer to this reation as the goba reation. 4 Perform the simutaneous spectra anaysis of the Lax pair, which yieds an integra representation of the soution qz, z in terms of the spectra functions ˆqk. 5 Given appropriate boundary conditions, use theinvariantsofthegobareationto eiminate the unknown boundary vaues appearing in the integra formua obtained in 4. 13

26 14. Linear eiptic equations in an equiatera triange. The impementation of the approach presented here has certain nove features. In particuar, it constructs the soution in terms of integras which invove integrands that have strong decay as k. This is to be contrasted with earier investigations see [3] where the soution was expressed in terms of a combination of an infinite series and integras with osciating kernes..1 The probems. We impement this approach to the Lapace, modified Hemhotz and generaized Hemhotz equations for some boundary vaue probems in the interior of an equiatera triange. a Fundamenta Domain Let D C be the interior of the equiatera triange depicted in Figure.1 and defined by its three vertices z 1,z,z 3, z 1 = 3 e iπ 3,z = z 1,z 3 = 3,.1.1 where is the ength of the side. The sides z 1,z, z,z 3, z 3,z 1 wibereferredassides1,,3. The compex variabe z, on each of the sides 1,,3, satisfies the foowing reations: 1 dz s =i, ds dz s =ia, ds 3 dz s =iā, a = e i 1 π 3 3 = ds + i, where s denotes the arcength. Integrating the above equations and using the boundary conditions

27 .1 The probems. 15 y z z x 3 z 1 Figure.1: The Equiatera Triange. z 1 = z 1, z = z,z 3 = z 3, we find the foowing expressions parametrizing each of the three sides: z 1 s = 3 + is, z s = 3 + is a, z 3 s = 3 + is ā, <s<.1.. b Formuation of the probems The equations investigated in this chapter are given by 1.1.1, where D denotes the interior of the equiatera triange. Using the transformation λ = β γ,withβ 0, γ C and γ =1,weobtainthefoowingformof1.1.1: q xx + q yy +4γβ q =0, x, y D..1.3

28 16. Linear eiptic equations in an equiatera triange. The cases {β =0}, {β >0,γ = 1} and {β >0,γ 1} correspond to the Lapace, the modified Hemhotz and the generaized Hemhotz equations respectivey. The probems anayzed in the first section of this chapter are: i The Symmetric Dirichet probem for the Lapace equation, i.e. the case with the boundary conditions q j s =gs, s [, ],j=1,, ii The Dirichet probem for the Lapace equation, i.e. the case with the boundary conditions q j s =g j s, s [, ], j =1,, The probems anayzed in the second section of this chapter are: i The Symmetric Dirichet probem for the modified Hemhotz equation, i.e. the case with the boundary conditions q j s =ds, s [, ],j=1,, ii The Poincaré probemforthemodifiedhemhotzequation,i.e.thecasewith the boundary conditions sin δ j q j N s+cosδ d j ds qj s χ j q j s =g j s, s [, ],j=1,, 3,.1.7 where δ 1 is a rea constant so that sin δ 1 0,δ and δ 3 satisfy sin δ 0and sin δ 3 0 and are given in terms of δ 1 by the expressions δ = δ 1 + nπ 3, δ 3 = δ 1 + mπ, m,n Z, whereas the rea constants χ j,j =1,, 3 satisfy the reations [χ 3β χ +einπ χ 1 3β χ 1 ] sin 3δ 1 =0.1.9

29 .1 The probems. 17 and [χ 3 3β χ 3 +eimπ χ 1 3β χ 1 ] sin 3δ 1 = Note that the assumption sin δ j 0 is without oss of generaity since if sin δ j = 0 then after integration the boundary condition can be rewritten as d ds qj s =d j s, which becomes the Dirichet probem. The probem anayzed in the third section of thischapteristhesymmetricdirichet probem for the generaized Hemhotz equation. It is assumed that the functions g j s havesufficientsmoothnessandthattheyarecompatibe at the vertices of the triange. Reca the foowing identities: a If z = x + iy, z = x iy, x, y R, then z = 1 x i y, z = 1 x + i y b If a side of a poygon is parametrica by s, then q z dz = 1 q + iq Nds, q z d z = 1 q iq Nds,.1.1 where q is the derivative aong the side, i.e. q = dqzs/ds and q N is the derivative norma to the side in the outward direction. Under the transformation.1.11 equation.1.3 can be written in this form q z z + γβ q =0, where z = x + iy..1.13

30 18. Linear eiptic equations in an equiatera triange.. The Lapace Equation. The substitution β = 0 in.1.13yieds the foowing form of the Lapace equation q z z =0...1 Hence, since q z z = 0, it foows that q is harmonic iff q z is an anaytic function on z. This impies that it is easier to obtain an integra representation for q z instead of q. In this respect we note that q satisfies the Lapace equation iff the foowing differentia form is cosed, W z, k =e ikz q z dz, k C... In what foows, we wi use the spectra anaysis of the differentia form d [ e ikz µz, k ] = e ikz q z dz, k C,..3 to obtain an integra representation for q z in the interior of a convex poygon Ω. Furthermore the foowing goba reations are vaid Ω e ikz q z dz =0, Ω e ik z q z d z =0,k C...4 If q is rea then the second equation comes from the Schwarz conjugate of the first of the equations..4. If q is compex, the second of the equations..4 is a consequence of the differentia form W z, k =e ik z q z d z, k C,..5

31 . The Lapace Equation. 19 which is aso cosed iff q satisfies the Lapace equation. The foowing theorem, which can be found sighty different in [9] and [4], gives the formuae for the goba reation and the integra representation for the Lapace s equations in the interior of a convex poygon. Theorem.1. Let Ω be the interior of a convex cosed poygon in the compex z-pane, with corners z 1,...,z n,z n+1 z 1. Assume that there exists a soution qz, z of the Lapace equation, i.e. of equation..1, vaidonω and suppose that this soution has sufficient smoothness on the boundary of the poygon. Then q z can be expressed in the form q z = 1 π 3 j=1 j e ikz ˆq j kdk,..6 where {ˆq j k} n 1 are defined by ˆq j k = zj+1 z j e ikz q z dz, k C, j =1,...,n..7 and { j } n 1 are the rays in the compex k-pane j = {k C : argk = argz j+1 z j }, j =1,...,n..8 oriented from zero to infinity. Furthermore, the foowing goba reations are vaid n ˆq j k =0, j=1 n q j k =0,k C,..9 j=1 where { q j k} n 1 are defined by q j k = zj+1 z j e ik z q z d z, k C, j =1,...,n...10

32 0. Linear eiptic equations in an equiatera triange. Proof. Integrating equation..3 we find µ j z, k = z The term exp[ikz ζ] is bounded as k for z j e ikz ζ q ζ dζ, z Ω, j =1,...,n arg k +argz ζ π...1 If z is inside the poygon and ζ is on a curve from z to z j,seefigure.,then argz j+1 z j argz ζ argz j 1 z j, j =1,...,n. z ζ z j 1 z j+1 side j z j Figure.: Part of the convex poygon. Hence, the inequaities..1 are satisfied provided that arg z j+1 z j arg k π arg z j 1 z j. Hence, the function µ j is an entire function of k which is bounded as k in the sector Σ j defined by Σ j = {k C, arg k [ arg z j+1 z j,π arg z j 1 z j ]}, j =1,...,n...13 The ange of the sector Σ j,whichwedenotebyψ j, equas ψ j = π arg z j 1 z j +argz j+1 z j =π φ j,..14 where φ j is the ange at the corner z j.hence n n ψ j = nπ φ j = nπ πn = π,..15 j=1 j=1

33 . The Lapace Equation. 1 thus the sectors {Σ j } n 1 precisey cover the compex k-pane. Hence, the function µ = µ j, z Ω, k Σ j, j =1,...,n,..16 defines a sectionay anaytic function in the compex k-pane. For the soution of the inverse probem, we note that integration by parts impies that µ j = O1/k ask in Σ j,i.e. µ = O 1, k...17 k Furthermore, by subtracting equation..11 and the anaogous equation for µ j+1 we find µ j µ j+1 = e ikz ˆq j k, z Ω, k j, j =1,...,n,..18 where {ˆq j k} n 1 are defined by equation..7 and j is the ray of overap of the sectors Σ j and Σ j+1. Using the identity π arg z j z j+1 = arg z j+1 z j mod π,..19 it foows that j is defined by equation..8. Furthermore, Σ j is to the eft of Σ j+1,see Figure.3. π argz j 1 z j Σ j argz j+1 z j π argz j z j+1 argz j+ z j+1 Σ j+1 Figure.3: The sectors Σ j and Σ j+1. The soution of the RH probem defined by equations is given by

34 . Linear eiptic equations in an equiatera triange. µ = 1 iπ n j=1 j e iz ˆq j d k, z Ω, k C\{ j} n Substituting this expression in equation..3, i.e. in the equation we find equation..6. µ z ikµ = q z, Using the definitions of {ˆq j } n 1 and of { q j} n 1, i.e. equations..7 and..10 respectivey, equations..4 yied the two goba reations..9. Substituting equations.1.1 in the definition of the function ˆq j k and q j k wefind the foowing expressions and ˆq j k = 1 q j k = 1 zj+1 e ikz z j zj+1 e ikz z j iq j N + qj ds, k C,..1 iq j N + qj ds, k C,.. where the index j denotesthevaueofthecorrespondingfunctionsonsidej. Observe that the soution..6 is given in terms of the spectra functions ˆq which invove both q and q n on the boundary, i.e. both known and unknown functions. In what foows the unknown functions wi be eiminated from the integra representation of the soution, by using appropriatey the goba reations...1 Symmetric Dirichet Probem. The probem anayzed here is the Symmetric Dirichet probem for the Lapace equation in the Equiatera TriangeΩ D, i.e. the case with the boundary conditions

35 . The Lapace Equation. 3 For convenience we define q j s =gs, s ds = qs, s [, ],j=1,, 3. [, ],j=1,, 3. It is aso assumed that the function ds has sufficient smoothness and that it is compatibe at the vertices of the triange, i.e. d =d. Appying the parametrization of the fundamenta domain given in equations.1., on equations..1 and.., we obtain the foowing expressions for the spectra functions {ˆq j k} 3 1 and { q jk} 3 1 : ˆq 1 k =ˆqk, ˆq k =ˆqak, ˆq 3 k =ˆqāk, with..3 ˆqk =E ik[iuk+dk] and q 1 k = qk, q k = qāk, q 3 k = qak, with..4 qk =Eik[ iuk+dk], where Ek =e k 1 3, Dk = e ks dsds, Uk = 1 e ks q N sds, k C. The function Dk is known, whereas the unknown function Uk contains the unknown Neumann boundary vaue q N. It turns out that, using agebraic manipuations of the goba reations and appropriate contour deformations of the { j } 3 1, it is possibe to eiminate the unknown functions Uk,

36 4. Linear eiptic equations in an equiatera triange. Uak, Uāk fromtherepresentationofthesoutionat..6. Inthiswaywewiobtain the foowing integra representation: qz z = 1 [ Ak, z, ze ik Dk+ Gk ] dk π 1 ak Ak, z, ze Gk iak π ak k dk 1 where 1 = {k C :argk = π }, 1 is the ray with π arg k π see Figure.4 and 6 Ak, z, z =e ikz +āe iākz + ae iakz, Gk = + akdk+dāk+ + kdak, k =ek e k, + k =ek+e k, ek =e k...6a..6b..6c D3 π 6 π 6 D D Figure.4: The contours j and j. Using the Goba Reations Appying..3 in the first of the goba reations..9 and mutipying by Eiāk we obtain the equation

37 . The Lapace Equation. 5 where e akuk+ekuak+uāk =ijk, k C,..7 Jk =e akdk+ekdak+dāk. Appying..4 in the second of the goba reations..9andmutipyingby E iāk weobtaintheequation eakuk+e kuak+uāk = ie kj k, k C,..8 where J k denotesthefunctionobtainedfromjk bytakingthecompexconjugateof a the terms in Jk exceptds. In this respect, note that if ds is a rea function, then equation..8 can be obtained by taking the Schwarz conjugate of..7 and mutipying by e k. Subtracting equations..7 and..8 we find the foowing equation which is vaid for a k C, where Gk =Jk+e kj k. akuk = kuak igk,..9 Substituting Uk in the expression of ˆqk in..3 we find ˆqk =E ikdk+ E ikgk k + i[e iāk E iak] Uak ak...30 The functions ˆq k andˆq 3 k canbeobtainedfrom..30byrepacingk with ak and āk respectivey. In what foows we wi show that the contribution of the unknown functions Uak, Uk anduāk canbecomputedintermsofthegivenboundaryconditions,usingthe

38 6. Linear eiptic equations in an equiatera triange. foowing basic facts. Basic facts 1. The zeros of the functions k, ak, āk occuronthefoowinginesrespectivey in the compex k-pane ir, e 5iπ 6 R, e iπ 6 R. Indeed, k =0 sinh k =0. Hence, the zeros of k occur on the imaginary axis and then the zeros of ak and āk canbeobtainedbyappropriaterotations.. The functions e ikz E iak, e ikz E ik, e ikz E iāk are bounded and anaytic for a z D, forargk in [ π, π ] [ π, 6 6, 5π ] [ 5π, 6 6, 3π respectivey, as shown in Figure.5. ], 3 e ikz E ik e ikz E iāk e ikz E iak 1 Figure.5: The domains of boundedness and anayticity. Indeed, et us consider the first exponentia e ikz E iak =e ikz z 1.Ifz D then π argz z 1 5π 6,

39 . The Lapace Equation. 7 thus if it foows that π arg k π 6, 0 arg[kz z 1 ] π. Hence the exponentia e iβkz z 1 is bounded. Simiary for the other two exponentias. 3. The functions Uk k, Uak ak and Uāk āk above ines where k, ak and āk havezeros. Indeed, regarding Uk k by e k for Rek < 0, hence are bounded and anaytic in C apart from the observe that k is dominated by ek forrek > 0and { Uk e kuk, Rek > 0 k ekuk, Rek < 0. Furthermore e kuk invoves e ks which is bounded for Rek 0andekUk invoves e ks+ which is bounded for Rek 0. The unknown Uak in the expression for ˆqk at..30,yiedsthecontributionc 1 z to the soution q given in..6, C 1 = i e ikz [E iāk E iak] Uak π 1 ak dk. The integra of the second term in the rhs of C 1 can be deformed from 1 to 1,where 1 is a ray with π arg k π 6. Hence, C 1 = i e ikz E iāk Uak π 1 ak dk i e ikz E iak Uak π 1 ak dk. In the second integra of the rhs of this equation we repace Uak byusing..9,i.e., akuk = kuak igk.

40 8. Linear eiptic equations in an equiatera triange. Hence C 1 = i e ikz E iāk Uak π 1 ak dk i e ikz E iak Uk π 1 k dk + 1 e ikz E Gk iak dk...31 π k ak 1 In summary the term ˆqk givesrisetothecontributionf 1 + Ũ1, whereũ1 denotes the first two terms of the rhs of..31 and F 1 is defined by F 1 = 1 π + 1 π [ E ikdk+ E ikgk ] dk ak e ikz E Gk iak dk...3 k ak 1 e ikz 1 The contributions to the soution of ˆq and ˆq 3 can be obtained from F 1 + Ũ1 with the aid of the substitutions 1 3, 1 3, k ak āk...33 The contribution of Ũj,j =1,, 3vanishduetoanayticity. Indeed,theintegrands e ikz E iāk Uak ak,eikz E ik Uāk āk,eikz E iak Uk k..34 occur in 1, 3, 3 1, and in the corresponding domains the above functions are bounded and anaytic. Hence, q = F 1 + F + F 3,..35

41 . The Lapace Equation. 9 where F and F 3 are obtained from F 1 using the substitutions..33. In order to derive the integra representation..5, we make the change of variabes k āk on the integras in F and the change of variabes k ak on the integras in F 3. In particuar, regarding F this eads to the foowing changes: 1. The differentia dk becomes ādk.. The rays and become 1 and 1 respectivey. 3. The exponentia e ikz becomes e iākz. 4. The remaining integrand is equa to the corresponding integrand in F 1. Simiar changes occur in F 3. The integrands appearing in the integras aong 1 and 1 defined in equation..5 contain terms which decay exponentiay. Regarding the integra aong 1,observethat Gk k ak is bounded in the domain of deformation of 1 see Figure.4. The function Ak, z, ze iak is aso exponentiay decaying. In particuar, the first term of the function e ikz E iak is an exponentia whose exponent has negative rea part in the domain D 1 see Figure.4. Simiar considerations are aso vaid for the other two remaining terms of the function Ak, z, ze iak. Hence we concude that the integrand of the integra aong 1 is an exponentiay decaying function. Regarding the integra aong 1,observethatDk and Gk ak are bounded for k 1 and since the function e ikz E ik is an exponentia whose exponent has negative rea part when k 1, we concude that the integrand of the integra aong 1 is an exponentiay decaying function. The above facts can be expicity verified in the foowing exampe. Exampe.1. Set = π and ds =cossi.e. q j s = sin s, j =1,, 3.

42 30. Linear eiptic equations in an equiatera triange. In this case, and [ Gk = 1+k + 1+ak Dk = 1 k 1+k cosh π ] cosh k π cosh ak π + where we have used that + k =cosh k π. āk 1+āk cosh π,..36 In order to check the convergence of a the integras appearing in the representation..5 observe the foowing: For the first integra aong 1, Rek =0and Imk < 0. Hence,ask : 1. e ikz E ik e ik Rez π 3 e Imk x π 3, x < π ; 3. Dk 1 3. k ; Gk ak 1 k. For the second integra, aong 1, arg k π, π.hence,ask : 6 [ 1. e ikz E iak exp x π cos ] φ + π 3 y + π sin φ + π,wherex< π and y> π and φ =argk. Hence the exponent is negative when arg k 3 π, π 6 i.e. in the domain of 1 deformation; Gk. 1. k ak k Simiar arguments are vaid for the other two terms of Ak, z, z... The Genera Dirichet Probem. We now consider the soution of the arbitrary Dirichet probem, i.e. of the probem with the boundary conditions q j s =d j s, s [, ],j=1,, 3, where the function d j s havesufficientsmoothnessandarecompatibeattheverticesof the triange, i.e. d j =d j+1,j=1,, 3, d 4s =d 1 s. The soution of this probem can be obtained in two different ways:

43 . The Lapace Equation. 31 i In the first approach we use the soution of the symmetric Dirichet probem, as we as the fact that the arbitrary Dirichet probem can be decomposed into three probems, which are soved in a way simiar to the symmetric Dirichet probem. ii In the second approach we use the invariants of the goba reations and foow the genera methodoogy used for the symmetric Dirichet probem. The second approach is more compicated, however, it has the advantage that it can be used to sove other probems that do not admit the decomposition mentioned in i above. Such probems are: a the Poincaré probem defined in equation.1.7; b the obique Robin probem defined in equation.1.7, with δ j = δ, and χ j = χ, j =1,, 3; c the Robin probem defined in equation.1.7, with δ j = π, and χ j = χ 0,j=1,, 3. The First Approach The genera Dirichet probem can be decomposed into the foowing three probems: 1. Let q satisfy the symmetric Dirichet probem for..1 in the domain D defined in.1.1, i.e. q j s =fs, s [, ],j=1,, 3, where fs is sufficienty smooth and compatibe at the corners of the triange i.e. f = f.. Let q satisfy..1 in the domain D defined in.1.1, with the foowing Dirichet boundary conditions q 1 s =gs, q s =ags, q 3 s =āgs, s [, ],j=1,, 3,

44 3. Linear eiptic equations in an equiatera triange. where gs is sufficienty smooth and compatibe at the corners of the triange i.e. g = ag. 3. Let q satisfy..1 in the domain D defined in.1.1, with the foowing Dirichet boundary conditions q 1 s =hs, q s =āhs, q 3 s =ahs, s [, ],j=1,, 3, where hs is sufficienty smooth and compatibe at the corners of the triange i.e. h =āh. A genera Dirichet boundary vaue probem can be written as the sum of above three boundary vaue probems. Indeed, suppose that the foowing Dirichet condition is vaid [ q j s =d j s, s, ],j=1,, 3. The matrix of the foowing 3 3agebraicsystemisnon-singuar: d 1 s fs d s = 1 a ā gs, Det[ 1 a ā ]=i d 3 s 1 ā a hs 1 ā a Due to uniqueness, the soution of the genera Dirichet probem is given by the sum of the three probems defined earier. The soution of the probems and 3 above can be obtained in a way simiar to the symmetric case. Indeed, et us consider probem, where the Dirichet conditions are given by q 1 s =ds, q s =ads, q 3 s =āds, s [, ],j=1,, 3. As mentioned earier the function ds has sufficient smoothness and is compatibe at the vertices of the triange, i.e. d =ad. Appying the parametrization of the fundamenta domain on equations..1 and..we obtain the foowing expressions for the spectra functions {ˆq j k} 3 1 and { q jk} 3 1 :

45 . The Lapace Equation. 33 ˆq 1 k =ˆqk, ˆq k =a ˆqak, ˆq 3 k =ā ˆqāk, with ˆqk = E ik[iuk+ Dk]..38 and q 1 k = qk, q k =ā qāk, q 3 k =a qak, with qk = Eik[ iuk+ Dk]..39 where Ek =e k 1 3, Dk = e ks dsds, Uk = 1 e ks q N sds, k C...40 The function Dk is known, whereas the unknown function Uk contains the unknown Neumann boundary vaue q N.Thesoutionofthisprobemcannowbeobtainedbyadopting the methodoogy of the symmetric case and aso making the foowing substitutions: Uk Uk, Uak auak, Uāk āuāk Dk Dk, Dak adak, Dāk ādāk Thus, the soution is given by qz = 1 Bk, z, ze ikdkdk z π Ak, z, ze ik Gk π 1 ak dk + 1 Ak, z, ze Gk iak π ak k dk where Ak, z, z =e ikz +āe iākz + ae iakz, Bk, z, z =e ikz + e iākz + e iakz, Gk = + akdk+ādāk+a + kdak, k =ek e k, + k =ek+e k, ek =e k,..43a..43b..43c..43d with Dk given in..40.

46 34. Linear eiptic equations in an equiatera triange. The Second Approach In what foows we iustrate a direct way to find the soution of the Dirichet probem in the interior of the equiatera triange, Figure.1. Appying the parametrization of the fundamenta domain, given in equations.1., on equations..1 and..we obtain the foowing expressions for the spectra functions {ˆq j k} 3 1 and { q j k} 3 1: ˆq 1 k =E ik[iu 1 k+d 1 k], ˆq k =E iak[iu ak+d ak], ˆq 3 k =E iāk[iu 3 āk+d 3 āk],..44 where Ek =e k 3, Dj k = 1 e ks d j sds, U j k = 1 e ks q j N sds, k C...45 Using agebraic manipuations of the goba reations and appropriate contour deformation, it is possibe to eiminate the unknown functions U 1 k, U ak, U 3 āk fromthe representation of the soution at..6 and thus, obtain the representation qz z = 1 e ikz E ik π e ikz E iak π + 1 e ikz E iāk π 3 [ D 1 k+i Γ 1ak ak [ D ak+i Γ āk āk [ D 3 āk+i Γ 3k k ] dk + i π ] dk + i ] dk + i π e ikz E iak Γ 13ak 1 k ak dk e ikz E Γ 31 āk iāk π ak āk dk e ikz E Γ 31 k ik āk k dk, where { j } 3 1, { j} 3 1 are depicted in Figure.4, k =e 3 k e 3 k, ek =e k,..47 Γ mn k =E 3 kγ k+γ m k+e 3 kγ n k,..48

47 . The Lapace Equation. 35 Γ 3 k =[E 3 iak+e 3 iak]e kd 1 k +[E 3 iak+e 3 iak]ekd k +[E 3 iāk+e 3 iāk]d 3 k..49 +e kd 1 ak+e kd ak+d 3 ak +e kd 1 āk+e kd āk+[e 3 k+e 3 k]d 3 āk, Γ 1 k is obtained by making the rotations 3 1, 1, 3onthesubscriptsofΓ 3 k and Γ k is obtained by making the rotations 3, 1, 1 3onthesubscriptsof Γ 3 k. Using the Goba Reations. Appying..44 in the first of the goba reations..9 we obtain the foowing equation E iku 1 k+e iaku ak+e iāku 3 āk =if 1 k, k C,..50 where F 1 k =E ikd 1 k+e iakd ak+e iākd 3 āk. Furthermore, appying..44 in the second of thegoba reationswe obtain the foowing equation EikU 1 k+eiāku āk+eiaku 3 ak = if k, k C,..51 where F k =EikD 1 k+eiākd āk+eiakd 3 ak. Appying the transformations k ak and k āk in both..50 and..51 we find an agebraic system of 6equationswhich invoves the foowing 9unknownfunctions: { } U 1 k, U 1 ak, U 1 āk,u k, U ak, U āk,u 3 k, U 3 ak, U 3 āk. Hence, we can sove this system for one of the unknown functions in terms of three other unknown functions and some known function. Hence, soving this system for U 3 āk in

48 36. Linear eiptic equations in an equiatera triange. terms of { U 1 k,u k,u 3 k } we obtain the foowing reationsee [3] ku 3 āk =[E 3 iak E 3 iak]e ku 1 k +[E 3 iak E 3 iak]eku k..5 +[E 3 iāk E 3 iāk]u 3 k+γ 3 k, where k is defined in..47 and {Γ j k} 3 1 invove the known functions {F 1 k, F k, F 1 ak, F ak, F 1 āk, F āk} which are defined in equation..49. Soving again the system of equation for U 1 k in terms of { U 1 ak,u ak,u 3 ak },orbysimpymaking the substitution k ak and the rotations 3 1, 1, 3onthesubscriptsof..5 we obtain the foowing reation aku 1 k =[E 3 ik E 3 ik]u 1 ak +[E 3 iāk E 3 iāk]e aku ak..53 +[E 3 iāk E 3 iāk]eaku 3 ak+γ 1 ak. Foowing the same pattern, we obtain the expression of U ak in terms of { U 1 āk, U āk, U 3 āk }, by substituting k āk and the rotations 1, 1 3, 3 onthe subscripts of the equation..5: āku ak =[E 3 ik E 3 ik]eāku 1 āk +[E 3 iak E 3 iak]u āk..54 +[E 3 ik E 3 ik]e āku 3 āk+γ āk. Rotating the subscripts of..53 we obtain aso the expressions of U k andu 3 k in terms of { U 1 ak,u ak,u 3 ak }.Theseexpressionsyiedthefoowingidentity E 3 ku 1 k+u k+e 3 ku 3 k k E 3 aku 1 ak+u ak+e 3 aku ak ak = + Γ 13ak k ak,..55 where k C / {k; k =0 ak =0} and Γ mn k aretheknownfunctionsdefined in..48. Furthermore, empoying the substitution k āk and the rotations 1 3,

49 . The Lapace Equation. 37 3, 1onthesubscriptsof..55,wefind U 1 āk+e 3 āku āk+e 3 āku 3 āk āk U 1 k+e 3 ku k+e 3 ku 3 k k = + Γ 31k k āk...56 Simiary the substitution k ak and the rotations 1, 3, 3 1onthesubscripts of..55 yied E 3 aku 1 ak+e 3 aku ak+u 3 ak ak E 3 āku 1 āk+e 3 āku āk+u āk āk = + Γ 31āk ak āk...57 Repacing in ˆq 1 k givenin..44thetermu 1 k with the expression given in..53 we find: ˆq 1 k =E ikd 1 k+ie ik Γ 1ak ak + i {[ ] E iāke 3 ak E iake 3 ak U 1 ak ak [ ] + E iāke 3 ak E iak U ak [ ] } + E iāk E iake 3 ak U 3 ak,..58 where we have used that a = 1 + i 3,ā = 1 i 3, Ek =ek 3 and ek =e k. Equation..58 can be rewritten in the foowing form ˆq 1 k =E ikd 1 k+ie ik Γ 1ak ak + i E iāk ak i E iak ak [ E 3 aku 1 ak+e 3 aku ak+u 3 ak [ ] E 3 aku 1 ak+u ak+e 3 aku 3 ak. ]..59 Hence, the unknown functions {U j ak} 3 1 in..59 yied the foowing contribution to the soution C 1 z = i e ikz E 1 iāk π 1 ak i e ikz E 1 iak π 1 ak [ ] E 3 aku 1 ak+e 3 aku ak+u 3 ak dk [ ] E 3 aku 1 ak+u ak+e 3 aku 3 ak dk...60