The Linearized Classical Boussinesq System on the Half-Line
TTHE LINEARIZED CLASSICAL BOUSSINESQ SYSTEM ON THE HALF-LINE
C. M. JOHNSTON, CLARENCE T. GARTMAN & DIONYSSIOS MANTZAVINOS ∗ DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, LAWRENCE, KS 66045
Abstract.
The linearization of the classical Boussinesq system is solved explicitly in the case ofnonzero boundary conditions on the half-line. The analysis relies on the unified transform methodof Fokas and is performed in two different frameworks: (i) by exploiting the recently introducedextension of Fokas’s method to systems of equations; (ii) by expressing the linearized classicalBoussinesq system as a single, higher-order equation which is then solved via the usual version ofthe unified transform. The resulting formula provides a novel representation for the solution of thelinearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergenceat the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system aswell as the prescribed initial and boundary data via a direct calculation. Introduction
The classical Boussinesq system v t + u x + ( vu ) x = 0 ,u t + v x + uu x − u xxt = 0 , (1.1)with v = v ( x, t ), u = u ( x, t ), is a central model in fluid dynamics that captures the propagationof small-amplitude, weakly nonlinear shallow waves on the free surface of an ideal irrotationalfluid under the effect of gravity. The system (1.1) was first derived by Boussinesq [B1, B2] asan asymptotic approximation (in the regime specified above) of the celebrated Euler equations ofhydrodynamics. As such, it has been the subject of several works in the literature [S, Am, BCS1,BCS2, AL, Ad, AD, L, MID, MTZ, LW] through various analytical as well as numerical techniques.The classical Boussinesq system (1.1) is nonlinear and hence challenging to study analytically.At the same time, the system is also dispersive. Hence, the existence and uniqueness of its solution(upon the prescription of suitable data) can be established via the powerful contraction mappingtechnique. A central role in the implementation of that technique is played by the solution map ofthe linear counterpart of the problem under consideration. More precisely, the (explicit) solutionformula of the forced linear problem inspires an implicit mapping for the solution of the nonlinearproblem; this mapping is then shown to be a contraction in an appropriate function space, therebyimplying a unique solution (namely, the unique fixed point of the contraction) for the nonlinearproblem. Therefore, the derivation of a linear solution formula which is effective for the purpose offunction estimates is crucial in the investigation of the solvability of the nonlinear system (1.1).
Date : September 20, 2020.
Revised : November 8, 2020. ∗ Corresponding author : [email protected]
Mathematics Subject Classification.
Primary: 35G46, 35G16. Secondary: 35G61, 35G31.
Key words and phrases. classical Boussinesq system, half-line, initial-boundary value problem, nonzero boundaryconditions, unified transform method of Fokas. More precisely, the system (1.1) was actually derived by Peregrine [P]; Boussinesq had derived a slightly modifiedversion of that system which is not well-posed. a r X i v : . [ m a t h . A P ] D ec The linearized classical Boussinesq system on the half-line
Moreover, there exists a particular aspect of the classical Boussinesq system (1.1) — and ofnonlinear dispersive systems in general — which has not been explored much in the literature,namely, their formulation as initial-boundary value problems (IBVPs) on domains that involve aboundary, as opposed to the fully unbounded domain associated with the initial value problem.One of the main reasons behind the slow progress in the rigorous analysis of nonlinear dispersiveIBVPs when compared to their associated initial value problems has been the absence of the Fouriertransform in the IBVP setting. Indeed, we recall that in the case of the initial value problem thelinear solution formulae, which are essential for obtaining the basic estimates used in the contractionmapping technique, are easily derived by simply applying a Fourier transform in the spatial variable.Nevertheless, once a boundary is introduced in the problem (e.g. in the case of the half-line { x > } )the spatial Fourier transform is no longer available and, even more, no classical transform existsthat can produce linear solution formulae which are effective for the purpose of estimates.Motivated by the above, in this work we consider the linearization about zero of the classicalBoussinesq system (1.1), i.e. we set v ( x, t ) = εr ( x, t ) and u ( x, t ) = εq ( x, t ) with 0 < ε (cid:28) r t + q x = 0 , q t + r x − q xxt = 0 , x > , t > , (1.2a) r ( x,
0) = r ( x ) , q ( x,
0) = q ( x ) , x (cid:62) , (1.2b) q (0 , t ) = g ( t ) , t (cid:62) , (1.2c)where, for the purpose of this work, we assume initial and boundary data with sufficient smoothnessand decay at infinity (e.g. in the Schwartz class). In particular, we assume compatibility of theinitial and boundary data at the origin, namely r (0) = g (0). We remark that the above IBVP issupplemented with just one boundary condition for q and no boundary condition for r . Althoughit is not a priori clear that this choice of data is admissible (and sufficient), our analysis will revealthat this is indeed the case.As noted earlier, no classical spatial transform can produce a solution formula for IBVP (1.2)which is appropriate for analyzing the corresponding nonlinear IBVP. We emphasize that this isnot a pathogeny of the linearized classical Boussinesq system, but rather a challenge which ispresent across the whole spectrum of linear dispersive IBVPs with nonzero boundary conditions.The resolution to this important obstacle in the study of linear (and, eventually, nonlinear) disper-sive IBVPs was provided in 1997 by Fokas [F1], who introduced the now well-established unifiedtransform method (UTM), also known in the literature as the Fokas method. This novel methodessentially provides the direct analogue of the Fourier transform in the IBVP framework. It relieson exploiting certain symmetries of the dispersion relation of the problem together with the abilityto deform certain paths of integration to appropriate contours in the complex spectral plane. Overthe last twenty years, UTM has been widely used for a plethora of linear as well as nonlinearevolution and elliptic equations, formulated on various domains in one or higher dimensions, andwith a broad range of (admissible) boundary conditions — see, for example, the research articles[FK, FI, AnF, FF, FFSS, SSF, FL, HM1, DSS, CFF, KO], the books [F2, FP2] and the reviewarticles [FS, DTV].Fairly recently, Deconinck, Guo, Shlizerman and Vasan extended the linear component of UTMto systems of equations [DGSV]. In this work, we shall exploit that recent progress in order to There exist works in the literature that study dispersive IBVPs in the case of rough data. This task, however,lies beyond the scope of the present article. . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 3 derive the novel, explicit representation (3.9) for the solution of the linearized classical BoussinesqIBVP (1.2). Our derivation will be done under the assumption of existence of solution. Nevertheless,taking advantage of one of the key features of UTM, namely the uniform convergence of its solutionformulae at the boundary, we shall explicitly demonstrate (via a direct calculation) that our novelformula does indeed satisfy IBVP (1.2) and is, in fact, a classical solution. Furthermore, we shall alsoprovide an alternative way of solving IBVP (1.2) by converting the linearized classical Boussinesqsystem into a single equation. A slight downside of the latter approach is that the resulting singleequation involves a second-order time derivative, which makes the analysis somewhat more tedious.At the same time, the latter approach illustrates that UTM as a method is equally effective in boththe single equation and the system frameworks.It should be noted that IBVP (1.2) has previously been considered by Fokas and Pelloni in[FP1]. However, the method employed in that paper was the nonlinear component of UTM, whichrelies on expressing the linearized classical Boussinesq system as a Lax pair and then using ideasinspired by the inverse scattering transform in order to associate the solution of IBVP (1.2) to thatof a (scalar) Riemann-Hilbert problem. This Riemann-Hilbert problem is then solved explicitlywith the help of Plemelj’s formulae to yield the solution to problem (1.2). Nevertheless, the highlytechnical aspects of the approach of [FP1] make it less accessible to the broader applied sciencescommunity. Furthermore, the solution representation produced in [FP1] involves certain principalvalue integrals, which arise as byproducts of the Plemelj formulae. This feature does not seemconvenient regarding (i) the derivation of linear estimates for the contraction mapping analysis ofthe nonlinear system (1.1), and (ii) numerical considerations. On the contrary, the new solutionrepresentation derived in the present work does not involve principal value integrals and, moreimportantly, relies solely on the linear component of UTM, which only requires knowledge of theFourier transform and of Cauchy’s theorem from complex analysis.
Organization of the article.
In Section 2, starting from the linearized classical Boussinesq IBVP(1.2) we derive an important spectral identity which plays a central role in UTM and is known asthe global relation. In Section 3, we combine the global relation with appropriate deformations inthe complex spectral plane in order to obtain the explicit solution formula of problem (1.2). Then,in Section 4, we revisit the problem by converting the linearized classical Boussinesq system into asingle equation which we then solve via the standard version of UTM. This offers a first, indirectway of corroborating our novel solution formula. The direct, explicit verification of our formula isthen presented in detail in Section 5. Finally, some concluding remarks are provided in Section 6.2.
Derivation of the global relation
We define the half-line Fourier transform byˆ f ( k ) = (cid:90) ∞ x =0 e − ikx f ( x ) dx, Im( k ) (cid:54) , (2.1a)with inverse f ( x ) = 12 π (cid:90) k ∈ R e ikx ˆ f ( k ) dk, x (cid:62) . (2.1b)We note that, unlike the standard Fourier transform, which is only valid on the real line, the half-line Fourier transform (2.1a) is valid on the closure of the lower half of the complex k -plane due tothe fact that x (cid:62)
0. Then, applying (2.1a) to the linearized classical Boussinesq system (1.2a) we
The linearized classical Boussinesq system on the half-line obtain ∂ t ˆ r ( k, t ) = − ik ˆ q ( k, t ) + g ( t ) , (cid:0) k (cid:1) ∂ t ˆ q ( k, t ) = − ik ˆ r ( k, t ) + h ( t ) − g (cid:48) ( t ) − ikg (cid:48) ( t ) , (2.2)where (cid:48) denotes differentiation with respect to t and we have introduced the notation g j ( t ) = ∂ jx q (0 , t ) , h j ( t ) = ∂ jx r (0 , t ) , j ∈ N ∪ { } . (2.3)For k (cid:54) = ± i , we may express the system of ordinary differential equations (2.2) in matrix form as ∂ t ˆ q ( k, t ) = A ( k ) ˆ q ( k, t ) + g ( k, t ) , (2.4)where ˆ q ( k, t ) = (cid:18) ˆ r ( k, t )ˆ q ( k, t ) (cid:19) , A ( k ) = − ik (cid:32) k (cid:33) , g ( k, t ) = (cid:32) g ( t ) k [ h ( t ) − g (cid:48) ( t ) − ikg (cid:48) ( t )] (cid:33) . (2.5)Next, recalling the definition of the matrix exponential e A := (cid:80) ∞ j =0 A j j ! , we integrate (2.4) to obtainthe following spectral identity which in the UTM terminology is known as the global relation (sinceit only involves integrals of the vector q = ( r, q ) T and its initial and boundary values): (cid:98) q ( k, t ) = e A ( k ) t (cid:98) q ( k ) + (cid:90) tτ =0 e A ( k )( t − τ ) g ( k, τ ) dτ, Im( k ) (cid:54) , k (cid:54) = − i. (2.6)We emphasize that the global relation is valid for all Im( k ) (cid:54) k (cid:54) = − i because the half-lineFourier transform (2.1a) makes sense for all Im( k ) (cid:54) A as A = P DP − (2.7)with P = (cid:32) − µ µ (cid:33) , D = (cid:18) iω − iω (cid:19) , ω = ω ( k ) := kµ ( k ) , (2.8)where the complex square root µ ( k ) := (cid:0) k (cid:1) (2.9)is made single-valued by taking a branch along the segment B := i [ − , (cid:0) k (cid:1) = (cid:112) | k | e i ( θ + θ − π ) / , k / ∈ B , (2.10)with the angles θ , θ ∈ [0 , π ) as shown in Figure 2.1 and for k ∈ B we identify µ ( k ) by its limitfrom the right.The above diagonalization allows us to write e At explicitly as a matrix: e At = P e Dt P − = 12 (cid:32) e iωt + e − iωt − µ (cid:0) e iωt − e − iωt (cid:1) − µ (cid:0) e iωt − e − iωt (cid:1) e iωt + e − iωt (cid:33) . (2.11) . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 5 θ θ k i − i B Figure 2.1.
The definition (2.10) of the complex square root µ ( k ) = (cid:0) k (cid:1) asa single-valued function by taking a branch cut along the segment B = i [ − , k ) (cid:54) k (cid:54) = − i the global relation (2.6) can be expressed in component form asˆ r ( k, t ) = 12 (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) (2.12a)+ 12 µ ( k ) (cid:26) − (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) andˆ q ( k, t ) = 12 (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) (2.12b)+ 12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) − µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) − (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) , where we have introduced the notation (cid:101) f ( ω, t ) = (cid:90) tτ =0 e − iωτ f ( τ ) dτ. (2.13)3. Elimination of the unknown boundary values
The global relations (2.12) involve three boundary values, one for r and two for q . As usual inthe context of UTM, escaping to the complex k -plane by means of Cauchy’s theorem will allow usto eliminate two of those boundary values and thereby derive an effective solution representation,in the sense that it will only involve the boundary value q (0 , t ) = g ( t ) prescribed as a datum inIBVP (1.2). This elimination procedure illustrates the ability of UTM to indicate which boundaryvalues are admissible as data for a well-posed problem.We begin by observing that the first of equations (1.2) evaluated at x = 0 implies g ( t ) = − h (cid:48) ( t ) . (3.1)The above formal evaluation is done (as noted earlier) under the assumption of existence of a smoothsolution; it can be verified a posteriori by direct evaluation of formula (3.9) using the methods of The linearized classical Boussinesq system on the half-line
Section 5. Hence, the boundary value g can be eliminated from the global relations (2.12), whichnow readˆ r ( k, t ) = 12 (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) (3.2a)+ 12 µ ( k ) (cid:26) − (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) − (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) andˆ q ( k, t ) = 12 (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) (3.2b)+ 12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) − µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) . Recall that the global relations (3.2) are valid for Im( k ) (cid:54) k (cid:54) = − i . Thus, using theseexpressions for k ∈ R in the Fourier inversion (2.1b) yields the following integral representationsfor the two components of system (1.2): r ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈ R e ikx µ ( k ) (cid:26) − (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) − (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk (3.3a)and q ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) dk + 12 π (cid:90) k ∈ R e ikx
12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) (cid:27) dk − µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (3.3b) Remark 3.1 (Crossing the branch cut) . Although ω inherits the branch cut B from µ , the in-tegrands involved in (3.3) are entire in k . Indeed, denoting by µ + and µ − the limits of µ as k approaches B from the left and from the right respectively, according to (2.10) we have µ + = − µ − ,i.e. µ changes sign across B . In turn, ω + = − ω − and hence the functions e iωt + e − iωt , e iωt − e − iωt µ , µ (cid:0) e iωt − e − iωt (cid:1) and µ are continuous across B and, therefore, the paths of integration in (3.3) areallowed to cross B .The integral representations (3.3) involve two boundary values, q (0 , t ) = g ( t ) and r (0 , t ) = h ( t ).However, only the first one is prescribed as a boundary condition in problem (1.2), which is thereason why (3.3) is not an effective solution formula. Next, we shall eliminate from (3.3) theunknown boundary value h ( t ) or, more precisely, the transforms (cid:101) h and (cid:102) h (cid:48)(cid:48) , by using a symmetryof the global relations (3.2). More specifically, we note that the transformation k (cid:55)→ − k leaves ω . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 7 invariant. This is because the definition (2.10) implies µ ( − k ) = − µ ( k ) and hence ω ( − k ) = ω ( k ).Thus, under the transformation k (cid:55)→ − k the global relations (3.2) yield the identitiesˆ r ( − k, t ) = 12 (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) (cid:3) (3.4a)+ 12 µ ( k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) andˆ q ( − k, t ) = 12 (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) (cid:21) (3.4b)+ 12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) . Since the global relations (3.2) are valid for Im( k ) (cid:54) k (cid:54) = − i , the identities (3.4) hold forIm( k ) (cid:62) k (cid:54) = i . Thus, they can be readily employed for eliminating the unknown transforms (cid:101) h and (cid:102) h (cid:48)(cid:48) . However, before doing so, it turns out useful to first deform the contours of integration ofthe integrals in (3.3) which involve the boundary values from the real axis to the complex k -plane.In particular, observe that, since x >
0, the entire function e ikx is bounded for Im( k ) (cid:62) e iω ( t − τ ) + e − iω ( t − τ ) and µ (cid:2) e iω ( t − τ ) + e − iω ( t − τ ) (cid:3) , which arise in the integrandsof (3.3) through the relevant transforms of the boundary values, are analytic and bounded for all k (cid:54) = ± i . Hence, using Cauchy’s theorem and Jordan’s lemma from complex analysis (e.g. seeLemma 4.2.2 in [AbF]), we are able to deform the contours of integration of the boundary valuesintegrals in (3.3) from R to the closed contour C encircling i (see Figure 3.1). We emphasize thatJordan’s lemma can be employed because of the uniform (in arg( k )) decay of the quantities k , µ ( k ) and k k as | k | → ∞ . Hence, the integral representations (3.3) can be written as r ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx µ ( k ) (cid:26) − (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) − (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) + µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk (3.5a)and q ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) dk + 12 π (cid:90) k ∈C e ikx
12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) − µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (3.5b) An easy way to see that µ ( − k ) = − µ ( k ) is to observe that the branch cut for µ is such that µ ( k ) (cid:39) k as | k | → ∞ . This deformation is inspired by the one of [VD] for the BBM equation; here, however, we have the additionalcomplication of a dispersion relation with branching.
The linearized classical Boussinesq system on the half-line i − i C k Figure 3.1.
The closed contour C for the integrals involving the boundary values.We are now ready to take advantage of the identities (3.4). Rearranging (3.4a), we have12 µ ( k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) − e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) (cid:27) (3.6a)= ˆ r ( − k, t ) − (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) (cid:3) − µ ( k ) (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) . Moreover, (3.4b) can be written as12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) (cid:27) (3.6b)= ˆ q ( − k, t ) − (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) (cid:21) −
12 (1 + k ) (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) . Since the expressions (3.6) are valid for all Im( k ) (cid:62) k (cid:54) = i , we employ them for k ∈ C andcombine them with the integral representations (3.5) to obtain r ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx (cid:26) − ˆ r ( − k, t ) + (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ikµ ( k ) (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk (3.7a) . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 9 and q ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) dk − π (cid:90) k ∈C e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) (cid:21) dk − π (cid:90) k ∈C e ikx (cid:26) ˆ q ( − k, t ) + 1 µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik k (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (3.7b)Of course, the above expressions still involve unknown quantities, namely the transforms ˆ r ( − k, t )and ˆ q ( − k, t ). However, both of these transforms, as well as the exponential e ikx , are analytic in theupper half-plane. Hence, by Cauchy’s theorem inside the region enclosed by C we conclude that (cid:90) k ∈C e ikx ˆ r ( − k, t ) dk = (cid:90) k ∈C e ikx ˆ q ( − k, t ) dk = 0 (3.8)for all x, t . Therefore, we arrive at the following UTM solution formula for the linearized classicalBoussinesq IBVP (1.2): r ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx (cid:26) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + iω ( k ) (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk (3.9a)and q ( x, t ) = 12 π (cid:90) k ∈ R e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) dk − π (cid:90) k ∈C e ikx (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) (cid:21) dk − π (cid:90) k ∈C e ikx (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik k (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (3.9b)We emphasize that, as explained in Remark 3.1, the integrands in the above formula are analytic functions of k in the respective domains of integration despite the fact that they involve the branchedfunction ω ( k ). Remark 3.2 (Deformation back to R ) . Thanks to analyticity and exponential decay, it it ispossible to deform the contour of integration C in formulae (3.9) back to R . However, the resultingexpression is not uniformly convergent at the boundary x = 0, and hence it is not suitable forexplicitly verifying that the UTM formulae (3.9) indeed satisfy IBVP (1.2). This fact is clearlyillustrated by the computations of Section 5. Remark 3.3 (Other types of boundary conditions) . The elimination procedure performed in thissection works in the same way for other types of admissible boundary data. For example, insteadof the Dirichlet condition (1.2c), one could prescribe the Neumann datum q x (0 , t ) = g ( t ) which,by integrating (3.1) and employing the compatibility condition h (0) = r (0), is equivalent to theprescription of the Dirichlet datum r (0 , t ) = h ( t ). Remark 3.4 (Solution for the forced linear problem) . Thanks to the Duhamel principle, the UTMsolution formula (3.9) can be easily adapted to the case of the forced counterpart of the linearizedclassical Boussinesq IBVP (1.2). The resulting formula can then be employed for studying thewell-posedness of the nonlinear classical Boussinesq system (1.1) on the half-line via contractionmapping techniques. 4.
Revisiting the problem as a single equation
In the previous section, we solved IBVP (1.2) for the linearized classical Boussinesq system usingthe recently introduced extension of UTM to systems [DGSV]. This extension provides a generalmethod that can in principle be applied to any linear system of evolution equations. However,specifically in the case of IBVP (1.2), it is also possible to convert the problem into one involving asingle equation and hence solve it via the standard UTM. Indeed, differentiating the first equationof system (1.2a) with respect to x and the second one with respect to t , we have r xt + q xx = 0 , q tt + r xt − q xxtt = 0 . (4.1)Thus, we can eliminate r and obtain a single equation for q : q tt − q xx − q xxtt = 0 . (4.2)In the remaining of this section, we will employ UTM to solve equation (4.2) in terms of the initialand boundary data of problem (1.2). Once an explicit solution formula for q is obtained, it will bestraightforward to deduce a corresponding formula for r since a simple integration of the first ofequations (4.1) yields r ( x, t ) = r ( x ) − (cid:90) tτ =0 q x ( x, τ ) dτ (4.3)with r ( x ) being the initial datum for r prescribed in problem (1.2).Applying the half-line Fourier transform (2.1a) to equation (4.2), we find ∂ t ˆ q ( k, t ) + k k ˆ q ( k, t ) = −
11 + k (cid:2) g ( t ) + ikg ( t ) + g (cid:48)(cid:48) ( t ) + ikg (cid:48)(cid:48) ( t ) (cid:3) , (4.4)where Im( k ) (cid:54) k (cid:54) = − i and we have denoted, as usual, g ( t ) = q (0 , t ) and g ( t ) = q x (0 , t ).The second-order ordinary differential equation (4.4) can be solved via variation of parameters.In particular, the general solution to the homogeneous counterpart of (4.4) isˆ q h ( k, t ) = c ( k ) e iωt + c ( k ) e − iωt (4.5)with ω defined by (2.8). The constants (with respect to t ) c ( k ) and c ( k ) in the homogeneoussolution (4.5) can be computed by enforcing the initial conditions of problem (1.2). First, note thatthe initial condition q ( x,
0) = q ( x ) readily impliesˆ q ( k,
0) = ˆ q ( k ) . (4.6) Recall that we are working with smooth functions and hence we are allowed to interchange the order of partialderivatives. . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 11
Furthermore, the second of equations (1.2a) evaluated at t = 0 and combined with the initialcondition r ( x,
0) = r ( x ) yields q t ( x, − q xxt ( x,
0) = − r (cid:48) ( x ) (4.7)and, therefore, taking the half-line Fourier transform (2.1a) we obtainˆ q t ( k,
0) = 11 + k (cid:2) r (0) − ik ˆ r ( k ) − g (cid:48) (0) − ikg (cid:48) (0) (cid:3) =: ˆ q ( k ) . (4.8)The conditions (4.6) and (4.8) must be satisfied by the homogeneous solution formula (4.5). Forthis, we must have c ( k ) = 12 (cid:20) ˆ q ( k ) + ˆ q ( k ) iω (cid:21) , c ( k ) = 12 (cid:20) ˆ q ( k ) − ˆ q ( k ) iω (cid:21) . (4.9)Furthermore, variation of parameters yields a particular solution of (4.4) in the formˆ q p ( k, t ) = e iωt (cid:90) tτ =0 e − iωτ ϕ ( τ )2 iω dτ − e − iωt (cid:90) tτ =0 e iωτ ϕ ( τ )2 iω dτ, (4.10)where ϕ ( t ) is simply the forcing on the right-hand side of (4.4): ϕ ( t ) = ϕ ( k, t ) = −
11 + k (cid:2) g ( t ) + ikg ( t ) + g (cid:48)(cid:48) ( t ) + ikg (cid:48)(cid:48) ( t ) (cid:3) . (4.11)Therefore, recalling the notation (cid:101) f ( ω, t ) = (cid:82) tτ =0 e − iωτ f ( τ ) dτ and noting that ˆ q = ˆ q h + ˆ q p , weoverall find the solution to (4.4) asˆ q ( k, t ) = 12 (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) + 12 iω (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k )+ 12 iω (1 + k ) (cid:26) − e iωt (cid:104)(cid:101) g ( ω, t ) + (cid:101) g (cid:48)(cid:48) ( ω, t ) + ik (cid:101) g ( ω, t ) + ik (cid:101) g (cid:48)(cid:48) ( ω, t ) (cid:105) + e − iωt (cid:104)(cid:101) g ( − ω, t ) + (cid:101) g (cid:48)(cid:48) ( − ω, t ) + ik (cid:101) g ( − ω, t ) + ik (cid:101) g (cid:48)(cid:48) ( − ω, t ) (cid:105) (cid:27) , (4.12)valid for Im( k ) (cid:54) k (cid:54) = − i . In fact, integrating by parts we have (cid:101) g (cid:48)(cid:48) ( ± ω, t ) = e ∓ iωt g (cid:48) ( t ) − g (cid:48) (0) ± iω (cid:101) g (cid:48) ( ± ω, t )and similarly for (cid:101) g (cid:48)(cid:48) ( ± ω, t ). Thus, substituting also for ˆ q via (4.8), we can write (4.12) in the formˆ q ( k, t ) = 12 (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) + 12 iω (1 + k ) (cid:0) e iωt − e − iωt (cid:1) [ r (0) − ik ˆ r ( k )] (4.13) − iω (1 + k ) (cid:26) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + iω (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) + ik (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − kω (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) . Expression (4.13) is the global relation corresponding to equation (4.2). Since it is valid forIm( k ) (cid:54) k (cid:54) = − i , it can be combined with the Fourier inversion formula (2.1b) to yield anintegral representation for q . This integral representation will involve two boundary values, g and g . Since only the former one is prescribed as a boundary condition in problem (1.2), the latterone will have to be eliminated. This is achieved by using the transformation k (cid:55)→ − k , which is asymmetry of ω , and then by exploiting analyticity and Cauchy’s theorem. The whole procedureis just like the one presented in detail in Section 3 and so we do not repeat it here. In fact, it is easy to convert the global relation (4.13) to the global relation (3.2b) that we obtained earlier inSection 3 using the UTM for systems approach.Indeed, integrating by parts and recalling (3.1), we find e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) = − e iωt (cid:102) h (cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48) ( − ω, t )= − e iωt (cid:104) e − iωt h ( t ) − h (0) + iω (cid:101) h ( ω, t ) (cid:105) + e − iωt (cid:104) e iωt h ( t ) − h (0) − iω (cid:101) h ( − ω, t ) (cid:105) = (cid:0) e iωt − e − iωt (cid:1) r (0) − iω (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) , (4.14)where the last equality follows from the compatibility at the origin: h (0) := r (0 ,
0) =: r (0). Inaddition, by (3.1) we have e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) = − e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) − e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) . (4.15)Therefore, (4.13) can also be written in the formˆ q ( k, t ) = 12 (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k )+ 12 (1 + k ) (cid:26) (cid:104) e iωt (cid:101) h ( ω, t ) + e − iωt (cid:101) h ( − ω, t ) (cid:105) + (cid:104) e iωt (cid:102) h (cid:48)(cid:48) ( ω, t ) + e − iωt (cid:102) h (cid:48)(cid:48) ( − ω, t ) (cid:105) − µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) − ik (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) , (4.16)which is precisely the global relation (3.2b) that we obtained in Section 3 via the UTM for systemsapproach. Therefore, from this point onwards, following the elimination procedure of Section 3 wearrive once again at formula (3.9b) for q . Once again, we emphasize that it is possible to obtainformula (3.9b) directly from the global relation (4.13), i.e. without converting that global relationinto (3.2b). 5. Explicit verification of the novel solution formula
Since the derivations of the previous sections were performed under the assumption of existenceof solution, we shall now verify that the resulting formulae do indeed satisfy IBVP (1.2).We begin with the linearized classical Boussinesq system (1.2a). Differentiating formula (3.9a)with respect to t , we have r t ( x, t ) = 12 π (cid:90) k ∈ R e ikx iω (cid:2)(cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx iω (cid:2)(cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx iω (cid:26) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + iω ( k ) (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk + 12 π (cid:90) k ∈C e ikx (cid:26) [ g ( t ) + g ( t )] + iω ( k ) (cid:2) g (cid:48) ( t ) − g (cid:48) ( t ) (cid:3) (cid:27) dk, (5.1) . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 13 while taking the derivative of (3.9b) with respect to x gives q x ( x, t ) = 12 π (cid:90) k ∈ R e ikx ik (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( k ) (cid:21) dk − π (cid:90) k ∈C e ikx ik (cid:20)(cid:0) e iωt + e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ r ( − k ) (cid:21) dk − π (cid:90) k ∈C e ikx ik (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik k (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) + e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (5.2)Note, importantly, that the last integral in (5.1) is zero, since the function e ikx is analytic insidethe region enclosed by C and the part of the integrand involving ω ( k ) vanishes identically. Thus,recalling that ωµ = k , we see that r t + q x = 0, i.e. the first equation of system (1.2a) is satisfied.Furthermore, using Cauchy’s residue theorem we compute (cid:90) k ∈C e ikx ik k dk = − πe − x and hence q t ( x, t ) = 12 π (cid:90) k ∈ R e ikx iω (cid:20)(cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) (cid:21) dk − π (cid:90) k ∈C e ikx iω (cid:20)(cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) (cid:21) dk − π (cid:90) k ∈C e ikx iω (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik k (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk + e − x g (cid:48) ( t ) (5.3)and q xxt ( x, t ) = 12 π (cid:90) k ∈ R e ikx − ik ω (cid:20)(cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) − µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) (cid:21) dk (5.4) − π (cid:90) k ∈C e ikx − ik ω (cid:20)(cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) + 1 µ ( k ) (cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) (cid:21) dk − π (cid:90) k ∈C e ikx (cid:0) − ik ω (cid:1) (cid:26) µ ( k ) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ik k (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk + e − x g (cid:48) ( t ) . Moreover, r x ( x, t ) = 12 π (cid:90) k ∈ R e ikx ik (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( k ) − µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx ik (cid:2)(cid:0) e iωt + e − iωt (cid:1) ˆ r ( − k ) + µ ( k ) (cid:0) e iωt − e − iωt (cid:1) ˆ q ( − k ) (cid:3) dk + 12 π (cid:90) k ∈C e ikx ik (cid:26) (cid:2) e iωt (cid:101) g ( ω, t ) + e − iωt (cid:101) g ( − ω, t ) (cid:3) + ikµ ( k ) (cid:104) e iωt (cid:101) g (cid:48) ( ω, t ) − e − iωt (cid:101) g (cid:48) ( − ω, t ) (cid:105) (cid:27) dk. (5.5) Therefore, observing that ω − kµ + k ω = 0, we deduce that q t + r x − q xxt = 0, as required by thesecond component of system (1.2a).Next, we verify the initial conditions (1.2b). Evaluating (3.9) at t = 0, we have r ( x,
0) = 12 π (cid:90) k ∈ R e ikx ˆ r ( k ) dk + 12 π (cid:90) k ∈C e ikx ˆ r ( − k ) dk (5.6)and q ( x,
0) = 12 π (cid:90) k ∈ R e ikx ˆ q ( k ) dk − π (cid:90) k ∈C e ikx ˆ q ( − k ) dk. (5.7)In view of the Fourier inversion formula (2.1b), the first integrals in the above expressions aresimply r ( x ) and q ( x ), respectively. Moreover, the second integrals are zero because of analyticityof the integrands inside the region enclosed by C . Hence, the initial conditions are satisfied.Finally, we verify the boundary condition (1.2c). This task is a bit more challenging that theprevious two verifications. We begin by observing thatˆ q ( k ) := (cid:90) ∞ y =0 e − iky q ( y ) dy = 1 ik (cid:104) q (0) + (cid:98) q (cid:48) ( k ) (cid:105) and, similarly, (cid:101) g (cid:48) ( ω, t ) = e − iωt g ( t ) − g (0) + iω (cid:101) g ( ω, t ) . Thus, noting that (cid:90) k ∈ R e ikx (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) dk = − (cid:90) k ∈ R e ikx (cid:0) e iωt + e − iωt (cid:1) ˆ q ( k ) dk since the integral on the left-hand side exists without the need for taking the principal value − (cid:82) , wehave q ( x, t ) = 12 iπ − (cid:90) k ∈ R e ikx cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( k ) (cid:105) dk + 12 iπ (cid:90) k ∈C e ikx cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( − k ) (cid:105) dk + 12 iπ (cid:90) k ∈ R e ikx sin( ωt ) µ ( k ) ˆ r ( k ) dk + 12 iπ (cid:90) k ∈C e ikx sin( ωt ) µ ( k ) ˆ r ( − k ) dk − π (cid:90) k ∈C e ikx k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) dk + g ( t ) iπ (cid:90) k ∈C e ikx k k dk − g (0)2 iπ (cid:90) k ∈C e ikx k k (cid:0) e iωt + e − iωt (cid:1) dk. (5.8)Now, note that all the integrals in (5.8) whose contour is C are uniformly convergent and hencewe can pass the limit x → R involving sin( ωt ),since the integrand is of O ( k − ) as | k | → ∞ . The first integral, i.e. the one along R involvingcos( ωt ), will be discussed separately below. Furthermore, (cid:90) k ∈C sin( ωt ) µ ( k ) ˆ r ( − k ) dk := (cid:90) ∞ y =0 r ( y ) (cid:90) k ∈C e iky sin( ωt ) µ ( k ) dkdy = (cid:90) ∞ y =0 r ( y ) (cid:90) k ∈ R e iky sin( ωt ) µ ( k ) dkdy = (cid:90) k ∈ R sin( ωt ) µ ( k ) ˆ r ( − k ) dk (5.9) Indeed, we have ω (cid:39) | k | → ∞ and so sin( ωt ) is bounded. Moreover, µ ( k ) (cid:39) k and, finally, integrating byparts yields ˆ r ( k ) = ik r (0) + ik (cid:98) r (cid:48) ( k ). . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 15 by applying Cauchy’s theorem and Jordan’s lemma along the upper semicircle of infinite radius inthe complex k -plane. Therefore, (5.8) becomes q (0 , t ) = 12 iπ lim x → − (cid:90) k ∈ R e ikx cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( k ) (cid:105) dk + 12 iπ (cid:90) k ∈C cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( − k ) (cid:105) dk + 12 iπ (cid:90) k ∈ R sin( ωt ) µ ( k ) ˆ r ( k ) dk + 12 iπ (cid:90) k ∈ R sin( ωt ) µ ( k ) ˆ r ( − k ) dk − π (cid:90) k ∈C k ) (cid:2) e iωt (cid:101) g ( ω, t ) − e − iωt (cid:101) g ( − ω, t ) (cid:3) dk + g ( t ) iπ (cid:90) k ∈C k k dk − g (0)2 iπ (cid:90) k ∈C k k (cid:0) e iωt + e − iωt (cid:1) dk, (5.10)and making the change of variable k (cid:55)→ − k to see that the two integrals involving ˆ r cancel (recallthat µ ( − k ) = − µ ( k )), we obtain q (0 , t ) = 12 iπ lim x → − (cid:90) k ∈ R e ikx cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( k ) (cid:105) dk + 12 iπ (cid:90) k ∈C cos( ωt ) k (cid:104) q (0) + (cid:98) q (cid:48) ( − k ) (cid:105) dk + 1 iπ (cid:90) tτ =0 g ( τ ) (cid:90) k ∈C sin( ω ( t − τ ))(1 + k ) dkdτ + g ( t ) iπ (cid:90) k ∈C k k dk − g (0) iπ (cid:90) k ∈C k cos( ωt )1 + k dk. (5.11)Next, we compute several integrals by exploiting the uniform convergence of Taylor series forcos and sin and using Cauchy’s residue theorem. First, we have (cid:90) k ∈C cos( ωt ) k q (0) dk = q (0) ∞ (cid:88) j =0 ( − j t j (2 j )! (cid:90) k ∈C k j − (1 + k ) j dk (cid:124) (cid:123)(cid:122) (cid:125) = iπ for j ∈ N and 0 for j =0 = iπq (0) (cos t − , (5.12)where the above integral has been evaluated by using the standard complex analysis formula forthe residue of a pole together with the Leibniz rule for the derivative of a product. Similarly, (cid:90) k ∈C sin( ω ( t − τ ))(1 + k ) dk = ∞ (cid:88) j =0 ( − j ( t − τ ) j +1 (2 j + 1)! (cid:90) k ∈C k j +1 (1 + k ) j +2 dk (cid:124) (cid:123)(cid:122) (cid:125) = 0 ∀ j ∈ N ∪{ } = 0 . (5.13)Also, (cid:82) k ∈C k k dk = iπ and, finally, (cid:90) k ∈C k cos( ωt )1 + k dk = ∞ (cid:88) j =0 ( − j t j (2 j )! (cid:90) k ∈C k j +1 (1 + k ) j +1 dk (cid:124) (cid:123)(cid:122) (cid:125) = iπ ∀ j ∈ N ∪{ } = iπ cos t. (5.14)Substituting the above computations in (5.11) and recalling the compatibility condition q (0) = g (0), we obtain q (0 , t ) = 12 iπ lim x → − (cid:90) k ∈ R e ikx cos( ωt ) k (cid:104) g (0) + (cid:98) q (cid:48) ( k ) (cid:105) dk + 12 iπ (cid:90) k ∈C cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk − g (0) (cos t + 1) + g ( t ) . (5.15) Crucial for the application of Jordan’s lemma is the uniform decay of the non-exponential part of the integrandthanks to µ ( k ) . Next, we discuss the first principal value integral in (5.15). We have − (cid:90) k ∈ R e ikx cos( ωt ) k dk = (cid:90) k ∈ Γ e ikx cos( ωt ) k dk − (cid:90) k ∈ C ε e ikx cos( ωt ) k dk, (5.16)where Γ is the closed, anti-clockwise contour consisting of ( −∞ , − ε ], C ε , [ ε, ∞ ) and C R , with C ε being the upper semicircle of radius ε centered at the origin and oriented clockwise, and with C R being the upper semicircle of radius R → ∞ centered at the origin and oriented anti-clockwise. Wenote that the deformation from ( −∞ , − ε ] ∪ C ε ∪ [ ε, ∞ ) to Γ is possible due to the fact that theintegral along C R vanishes thanks to Jordan’s lemma. Now, by Cauchy’s theorem and (5.12) wehavelim x → (cid:90) k ∈ Γ e ikx cos( ωt ) k dk = lim x → (cid:90) k ∈C e ikx cos( ωt ) k dk = (cid:90) k ∈C cos( ωt ) k dk = iπ (cos t − . (5.17)Furthermore, noting that C ε is oriented clockwise, we computelim x → (cid:90) k ∈ C ε e ikx cos( ωt ) k dk = (cid:90) k ∈ C ε cos( ωt ) k dk = − iπ Res (cid:20) cos( ωt ) k , k = 0 (cid:21) = − iπ. (5.18)Thus, lim x → − (cid:90) k ∈ R e ikx cos( ωt ) k dk = iπ cos t (5.19)and, in turn, (5.15) becomes q (0 , t ) = 12 iπ − (cid:90) k ∈ R cos( ωt ) k (cid:98) q (cid:48) ( k ) dk + 12 iπ (cid:90) k ∈C cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk − g (0) + g ( t ) , (5.20)where we have passed the limit x → e ikx ) thanks to the fact that, as | k | → ∞ , ω (cid:39) ωt ) is bounded) and (cid:98) q (cid:48) ( k ) = ik q (cid:48) (0) + ik (cid:98) q (cid:48)(cid:48) ( k ) (recall that q belongs to theSchwartz class).Finally, invoking Cauchy’s theorem and Jordan’s lemma once again, we have (cid:90) k ∈C cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk = − (cid:90) k ∈ R cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk + (cid:90) k ∈ C ε cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk. (5.21)Therefore, using the change of variable k (cid:55)→ − k as appropriate, we find q (0 , t ) = 12 iπ − (cid:90) k ∈ R cos( ωt ) k (cid:98) q (cid:48) ( k ) dk + 12 iπ − (cid:90) k ∈ R cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk + 12 iπ (cid:90) k ∈ C ε cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk − g (0) + g ( t )= 12 iπ − (cid:90) k ∈ R cos( ωt ) k (cid:98) q (cid:48) ( k ) dk + 12 iπ − (cid:90) k ∈ R cos( ωt ) − k (cid:98) q (cid:48) ( k ) dk + 12 iπ (cid:90) k ∈ C ε cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk − g (0) + g ( t )= 12 iπ (cid:90) k ∈ C ε cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk − g (0) + g ( t ) . (5.22) Note here the importance of the uniform decay of k in the integrand (recall that ω (cid:39) | k | → ∞ so cos( ωt ) isbounded at infinity). The exponential decay required for Jordan’s lemma is provided by the half-line Fourier transform (cid:98) r (cid:48) ( − k ) := (cid:82) ∞ x =0 e ikx r (cid:48) ( x ) dx . . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 17 Finally, by Cauchy’s residue theorem we compute12 iπ (cid:90) k ∈ C ε cos( ωt ) k (cid:98) q (cid:48) ( − k ) dk = −
12 Res (cid:20) cos( ωt ) k (cid:98) q (cid:48) ( − k ) , k = 0 (cid:21) = − (cid:98) q (cid:48) (0) := − (cid:90) ∞ x =0 q (cid:48) ( x ) dx = 12 q (0) = 12 g (0) (5.23)with the last equality due to the compatibility condition q (0) = q (0 ,
0) = g (0). Hence, we overallconclude that if q ( x, t ) is defined by the UTM formula (3.9b) then q (0 , t ) = g ( t ), i.e. the boundarycondition (1.2c) is satisfied. Remark 5.1 (Uniform convergence at the boundary) . As a final remark, we emphasize the im-portance of the complex contour C in the UTM solution formulae (3.9). Indeed, as illustrated bythe above computations, had we deformed from C back to R it would not have been possible toexplicitly verify that our formulae satisfy IBVP (1.2) (and, in particular, the boundary condition)due to the loss of uniform convergence at the boundary induced by the deformation to the real axis.6. Conclusion
A novel solution formula for the linearized classical Boussinesq system on the half-line withDirichlet boundary data was derived by employing the unified transform method of Fokas. Moreprecisely, the analysis utilized the recently formulated extension of the linear component of Fokas’smethod from single equations to systems [DGSV], as well as fundamental ideas of the method suchas escaping to the complex spectral plane and exploiting the symmetries of a central identity knownas the global relation.The resulting solution formula (3.9) has the important advantage of uniform convergence at theboundary x = 0, thereby allowing for its explicit verification against the linearized classical Boussi-nesq IBVP (1.2) through a direct calculation (see Section 5). Beyond uniform convergence, the novelformula enjoys exponentially decaying integrands and hence, as usual with formulae derived viathe unified transform, it is expected to be particularly effective regarding numerical considerations.Indeed, only a few lines of code in Mathematica result in the three plots of Figure 6.1. * * * * * * * * * * * * * * * * * * * * * ( x,0 ) * * * * * * * * * * * * * * * * * * * * * - - ( ) Figure 6.1.
Evaluation of the novel solution formula (3.9b) for the q -componentof IBVP (1.2) with initial data q ( x ) = x e − x and boundary data g ( t ) = sin(5 t ). Left panel:
The solution q as a function of ( x, t ) ∈ (0 , × (0 , Center panel:
Thesolution q at t = 0 as a function of x ∈ (0 , Right panel:
The solution q at x = 0as a function of t ∈ (0 , precise values of the data q ( x ) and g ( t ), respectively, showing perfect agreementof formula (3.9b) with the prescribed data. Of particular importance for the thirdplot is the uniform convergence of formula (3.9b) at the boundary x = 0. Furthermore, formula (3.9b) was rederived in Section 4 from a different starting point, namelyby reducing system (1.2a) to a single equation. This reduction is not possible for general dispersivesystems, which is the main reason why the systems approach of Section 2 is preferable; however,the second approach demonstrates the versatility of the unified transform method and offers adifferent perspective concerning the types of admissible boundary data. Moreover, to the best ofour knowledge, the analysis of Section 4 signifies the first time that the unified transform methodhas been applied to an equation whose dispersion relation is a quotient involving a complex squareroot (and hence branching), due to the presence of the mixed derivative (cid:0) − ∂ x (cid:1) ∂ t .We emphasize that the present article is not the first one devoted to the linearized classicalBoussinesq IBVP (1.2) via the unified transform method. Indeed, Fokas and Pelloni had previouslyconsidered the same problem in [FP1]. Importantly, however, their approach was entirely different,as they utilized the nonlinear component of the unified transform method, which relies on formu-lating IBVP (1.2) as a Lax pair and then integrating it by using ideas inspired from the inversescattering transform, namely by associating it with a Riemann-Hilbert problem which is then solvedvia Plemelj’s formulae. The complexity of those techniques naturally limits the accessibility of thederivation of [FP1] to a very specialized audience. In contrast, the present work employs the linear component of the unified transform method, which only requires knowledge of Fourier transformand Cauchy’s theorem from complex analysis and hence is accessible to a broad audience withinthe applied sciences.Another significant difference between the present work and [FP1] is the fact that, due tothe nature of the Riemann-Hilbert problem, the resulting solution formula in [FP1] involves certainprincipal value integrals. The presence of these integrals seems unsuitable for the purpose of effectivenumerical implementations. Furthermore, it will most likely pose an issue when attempting to usethat formula for establishing well-posedness of the original, nonlinear classical Boussinesq system(1.1) via the contraction mapping approach (see relevant discussion in the introduction). On theother hand, the novel solution formula derived in this article does not involve any singularities andhence is expected to be effective for showing well-posedness of the classical Boussinesq system (1.1)on the half-line via the contraction mapping approach, along the lines of [FHM1, FHM2, HM2,HM3]. Acknowledgements.
The first author would like to thank the Department of Mathematics ofthe University of Kansas for partially supporting their research through an undergraduate researchaward. All three authors are grateful to Andre Kurait for inspiring discussions during the 2018-19academic year that paved the way to the present work. Finally, the authors are thankful to thereviewers of the manuscript for useful remarks and suggestions.
References [AbF] M. Ablowitz and A. Fokas,
Complex variables: introduction and applications , Cambridge University Press,2003 (2nd edition).[Ad] K. Adamy,
Existence of solutions for a Boussinesq system on the half line and on a finite interval.
DiscreteContin. Dyn. Syst. (2011), 25-49.[AL] B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics.
Invent.Math. (2008), 485-541.[Am] C. Amick,
Regularity and uniqueness of solutions to the Boussinesq system of equations.
J. DifferentialEquations (1984), 231-247.[AnF] Y. Antipov and A. Fokas, The modified Helmholtz equation in a semi-strip.
Math. Proc. Cambridge Philos.Soc. (2005), 339-365. . M. Johnston, Clarence T. Gartman & Dionyssios Mantzavinos 19 [AD] D. Antonopoulos and V. Dougalis,
Numerical solution of the “classical” Boussinesq system.
Math. Comput.Simul. (2012), 984-1007.[BCS1] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long wavesin nonlinear dispersive media I: Derivation and linear theory.
J. Nonlinear Sci. (2002), 283-318.[BCS2] J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long wavesin nonlinear dispersive media II: The nonlinear theory.
Nonlinearity (2004), 925-952.[B1] J. Boussinesq, Th´eorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal,en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond .J. Math. Pures Appl. (1872), 55-108.[B2] J. Boussinesq, Essai sur la th´eorie des eaux courantes . M´emoires pr´esent´es par divers savants `a l’Acad´emiedes Sciences (1877), 1-680.[CFF] M. Colbrook, N. Flyer and B. Fornberg, On the Fokas method for the solution of elliptic problems in bothconvex and non-convex polygonal domains.
J. Comput. Phys. (2018), 996-1016.[DGSV] B. Deconinck, Q. Guo, E. Shlizerman and V. Vasan,
Fokas’s unified transform method for linear systems .Quart. Appl. Math. (2018), 463-488.[DSS] B. Deconinck, N. Sheils and D. Smith, The linear KdV equation with an interface.
Comm. Math. Phys. (2016), 489-509.[DTV] B. Deconinck, T. Trogdon and V. Vasan,
The method of Fokas for solving linear partial differential equa-tions . SIAM Rev. (2014), 159-186.[F1] A. Fokas, A unified transform method for solving linear and certain nonlinear PDEs.
Proc. R. Soc. A (1997), 1411-1443.[F2] A. Fokas,
A unified approach to boundary value problems , CBMS-NSF Regional Conference Series in AppliedMathematics , SIAM, Philadelphia, PA, 2008.[FFSS] A. Fokas, N. Flyer, S. Smitheman and E. Spence, A semi-analytical numerical method for solving evolutionand elliptic partial differential equations.
J. Comput. Appl. Math. (2009), 59-74.[FHM1] A. Fokas, A. Himonas and D. Mantzavinos,
The Korteweg-de Vries equation on the half-line . Nonlinearity (2016), 489-527.[FHM2] A. Fokas, A. Himonas and D. Mantzavinos, The nonlinear Schr¨odinger equation on the half-line . Trans.Amer. Math. Soc. (2017), 681-709.[FI] A. Fokas and A. Its,
The nonlinear Schr¨odinger equation on the interval . J. Phys. A: Math. Gen. (2004),6091-6114.[FK] A. Fokas and A. Kapaev, On a transform method for the Laplace equation in a polygon.
IMA J. Appl.Math. (2003), 355-408.[FL] A. Fokas and J. Lenells, The unified method: I. Nonlinearizable problems on the half-line.
J. Phys. A (2012), 195201.[FF] N. Flyer and A. Fokas, A hybrid analytical numerical method for solving evolution partial differentialequations. I: the half-line.
Proc. R. Soc. (2008), 1823-1849.[FP1] A. Fokas and B. Pelloni,
Boundary value problems for Boussinesq type systems.
Math. Phys. Anal. Geom. (2005), 59-96.[FP2] A. Fokas and B. Pelloni, Unified transform for boundary value problems: applications and advances , SIAM,Philadelphia, PA, 2015.[FS] A. Fokas and E. Spence,
Synthesis, as opposed to separation, of variables. SIAM Rev. (2012), 291-324. [HM1] A. Himonas and D. Mantzavinos, On the initial-boundary value problem for the linearized Boussinesqequation . Stud. Appl. Math. (2015), 62-100.[HM2] A. Himonas and D. Mantzavinos,
The “good” Boussinesq equation on the half-line . J. Differential Equations (2015), 3107-3160.[HM3] A. Himonas and D. Mantzavinos,
Well-posedness of the nonlinear Schr¨odinger equation on the half-plane .Nonlinearity (2020), 5567-5609.[KO] K. Kalimeris and T. Ozsari, An elementary proof of the lack of null controllability for the heat equation onthe half line.
Appl. Math. Lett. (2020), 106241.[L] D. Lannes,
The water waves problem: mathematical analysis and asymptotics , Amer. Math. Soc., 2013.[LW] D. Lannes and L. Weynans,
Generating boundary conditions for a Boussinesq system . Nonlinearity (2012), 6868-6889. [MID] D. Mitsotakis, B. Ilan and D. Dutykh, On the Galerkin/finite-element method for the Serre equations.
J.Sci. Comput. (2014), 166-195.[MTZ] L. Molinet, R. Talhouk and I. Zaiter, The classical Boussinesq system revisited. arXiv :2001.11870v1 (2020).[P] D. Peregrine,
Long waves on a beach.
J. Fluid Mech. (1967), 815-827.[S] M. Schonbeck, Existence of solutions for the Boussinesq system of equations.
J. Differential Equations (1981), 325-352.[SSF] S. Smitheman, E. Spence and A. Fokas, A spectral collocation method for the Laplace and modified Helmholtzequations in a convex polygon.
IMA J. Numer. Anal. (2010), 1184-1205.[VD] V. Vasan and B. Deconinck, Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation.
Discrete Contin. Dyn. Syst.33