# Boundary value problems for linear and nonlinear wave equations

Contents Ac knowledgments iv Abstract v 1 IBVPs for linear evolution equations 1 1.1 Lax pair formulation of linear evolution equations.......................1 1.2 IVPs for linear evolution equations via spectral methods...................3 1.3 The linear Schrödinger equation................................4 1.3.1 IBVPs via Fourier transforms.............................4 1.3.2 IBVP via spectral method:Dirichlet boundary condition...............5 1.3.3 Other IBVPs:Neumann and Robin boundary conditions...............9 1.4 The linear Korteweg-deVries equation.............................12 1.4.1 IBVPs:Dirichlet and Neumann boundary conditions.................12 1.4.2 Other IBVPs:Robin boundary conditions.......................16 1.5 IBVPs for general linear evolution equations..........................18 2 IBVPs for discrete linear evolution equations 22 2.1 IVP for discrete linear evolution equations via Fourier methods................23 2.2 Discrete linear Schrödinger equation:IBVP via Fourier methods...............24 2.3 A Lax pair for the discrete linear Schrödinger equation....................25 2.4 IVP for the discrete linear Schrödinger equation via spectral analysis of the Lax pair.....26 2.5 IBVP for the discrete linear Schrödinger equation via spectral analysis of the Lax pair....27 2.6 IBVP with Robin-type boundary conditions..........................36 2.7 Lax pair for discrete linear evolution equations........................38 3 IVP and IBVPs for the nonlinear Schrödinger equation 40 viii

3.1 IVP for the nonlinear Schrödinger equation..........................40 3.2 Soliton solutions........................................46 3.3 IBVP for the nonlinear Schrödinger equation.........................49 4 IVP and IBVPs for the Ablowitz-Ladik system 58 4.1 The Ablowitz-Ladik systemon the integers..........................59 4.2 The Ablowitz-Ladik systemon the naturals..........................68 4.3 Elimination of the unknown boundary datum.........................78 4.4 Linearizable boundary conditions................................80 4.5 Discrete spectrumand soliton solutions............................82 4.6 Asymptotics of the eigenfunctions...............................85 4.7 Independence of the solution on T...............................87 4.8 Concluding remarks.......................................89 5 Soliton behavior in IBVPs 91 5.1 IBVPs for the linear Schrödinger equation via the method of images.............91 5.2 IBVPs for the nonlinear Schrödinger equation via the method of images...........94 5.3 Characterization of the discrete eigenvalues..........................98 5.4 Relations between discrete eigenvalues and norming constants................101 5.5 Soliton reﬂection........................................103 5.6 Reﬂection-induced shift.....................................107 5.7 Concluding remarks.......................................110 Conclusions 112 Appendix:Notation and frequently used formulae 115 Bibliography 118 ix

Chapter 1 IBVPs for linear evolution equations A general approach to solve initial-boundary value problems (IBVPs) for linear and nonlinear evolution equations was developed by A.S.Fokas (e.g.,see [40,41,43-45,48,50,55,56,59] and references therein), sometimes called the Fokas method.Here we present the method to solve IBVPs for linear evolution equa- tions (LEEs) on the half line.We ﬁrst discuss the formulation of Lax pair for LEEs in section 1.1 and solve IVPs for LEEs via spectral method in section 1.2.We then show in detail how the Fokas method works for the linear Schrödinger (LS) equation in section 1.3 and the linear Korteweg-deVries equation (LKdV) in section 1.4.In section 1.5 we brieﬂy present the general method to solve IBVPs for general LEEs. 1.1 Lax pair formulation of linear evolution equations Let us consider the initial value problem(IVP) for a general LEE,namely, iq t −ω(−i∂ x )q =0,−∞< x < ∞,t > 0,(1.1.1) where an initial condition q(x,0) is given,ω(k) is a polynomial in k,called the linear dispersion relation,and the subscripts x and t denote partial derivatives.To avoid technical complications,we assume that q(x,0) belongs to the Schwartz class,which we denote by S(R).The IVP for (1.1.1) is well posed only if Imω(k) is bounded fromabove ∀k ∈ R,and is solvable by the Fourier transform(FT) pair given by ˆq(k,t) = ∞

−∞ e −ikx q(x,t)dx,(1.1.2a) and its inverse Fourier transform 1

q(x,t) = 1 2π ∞

−∞ e ik x ˆq(k,t)dk.(1.1.2b) The Fourier transform is summarized as follows.First,the given initial condition q(x,0) is transformed to the spectral function ˆq(k,0),which is also known as the scattering datum.The Fourier transform of (1.1.1) is then used to determine the time evolution of the scattering datumgiven by i ∂ˆq ∂t −ω(k)ˆq =0 ,(1.1.3) which can be solved by ˆq(k,t) =e −iω(k)t ˆq(k,0).Thus the solution of the IVP in Ehrenpreis form[41,45] is given by q(x,t) = 1 2π ∞

−∞ e i(k x−ω(k)t) ˆq(k,0)dk.(1.1.4) As we will show,an alternative method to solve IVPs for LEEs is use of Lax pair for LEEs (cf.[41,43,50]). The equation (1.1.1) can be considered as a compatibility condition of an overdetermined linear system, called Lax pair: µ x −ikµ =q,µ t +iω(k)µ =H,(1.1.5) where the complex number k is called the scattering (or spectral) parameter,q is called the potential of the Lax pair and H will be given in (1.1.8).The x-part and the t-part of the Lax pair are referred to as the scattering problem and the time-dependence,respectively.With compatibility condition µ xt =µ tx ,one can verify that the Lax pair are equivalent to the fact that the potential q satisﬁes the given LEE.The Lax pair formulation was introduced to solve nonlinear PDEs ﬁrst,however the formulation of Lax pair for any LEEs was discovered later (for example,see [40]) and used widely to solve both IVPs and IBVPs for LEEs,as well as integrable nonlinear evolution equations (NLEEs). To derive H(x,t,k) in the Lax pair,ﬁrst note that the compatibility condition µ xt = µ tx with the Lax pair implies iq t −ω(k)q−kH−iH x =0.(1.1.6) From(1.1.1),one then now have H =−[k +i∂ x ] −1 [ω(k) −ω(−i∂ x )]q.(1.1.7) 2

Since ω(k) −ω(s) is always divisible by (k −s),we can write H as H =− ω(k) −ω(s) k −s

s=−i∂ x q.(1.1.8) 1.2 IVPs for linear evolution equations via spectral methods Now we present the solution of the IVP for (1.1.1) by using the spectral analysis of the Lax pair,which is known as linear analogue of the inverse scattering (spectral) transform for integrable NLEEs.It is con- venient to introduce the modiﬁed eigenfunction µ(x,t,k) =e ikx−ω(k)t ψ(x,t,k),so that ψ(x,t,k) satisﬁes the simpliﬁed Lax pair ψ x =e −i(kx−ω(k)t) q,ψ t =e −i(kx−ω(k)t) H.(1.2.1) It is then easy to obtain the solutions of (1.1.5) which decay as x →∓∞respectively as: µ + (x,t,k) = x

−∞ e ik(x−x

) q(x

,t)dx

,(1.2.2a) µ − (x,t,k) =− ∞

x e ik(x−x

) q(x

,t)dx

.(1.2.2b) Note that the Fourier transform is only deﬁned for k ∈ R,however,µ ± can be extended to the complex k-plane.That is,µ ± (x,t,k) are analytic for Imk > < 0,since x −x

> 0 in (1.2.2a) and x −x

< 0 in (1.2.2b), respectively.Also,on Imk =0 one has µ + (x,t,k) −µ − (x,t,k) =e ikx ˆq(k,t),(1.2.3) where ˆq(k,t) is the Fourier transform of q(x,t) given in (1.1.2a).Note however that the diﬀerence µ + −µ − solves the homogeneous version of (1.1.5) and hence it depends on x and t only through the exponential phase ikx−iω(k)t.Therefore µ + (x,t,k) −µ − (x,t,k) =e i(kx−ω(k)t) s(k),for some appropriate function s(k). Evaluating the right-hand side (RHS) of (1.2.3) at (x,t) =(0,0) we then obtain s(k) = ˆq(k,0). Also,integration by parts implies µ ± (x,t,k) =O(1/k) as k →∞in their respective half planes.Thus (1.2.3) deﬁnes a scalar Riemann-Hilbert problem (RHP) [68] which is trivially solved via the standard Cauchy projectors P ± P ± ( f )(k) = 1 2πi ∞

−∞ f (k

) k

−(k ±0i) dk

,(1.2.4) 3

with use of Plemelj formulae [4]:if f ± is analytic for Imk > < 0,then P ± ( f ± )(k) =±f ± (k),P ± ( f ∓ ) =0.(1.2.5) Hence we can write the solution of the RHP as µ(x,t,k) = 1 2πi ∞

−∞ e i(k

x−ω(k)t) ˆq(k

, 0) k

−k dk

.(1.2.6) Inserting (1.2.6) into the x-part of the Lax pair (1.1.5) then yields (1.1.4) as the solution of the IVP. 1.3 The linear Schrödinger equation We consider IBVPs for the LS equation on the half line: iq t +q xx =0,0 < x < ∞,0 < t < T,(1.3.1) where q(x,0) is given and one of boundary conditions (BCs) at x =0 is imposed.For the later reference, here we describe BCs that will be considered throughout this thesis. • Dirichlet boundary condition:q(0,t) =h(t); • Neumann boundary condition:q x (0,t) =h(t); • Robin boundary condition:aq(0,t) +q x (0,t) =h(t). Note that as a →∞,or when a =0,Robin BC leads to Dirchlet BC or Neumann BC,respectively. 1.3.1 IBVPs via Fourier transforms We brieﬂy review the Fourier transform for the LS equation on the half line with Dirichlet BCs.As we mentioned in section 1.1,the IVP for the LS equation is trivially solved by use of the Fourier transformpair (1.1.2): q(x,t) = 1 2π ∞

−∞ e i(k x−k 2 t) ˆq(k,0)dk.(1.3.2) 4

One can also solve IBVP for the LS equation (1.3.1) by employing the sine transformpair ˆq (s) (k,t) = ∞

0 sin(kx)q(x,t)dx,(1.3.3a) q(x,t) = 2 π ∞

0 sin(k x)ˆq(k,t)dk.(1.3.3b) Use of (1.3.3) into (1.3.1) implies i ∂ˆq (s) ∂t −k 2 ˆq (s) =−k q(0,t),(1.3.4a) which is solved by ˆq (s) (k,t) =e −ik 2 t ˆq (s) (k,0) +ik t

0 e −ik 2 (t−t

) q(0,t

)dt

.(1.3.4b) Thus the solution of the IBVP is q(x,t) = 2 π ∞

0 e −ik 2 t sin(k x) ˆq (s) (k,0)dk + 1 π ∞

0 e −ik 2 t sin(k x)ˆg(k,t)dk,(1.3.5a) where ˆg(k,t) =2ik t

0 e ik 2 t

q(0,t

)dt

.(1.3.5b) The reason why the sine transform approach works of course is that it eliminates the unknown boundary datum q x (0,t) from (1.3.4a).A similar approach using the cosine transform works for the IBVP with Neumann BCs.A more general transform is needed,however,for Robin boundary conditions.Moreover, the particular transform to be used depends on the coeﬃcients of the assigned boundary conditions,which is clearly undesirable.The sine-cosine transform approach only works for some linear evolution equations, and there are many examples where no such transforms exist,as discussed in [59]. 1.3.2 IBVP via spectral method:Dirichlet boundary condition Let us now consider the solution of the IBVP on the half line via the spectral method based on the analysis of the Lax pair.Unlike the spectral method that we mentioned in section 1.1,we treat simultaneously both parts of the Lax pair when we deﬁne eigenfunctions of the scattering problem. 5

An algorithmic method to obtain the Lax pair of LEEs was given in section 1.1.However,one can also obtain a Lax pair for the LS equation via the linear limit of the Lax pair of the nonlinear Schrödinger equation (NLS) [8,90],namely Φ x −ikσ 3 Φ=QΦ,(1.3.6a) Φ t +2ik 2 σ 3 Φ=HΦ,(1.3.6b) where Φ(x,t,k) is a 2-component vector and the 2 ×2 matrices Q,H and σ 3 are deﬁned in (A.3).To obtain the linear limit,let Q =O(ε) and take Φ(x,t,k) =v(x,t,k).Then,to leading order it is v(x,t,k) = e i(kx−2k 2 t) ˆσ 3 v o ,where v o =(v 1,o ,v 2,o ) t is an arbitrary constant vector.Choosing v 2,o =1 and substituting into the RHS of (1.3.6) then yields the following equations for µ(x,t,k) =e i(kx−2k 2 t) v 1 (x,t,k) up to O(ε 2 ) terms: µ x −ik

µ =q,µ t +ik 2 µ =iq x −k

q,(1.3.7) where k

=2k.One can nowverify that enforcing the compatibility (namely,µ xt =µ tx ) of the Lax pair (1.3.7) yields (1.3.1).Hereafter,for convenience,we will omit the primes.Note that the Lax pair (1.3.7) will be used to solve both IVPs and IBVPs. We ﬁrst solve (1.3.1) with Dirchlet BCs,that is,(1.3.1) with 0 < x,0 < t < T,and q(x,0) and q(0,t) given. To do so,we deﬁne simultaneous solutions of both the x-part and the t-part of the Lax pair using the function ψ(x,t,k) deﬁned earlier: µ ( j) (x,t,k) = (x,t)

(x j ,t j ) e ik(x−x

)−ik 2 (t−t

)

q(x

,t

)dx

+

iq x

(x

,t

) −kq(x

,t

)

dt

.(1.3.8) In particular,consider the three eigenfunctions µ ( j) (x,t,k),j = 1,2,3,deﬁned by the choices (x 1 ,t 1 ) = (0,0),(x 2 ,t 2 ) =(∞,t) and (x 3 ,t 3 ) =(0,T) (cf.ﬁgure 1.1): µ (1) (x,t,k) = x

0 e ik(x−x

) q(x

,t)dx

+ t

0 e ikx−ik 2 (t−t

)

iq x (0,t

) −kq(0,t

)

dt

,(1.3.9a) µ (2) (x,t,k) =− ∞

x e ik(x−x

) q(x

,t)dx

,(1.3.9b) µ (3) (x,t,k) = x

0 e ik(x−x

) q(x

,t)dx

− T

t e ikx−ik 2 (t−t

)

iq x (0,t

) −kq(0,t

)

dt

.(1.3.9c) 6

Figure 1.1:The distinguished points for the eigenfunctions µ (1) ,µ (2) and µ (3) . Note that µ (2) coincides with the eigenfunction in the IVP.As for µ (1) and µ (3) ,they are entire functions of k.It should be clear fromthe exponentials that,for absolutely integrable potentials,these eigenfunctions have the following domains of boundedness: µ (1) :k ∈ ¯ C II ,µ (2) :k ∈ ¯ C III+IV ,µ (3) :k ∈ ¯ C I ,(1.3.10) where C III+IV denotes the lower-half plane.Note that Imk 2 ≥ 0 on the ﬁrst and third quadrant of the complex k-plane.The two jumps on Imk =0 and the jump on Re k =0 ∧Imk ≥0 then deﬁne a scalar RHP which allows us to obtain the solution of the scattering problemas well as reconstruct the potential in terms of the scattering data.As before,since the µ ( j) (x,t,k) solve both parts of the Lax pair (1.3.7),however, any diﬀerence µ ( j) −µ ( j

) solves the homogeneous version of (1.3.7).That is,µ ( j) (x,t,k) −µ ( j

) (x,t,k) = e ikx−ik 2 t s j j

(k),where s j j

(k) is independent of x and t.Evaluating (1.3.9) at (x,t) = (0,0) we can then write the jumps as (of course,any two of these jumps uniquely determine the third one): µ (1) (x,t,k) −µ (3) (x,t,k) =e ikx−ik 2 t ˆ F(k,T),Re k =0 ∧ Imk ≥ 0,(1.3.11a) µ (1) (x,t,k) −µ (2) (x,t,k) =e ikx−ik 2 t ˆq(k,0),Imk =0 ∧ Re k ≤ 0,(1.3.11b) µ (3) (x,t,k) −µ (2) (x,t,k) =e ikx−ik 2 t

ˆq(k,0) − ˆ F(k,T)

,Imk =0 ∧ Re k ≥ 0.(1.3.11c) Here ˆ F(k,t) =i ˆ f 1 (k,t) −k ˆ f 0 (k,t),and with ˆq(k,t) = ∞

0 e −ikx q(x,t)dx, ˆ f n (k,t) = t

0 e ik 2 t

∂ n x q(x,t

)| x=0 dt

.(1.3.12) Unlike the usual FT,the spectral functions (1.3.12) enjoy some nice properties.Namely,the one-sided Fourier transform ˆq(k,t) is analytic for Imk < 0 and bounded for Imk ≤ 0,while the transforms ˆ f n (k,t) of 7

the boundary data are entire,and are bounded for Imk 2 ≥ 0.Moreover,ˆq(k,t) →0 as k →∞with Imk < 0, and ˆ f n (z,t) →0 as k →∞with Imk 2 < 0.The solution of the RHP deﬁned by (1.3.11) is thus given by µ(x,t,k) = 1 2πi ∞

−∞ e ik

x−ik 2 t ˆq(k

, 0) k

−k dk

− 1 2πi

∂C I e ik

x−ik 2 t ˆ F(k

,T) k

−k dk

, where the orientation of ∂C I = (i∞,0] ∪[0,∞) is chosen to leave C I to its left,as usual.Inserting (1.3.2) into the x-part of the Lax pair (1.3.7) then yields the reconstruction formula: q(x,t) = 1 2π ∞

−∞ e ik x−ik 2 t ˆq(k,0)dk − 1 2π

∂C I e ik x−ik 2 t ˆ F(k,T)dk.(1.3.13) Up to this point the solution still depends on the unknown boundary data q x (0,t) via ˆ f 1 (k,t).Our last task is then to obtain the Dirichlet-to-Neumann map,namely to determine the unknown boundary datum in terms of the given one.To this end we note that enforcing the compatibility of (1.3.7) by integrating from (0,0) to (0,T) to (∞,T) to (∞,0) and back to (0,0) yields the global relation T

0 e ik 2 t

iq x (0,t) −kq(0,t)

dt +e ik 2 T ∞

0 e −ikx q(x,T)dx = ∞

0 e −ikx q(x,0)dx,(1.3.14) which holds for Imk ≤ 0 ∧ Imk 2 ≤ 0,i.e.,k ∈ C III .In terms of the spectral data: i ˆ f 1 (k,T) −k ˆ f 0 (k,T) +e ik 2 T ˆq(k,T) = ˆq(k,0),∀k ∈ ¯ C III .(1.3.15a) Using the the transformation k →−k,which leaves ˆ f n (k,t) invariant,from(1.3.15a) we obtain i ˆ f 1 (k,T) +k ˆ f 0 (k,T) +e ik 2 T ˆq(−k,T) = ˆq(−k,0) ∀k ∈ ¯ C I .(1.3.15b) We then solve for ˆ f 1 (k,T) and insert the result in (1.3.13).Note that the third term in the RHS of (1.3.15b) yields a zero contribution to the solution.Thus,the solution of the IBVP is given by q(x,t) = 1 2π ∞

−∞ e ik x−ik 2 t ˆq(k,0)dk − 1 2π

∂C I e ik x−ik 2 t

ˆq(−k,0) −2k ˆ f 0 (k,T)

dk. Note that (1.3.2) expresses the solution of the IBVP in Ehrenpreis form.Indeed,one can replace ˆ f 0 (k,t) with ˆ f 0 (k,T),for any T > t.Since ˆ f 0 (k,T) can be written as ˆ f 0 (k,T) = t

0 e ik 2 t

q(0,t

)dt

+ T

t e ik 2 t

q(0,t

)dt

,(1.3.16) 8

the additional portion with e ikx−ik 2 t is e ikx T

t e ik 2 (t

−t) q(0,t

)dt

.(1.3.17) This function is entire and bounded on ¯ C I+III decaying exponentially as k →∞,since t

−t > 0.Therefore the additional portion in the reconstruction formula vanishes and hence it yields zero contribution to the solution.Note also that the second integrand in (1.3.2) is analytic and bounded for Imk ≥ 0 ∧ Imk 2 ≤ 0. Thus,we can deform the integration contour on the second integral to be along the real k-axis,thereby obtaining a modiﬁed reconstruction formula: q(x,t) = 1 2π ∞

−∞ e ik x−ik 2 t

ˆq(k,0) − ˆq(−k,0) +2k ˆ f 0 (k,t)

dk.(1.3.18) Fromhere one can then recover the sine transformsolution given in (1.3.5a).Indeed,the one-sided Fourier transform in (1.3.12) is ˆq(k,t) = ˆq (c) (k,t) −i ˆq (s) (k,t),where ˆq (s) (k,t) and ˆq (c) (k,t) are the sine and co- sine transforms.Moreover,ˆg(k,t) = ik ˆ f 0 (k,t).Unlike the sine transform approach,however,the method described here can be immediately applied to the solution of IBVPs with more complicated boundary con- ditions,by simply solving the global relations (1.3.15a) and (1.3.15b) for any unknown boundary data,as we show next. 1.3.3 Other IBVPs:Neumann and Robin boundary conditions In this section we will derive the solutions of the IBVPs for the LS equation with other BCs.Fromthe global relation (1.3.15b) and the deﬁnition of ˆ F(k,T),we obtain k ˆ f 0 (k,T) +i ˆ f 1 (k,T) = ˆq(−k,0) −e ik 2 T ˆq(−k,T),(1.3.19a) −k ˆ f 0 (k,T) +i ˆ f 1 (k,T) = ˆ F(k,T).(1.3.19b) Solving (1.3.19) for ˆ f 0 (k,T) and ˆ f 1 (k,T) gives −2k ˆ f 0 (k,T) = ˆ F(k,T) − ˆq(−k,0) +e ik 2 T ˆq(−k,T),(1.3.20a) 2i ˆ f 1 (k,T) = ˆ F(k,T) +ˆq(−k,T) −e ik 2 T ˆq(−k,T).(1.3.20b) 9

Thus, if we consider Neumann BCs,that is,q x (0,t) is given,we can obtain ˆ F(k,T) in the reconstruction formula as ˆ F(k,T) =2i ˆ f 1 (k,T) − ˆq(−k,0),(1.3.21) which yields the solution of IBVP given by q(x,t) = 1 2π ∞

−∞ e ik x−ik 2 t ˆq(k,0)dk − 1 2π

∂C I e ik x−ik 2 t

2i ˆ f 1 (k,t) −q(−k,0)

dk.(1.3.22) Note that the IBVP for the LS equation with Neumann BCs can be solved via the Fourier cosine transform, however,in general,the Fourier transformcannot be applied to solve the IBVP for the LS with complicated BCs,as we discuss next. Let us consider Robin BCs,namely, aq(0,t) +q x (0,t) =h(t),a 0.(1.3.23) The transformof (1.3.23) is a ˆ f 0 (k,T) + ˆ f 1 (k,T) = ˆ h(k,T),(1.3.24) where ˆ h(k,t) = t

0 e ik 2 t

h(t

)dt

. From (−ia) ×(1.3.20a) +k ×(1.3.20b),it follows that (k −ia) ˆ F(k,T) =2ik ˆ h(k,T) −(k +ia)ˆq(−k,0) +e ik 2 T (k +ia)ˆq(−k,T).(1.3.25) Thus we can obtain ˆ F(k,T) in the reconstruction formula ˆ F(k,T) = ˆ G(k,T) k −ia +e ik 2 T k +ia k −ia ˆq(−k,T),k ia,(1.3.26) where ˆ G(k,T) =2ik ˆ h(k,T) −(k +ia)ˆq(−k,0).(1.3.27) First we suppose that π/2 < arg(ia) < 2π.In this case,k =ia lies outside ¯ C I ,and ˆ F(k,T) is analytic and bounded for k ∈ ¯ C I .By Jordan’s lemma,the second part of ˆ F(k,T) in (1.3.26) does not contribute to the 10

solution. Therefore,for 0 < arg(a) < 3π/2,the solution of the IBVP is given by q(x,t) = 1 2π ∞

−∞ e ik x−ik 2 t ˆq(k,0)dk − 1 2π

∂C I e ik x−ik 2 t ˆ G(k,t) k −ia dk.(1.3.28) Ne xt suppose that −π/2 ≤ arg(a) ≤ 0.Now it appears that ˆ F(k,T) deﬁned by (1.3.26) has a simple pole at k =ia which lies in ¯ C I .However,since ˆ G(k 0 ,T) +2k 0 e ik 2 0 T ˆq(−k 0 ,T) =0 for k 0 =ia by (1.3.25), ˆ F(k,T) can be redeﬁned to be analytic and bounded in ¯ C I .Let us consider the integral 1 2π

∂C I e ik x−ik 2 t ˆ G(k,T) k −ia +e ik 2 T k +ia k −ia ˆq(−k,T) dk.(1.3.29) Using analyticity,we can deformthe contour to pass to the right of point k =ia.We denote any such contour as ∂ ˜ C I .The second term vanishes by analyticity in ˜ C I .Taking into account the residue,we can express the integral in (1.3.29) as 1 2π

∂ ˜ C I e ik x−ik 2 t ˆ G(k,T) k −ia dk = 1 2π

∂C I e ik x−ik 2 t ˆ G(k,T) k −ia dk −iν a e ik 0 x−ik 2 0 t ˆ G(k 0 ,t),(1.3.30) where ν a =1 for − π 2 < ar g(a) < 0,and ν a = 1 2 for ar g(a) =0,− π 2 , and where the integral along ∂C I is to be taken as a principal value when arg(a) =0,− π 2 . Thus the solution of the IBVP when − π 2 ≤ ar g(a) ≤ 0 is given by q(x,t) = 1 2π ∞